Finite Group Databasebeta

← #33108451

Subgroups — #33108451

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988688C211yesmaximal, characteristic
2C2×SD16×Dic73C211yesmaximal, characteristic
3#22612991C211yesmaximal, characteristic, complemented
4#22612841C211yesmaximal, characteristic
5#22610752C211yesmaximal, characteristic, complemented
6#22610701C211yesmaximal, characteristic, complemented
7#22610623C211yesmaximal, characteristic, complemented
8#22610609C211yesmaximal, characteristic, complemented
9#22610267C211yesmaximal, characteristic
10#22610267C211yesmaximal, characteristic
11#22610201C211yesmaximal, characteristic, complemented
12#22610187C211yesmaximal, characteristic, complemented
13#22609808C211yesmaximal, characteristic, complemented
14#22609756C211yesmaximal, characteristic
15C2×#461×C73C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19C2×D4×Dic73C2211yescharacteristic
20#17988688—24no—
21SD16×Dic73—24no—
22C2×#14556525C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556515C2211yescharacteristic
25C2×#14556515C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556512C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×#14556502C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33#17988494C2211yescharacteristic
34#17988494C2211yescharacteristic
35#17988486—21no—
36#17988486C2211yescharacteristic
37#17988486C2211yescharacteristic
38#17988473—21no—
39#17988473—21no—
40#17988462—21no—
41#17988462—21no—
42#17988453C2211yescharacteristic
43#17988447—21no—
44C4⋊C4×Dic73—21no—
45C4×C4⋊Dic73C2211yescharacteristic
46C4×C4⋊Dic73C2211yescharacteristic
47C2×#14556498—21no—
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988341C2211yescharacteristic
53#17988341C2211yescharacteristic
54C22×SD16×C73C2211yescharacteristic
55C22×SD16×C73C2211yescharacteristic
56C2×C4⋊Q8×C73C2211yescharacteristic
57C2×C4×D4×C73C2211yescharacteristic
58#461×C73C418yescomplemented
59C2×C4⋊C8×C73C2211yescharacteristic
60C2×#103×C73C2211yescharacteristic
61C2×#103×C73C2211yescharacteristic
62C23×Dic73—21no—
63C23×Dic73—21no—
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66Q8×Dic73—22no—
67Q8×Dic73—22no—
68Q8×Dic73—41no—
69Q8×Dic73—41no—
70D4×Dic73—22no—
71D4×Dic73—22no—
72D4×Dic73—41no—
73D4×Dic73—41no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73—21no—
78C2×C4⋊Dic73D411yescharacteristic
79C2×C4⋊Dic73D411yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×C22⋊Dic73—21no—
85C2×C22⋊Dic73—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91#14556525—22no—
92#14556523—22no—
93#14556515—22no—
94#14556515—22no—
95#14556515—41no—
96#14556515—41no—
97#14556512—22no—
98#14556512D414yes—
99#14556502—22no—
100#14556501—22no—
101C8×Dic73—22no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498D411yescharacteristic
112#14556498D411yescharacteristic
113C22×Q8×C73C2311yescharacteristic
114C22×Q8×C73C2311yescharacteristic
115C22×D4×C73C2311yescharacteristic
116C23×C4×C73—21no—
117C2×SD16×C73—22no—
118C2×SD16×C73—22no—
119C2×SD16×C73—22no—
120C2×SD16×C73—22no—
121C2×SD16×C73C2×C414yes—
122C2×SD16×C73C2×C414yes—
123C22×C8×C73C2311yescharacteristic
124C22×C8×C73C2311yescharacteristic
125C4⋊Q8×C73—22no—
126C4⋊Q8×C73C2×C414yescomplemented
127C4×D4×C73—22no—
128C4×D4×C73C2×C414yes—
129C2×C4⋊C4×C73—21no—
130C2×C4⋊C4×C73—21no—
131C2×C4⋊C4×C73C2311yescharacteristic
132C2×C22⋊C4×C73—21no—
133C2×C42×C73C2311yescharacteristic
134C4⋊C8×C73C2×C414yescomplemented
135#103×C73C2×C414yescomplemented
136#103×C73C2×C414yescomplemented
