Finite Group Databasebeta

← #33104349

Subgroups — #33104349

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988688C211yesmaximal, characteristic
2C2×SD16×Dic73C211yesmaximal, characteristic
3Q8×C4⋊Dic73C211yesmaximal, characteristic, complemented
4#22612841C211yesmaximal, characteristic
5#22610670C211yesmaximal, characteristic, complemented
6#22610665C211yesmaximal, characteristic, complemented
7#22610201C211yesmaximal, characteristic, complemented
8#22610194C211yesmaximal, characteristic
9#22610187C211yesmaximal, characteristic, complemented
10#22610180C211yesmaximal, characteristic
11#22610099C211yesmaximal, characteristic, complemented
12#22609946C211yesmaximal, characteristic, complemented
13#22609931C211yesmaximal, characteristic
14#22609803C211yesmaximal, characteristic, complemented
15C2×#440×C73C211yesmaximal, characteristic
16C2×Q8×Dic73C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×D4×Dic73C2211yescharacteristic
19C2×D4×Dic73C2211yescharacteristic
20#17988688—24no—
21SD16×Dic73—24no—
22C2×#14556525C2211yescharacteristic
23C2×#14556525C2211yescharacteristic
24C2×#14556523C2211yescharacteristic
25C2×#14556523C2211yescharacteristic
26C2×C4⋊#10596831C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33#17988486—21no—
34#17988486—21no—
35#17988486C2211yescharacteristic
36#17988486C2211yescharacteristic
37#17988486C2211yescharacteristic
38#17988486C2211yescharacteristic
39#17988462—21no—
40#17988462—21no—
41#17988457—21no—
42#17988457C2211yescharacteristic
43#17988453C2211yescharacteristic
44#17988447—21no—
45C4×C4⋊Dic73—21no—
46C4×C4⋊Dic73C2211yescharacteristic
47C2×#14556498—21no—
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988341C2211yescharacteristic
53#17988341C2211yescharacteristic
54C22×SD16×C73C2211yescharacteristic
55C2×C4×Q8×C73C2211yescharacteristic
56C2×C4×D4×C73C2211yescharacteristic
57#440×C73C418yescomplemented
58C2×#107×C73C2211yescharacteristic
59C2×#103×C73C2211yescharacteristic
60C2×#102×C73C2211yescharacteristic
61C2×#97×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⋊Dic73—21no—
79C2×C4⋊Dic73—21no—
80C2×C4⋊Dic73—21no—
81C2×C4⋊Dic73—21no—
82C2×C4⋊Dic73D411yescharacteristic
83C2×C4⋊Dic73D411yescharacteristic
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×C4⋊Dic73C2311yescharacteristic
87C2×C4⋊Dic73C2311yescharacteristic
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73—21no—
92C2×C4×Dic73C2311yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94#14556525—22no—
95#14556525—41no—
96#14556523—22no—
97#14556523—41no—
98C4⋊#10596831—22no—
99#14556515—22no—
100#14556515—22no—
101#14556512—22no—
102#14556512—22no—
103#14556501—22no—
104C8×Dic73—22no—
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×D4×C73C2311yescharacteristic
115C23×C4×C73—21no—
116C2×SD16×C73C2×C414yes—
117C2×SD16×C73D414yes—
118C2×SD16×C73Q814yes—
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C4×Q8×C73—22no—
122C4×Q8×C73C2×C414yescomplemented
123C4×D4×C73—22no—
124C4×D4×C73C2×C414yes—
125C2×C4⋊C4×C73—21no—
126C2×C4⋊C4×C73C2311yescharacteristic
127C2×C4⋊C4×C73C2311yescharacteristic
128C2×C22⋊C4×C73—21no—
129C2×C42×C73—21no—
130C2×C42×C73C2311yescharacteristic
131#107×C73C2×C414yescomplemented
132#103×C73C2×C414yescomplemented
133#102×C73C2×C414yes—
134#97×C73C2×C414yescomplemented
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—42no—
140C22×Dic73—42no—
141C22×Dic73—42no—
142C22×Dic73—42no—
143C22×Dic73—81no—
144C22×Dic73—81no—
145C22×Dic73C2×D411yescharacteristic
146C22×Dic73C2×D411yescharacteristic
147C22×Dic73C2×D411yescharacteristic
148C22×Dic73C2×D411yescharacteristic
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C4⋊Dic73—22no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—41no—
165C4⋊Dic73—42no—
166C4⋊Dic73—42no—
167C4⋊Dic73—42no—
168C4⋊Dic73—81no—
169C4⋊Dic73—81no—
170C4⋊Dic73—81no—
171C4⋊Dic73—81no—
172C4⋊Dic73—81no—
173C22⋊Dic73—42no—
174C22⋊Dic73—42no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—21no—
178C4×Dic73—21no—
179C4×Dic73—42no—
180C4×Dic73—42no—
181C24×C73—21no—
182C2×Q8×C73—21no—
183C2×Q8×C73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73C22×C412yes—
186C2×Q8×C73C2×D412yes—
187C2×Q8×C73C2×Q812yes—
188C2×D4×C73—21no—
189C2×D4×C73—21no—
190C2×D4×C73—21no—
191C2×D4×C73C22×C412yes—
192C2×D4×C73C2×D412yes—
193C2×D4×C73C2×Q812yes—
194C22×C4×C73—21no—
195C22×C4×C73—21no—
196C22×C4×C73—21no—
197C22×C4×C73—24no—
198C22×C4×C73—81no—
199C22×C4×C73—81no—
200C22×C4×C73C2×D411yescharacteristic
201C22×C4×C73C2×D411yescharacteristic
202C22×