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—42no—
142C22×Dic73—42no—
143C22×Dic73—42no—
144C22×Dic73—42no—
145C22×Dic73—81no—
146C22×Dic73—81no—
147C22×Dic73C2×D411yescharacteristic
148C22×Dic73C2×D411yescharacteristic
149C22×Dic73C2×D411yescharacteristic
150C22×Dic73C2×D411yescharacteristic
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831C2×D412yes—
157C4⋊Dic73—21no—
158C4⋊Dic73—22no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—41no—
166C4⋊Dic73—42no—
167C4⋊Dic73—42no—
168C4⋊Dic73—81no—
169C4⋊Dic73—81no—
170C4⋊Dic73C2×D412yes—
171C22⋊Dic73—42no—
172C22⋊Dic73—42no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—42no—
178C4×Dic73—42no—
179C4×Dic73—42no—
180C24×C73—21no—
181C2×Q8×C73—21no—
182C2×Q8×C73—21no—
183C2×Q8×C73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73—21no—
186C2×Q8×C73—21no—
187C2×Q8×C73—21no—
188C2×Q8×C73—21no—
189C2×Q8×C73—21no—
190C2×Q8×C73—21no—
191C2×Q8×C73C22×C412yes—
192C2×Q8×C73C22×C412yes—
193C2×D4×C73—21no—
194C2×D4×C73—21no—
195C2×D4×C73—21no—
196C2×D4×C73C22×C412yes—
197C2×D4×C73C2×D412yes—
198C2×D4×C73#4112yes—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—21no—
202C22×C4×C73—24no—
203C22×C4×C73—81no—
204C22×C4×C73—81no—
205C22×C4×C73C2×D411yescharacteristic
206C22×C4×C73C2×D411yescharacteristic
207C22×C4×C73C2411yescharacteristic
208SD16×C73—28no—
209SD16×C73—28no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73—21no—
213C2×C8×C73—21no—
214C2×C8×C73C22×C412yes—
215C2×C8×C73C22×C412yes—
216C4⋊C4×C73—24no—
217C4⋊C4×C73—24no—
218C4⋊C4×C73—21no—
219C4⋊C4×C73C22×C412yes—
220C22⋊C4×C73—24no—
221C42×C73C22×C412yes—
222C42×C73#4112yes—
223C23×C73—22no—
224C23×C73—22no—
225C23×C73—22no—
226C23×C73—41no—
227C23×C73—41no—
228C23×C73—41no—
229C23×C73—41no—
230C23×C73C22×D411yescharacteristic
231Q8×C73—22no—
232Q8×C73—22no—
233Q8×C73—22no—
234Q8×C73—22no—
235Q8×C73—41no—
236Q8×C73—41no—
237Q8×C73—41no—
238Q8×C73—41no—
239D4×C73—22no—
240D4×C73—22no—
241D4×C73—22no—
242D4×C73C4×D414yes—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—22no—
248C2×C4×C73—22no—
249C2×C4×C73—22no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—41no—
254C2×C4×C73—41no—
255C2×C4×C73—41no—
256C2×C4×C73—84no—
257C2×C4×C73C4×D412yes—
258C2×C4×C73C4×D412yes—
259C2×C4×C73C23×C411yescharacteristic
260C2×C4×C73C22×D411yescharacteristic
261C2×C4×C73C2×#4111yescharacteristic
262C2×C4×C73C2×#4111yescharacteristic
263C2×C4×C73H(5,𝔽2)11yescharacteristic
264C2×C4×C732−1+411yescharacteristic
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—22no—
271C2×Dic73—21no—
272C2×Dic73—21no—
273C2×Dic73—21no—
274C2×Dic73—21no—
275C2×Dic73—21no—
276C2×Dic73—21no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73—41no—
284C2×Dic73—41no—
285C2×Dic73—41no—
286C2×Dic73—41no—
287C2×Dic73—41no—
288C2×Dic73—84no—
289C2×Dic73—84no—
290C2×Dic73#13712yes—
291C8×C73—22no—
292C8×C73—22no—
293#10596831—22no—
294#10596831—41no—
295C22×C73—24no—
296C22×C73—42no—
297C22×C73—42no—
298C22×C73—42no—
299C22×C73—42no—
300C22×C73—42no—
301C22×C73—42no—
302C22×C73C2×C4×D411yescharacteristic
303C22×C73D4211yescharacteristic
304C22×C73#54811yescharacteristic
305C22×C73C2×#13711yescharacteristic
306C22×C73#57511yescharacteristic
307C22×C73#57711yescharacteristic
308C22×C73#57711yescharacteristic
309C4×C73—22no—
310C4×C73—22no—
311C4×C73—42no—
312C4×C73—42no—
313C4×C73—42no—
314C4×C73C2×C4×D411yescharacteristic
315C4×C73#52011yescharacteristic
316C4×C73#54711yescharacteristic
317C4×C73#55311yescharacteristic
318Dic73—22no—
319Dic73—41no—
320Dic73—41no—
321Dic73—41no—
322Dic73—42no—
323Dic73—81no—
324Dic73—81no—
325Dic73—81no—
326#23012—731nomaximal, Sylow
327C2×C73—44no—
328C2×C73#236111yescharacteristic
329C2×C73C4×#13711yescharacteristic
330C2×C73#294911yescharacteristic
331C2×C73#326611yescharacteristic
332C2×C73#326711yescharacteristic
333C2×C73#337211yescharacteristic
334C2×C73#337611yescharacteristic
335C2×#461D7311yescharacteristic
336C2×#440—731no—
337C2×C4×SD16—731no—
338#2365—731no—
339#2349—731no—
340#1960—731no—
341#1949—731no—
342#1935—731no—
343#1933—731no—
344#1877—731no—
345#1877—731no—
346#1865—731no—
347#1863—731no—
348#1771—731no—
349#1762—731no—
350C73#2301211yescharacteristic, complemented, Sylow
351C22×SD16C2×D7311yescharacteristic
352C22×SD16C2×D7311yescharacteristic
353C2×C4⋊Q8C2×D7311yescharacteristic
354C2×C4×Q8—731no—
355C2×C4×Q8—731no—
356C2×C4×D4—731no—
357C2×C4×D4—731no—
358C2×C4×D4C2×D7311yescharacteristic
359#461Dic7318yescomplemented
360#440—1464no—
361C4×SD16—1464no—
362C2×#107—731no—
363C2×#106—731no—
364C2×C4⋊C8C2×D7311yescharacteristic
365C2×#103—731no—
366C2×#103—731no—
367C2×#103—731no—
368C2×#103—731no—
369C2×#103C2×D7311yescharacteristic
370C2×#103C2×D7311yescharacteristic
371C2×#102—731no—
372C2×#102—731no—
373C2×#97—731no—
374C2×#97—731no—
375C2×C4×C8—731no—
376#391—731no—
377#391—731no—
378#389—731no—
379#389—731no—
380#389—1461no—
381#387—1461no—
382#387—1461no—
383#385—1461no—
384#385—1461no—
385#382—731no—
386#380—1461no—
387C4×C4⋊C4—731no—
388C4×C4⋊C4—731no—
389C4×C4⋊C4—1461no—
390C2×#95—1461no—
391#340—731no—
392#340—731no—
393#340—731no—
394#340—731no—
395#336—731no—
396#336—731no—
397C22×Q8C22×D7311yescharacteristic
398C22×Q8C22×D7311yescharacteristic
399C22×D4C22×D7311yescharacteristic
400C23×C4—21no—
401C23×C4—1461no—
402C23×C4—1461no—
403C2×SD16—22no—
404C2×SD16—22no—
405C2×SD16—22no—
406C2×SD16—22no—
407C2×SD16C2×Dic7314yes—
408C2×SD16C2×Dic7314yes—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8C22×D7311yescharacteristic
412C22×C8C22×D7311yescharacteristic
413C4⋊Q8—22no—
414C4⋊Q8C2×Dic7314yescomplemented
415C4×Q8—2921no—
416C4×Q8—2921no—
417C4×Q8—1462no—
418C4×Q8—1462no—
419C4×D4—22no—
420C4×D4—2921no—
421C4×D4—2921no—
422C4×D4—1462no—
423C4×D4—1462no—
424C4×D4C2×Dic7314yes—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—731no—
428C2×C4⋊C4—731no—
429C2×C4⋊C4—731no—
430C2×C4⋊C4—731no—
431C2×C4⋊C4—731no—
432C2×C4⋊C4—731no—
433C2×C4⋊C4—1461no—
434C2×C4⋊C4—1461no—
435C2×C4⋊C4—1461no—
436C2×C4⋊C4—1461no—
437C2×C4⋊C4C22×D7311yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C42—731no—
442C2×C42—731no—
443C2×C42—1461no—
444C2×C42—1461no—
445C2×C42—1461no—
446C2×C42C22×D7311yescharacteristic
447#107—1462no—
448#106—1462no—
449C4⋊C8C2×Dic7314yescomplemented
450#103—2921no—
451#103—2921no—
452#103—1462no—
453#103—1462no—
454#103C2×Dic7314yescomplemented
455#103C2×Dic7314yescomplemented
456#102—734no—
457#102—1462no—
458#97—1462no—
459#97—1462no—
460C4×C8—1462no—
461#95—731no—
462#95—731no—
463#95—1461no—
464#95—1461no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470#95—1461no—
471#95—1461no—
472C24—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8C22×Dic7312yes—
484C2×Q8C22×Dic7312yes—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C22×Dic7312yes—
489C2×D4D4×D7312yes—
490C2×D4#1192402712yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—24no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4—2922no—
498C22×C4—2922no—
499C22×C4—2922no—
500C22×C4—2922no—
501C22×C4—5841no—
502C22×C4—5841no—
503C22×C4—731no—
504C22×C4—731no—
505C22×C4—731no—
506C22×C4—731no—
507C22×C4—1461no—
508C22×C4—1461no—
509C22×C4—1461no—
510C22×C4—1461no—
511C22×C4D4×D7311yescharacteristic
512C22×C4D4×D7311yescharacteristic
513C22×C4C23×D7311yescharacteristic
514SD16—28no—
515SD16—28no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—732no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8—1461no—
525C2×C8—1461no—
526C2×C8C22×Dic7312yes—
527C2×C8C22×Dic7312yes—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—21no—
531C4⋊C4—2921no—
532C4⋊C4—2922no—
533C4⋊C4—2922no—
534C4⋊C4—5841no—
535C4⋊C4—5841no—
536C4⋊C4—732no—
537C4⋊C4—1462no—
538C4⋊C4—1461no—
539C4⋊C4—1461no—
540C4⋊C4—1461no—
541C4⋊C4—1461no—
542C4⋊C4—1461no—
543C4⋊C4—1461no—
544C4⋊C4—1461no—
545C4⋊C4C22×Dic7312yes—
546C22⋊C4—24no—
547C22⋊C4—2922no—
548C22⋊C4—2922no—
549C42—2922no—
550C42—2922no—
551C42—2922no—
552C42—1461no—
553C42—1461no—
554C42—1461no—
555C42—1461no—
556C42C22×Dic7312yes—
557C42#1192402712yes—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C2×D4×D7311yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4D4×Dic7314yes—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—2921no—
595C2×C4—2921no—
596C2×C4—2921no—
597C2×C4—2921no—
598C2×C4—2921no—
599C2×C4—2921no—
600C2×C4—2921no—
601C2×C4—2921no—
602C2×C4—2921no—
603C2×C4—5844no—
604C2×C4—5844no—
605C2×C4—732no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1462no—
614C2×C4—1461no—
615C2×C4—1461no—
616C2×C4—1461no—
617C2×C4—1461no—
618C2×C4D4×Dic7312yes—
619C2×C4D4×Dic7312yes—
620C2×C4C23×Dic7311yescharacteristic
621C2×C4C2×D4×D7311yescharacteristic
622C2×C4C2×#1192402711yescharacteristic
623C2×C4C2×#1192402711yescharacteristic
624C2×C4#1455663711yescharacteristic
625C2×C4#1455663811yescharacteristic
626C8—22no—
627C8—22no—
628C8—2921no—
629C8—1462no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22C2×D4×Dic7311yescharacteristic
638C22#1798922111yescharacteristic
639C22#1798924311yescharacteristic
640C22#137×D7311yescharacteristic
641C22#1798938611yescharacteristic
642C22#1798939311yescharacteristic
643C22#1798939311yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—2921no—
650C4—2922no—
651C4—2921no—
652C4—2921no—
653C4—5841no—
654C4—5841no—
655C4—5841no—
656C4—1462no—
657C4C2×D4×Dic7311yescharacteristic
658C4#1798907011yescharacteristic
659C4#1798923711yescharacteristic
660C4#1798928911yescharacteristic
661C2—44no—
662C2#2261295111yescharacteristic
663C2#137×Dic7311yescharacteristic
664C2#2261879111yescharacteristic
665C2#2262144711yescharacteristic
666C2#2262146211yescharacteristic
667C2#2262270711yescharacteristic
668C2#2262275111yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#461×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC2×#461D73
hypocenterC73#23012
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22612991C2semidirect
5. #22610752C2semidirect
6. #22610701C2semidirect
7. #22610623C2semidirect
8. #22610609C2semidirect
11. #22610201C2semidirect
12. #22610187C2semidirect
13. #22609808C2semidirect
58. #461×C73C4semidirect
126. C4⋊Q8×C73C2×C4semidirect
134. C4⋊C8×C73C2×C4semidirect
135. #103×C73C2×C4semidirect
136. #103×C73C2×C4semidirect
350. C73#23012semidirect
359. #461Dic73semidirect
414. C4⋊Q8C2×Dic73semidirect
449. C4⋊C8C2×Dic73semidirect
454. #103C2×Dic73semidirect
455. #103C2×Dic73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179886882
C2×SD16×Dic732
#226129912
#226128412
#226107522
#226107012
#226106232
#226106092
#226102672
#226102012
#226101872
#226098082
#226097562
C2×#461×C732
#2301273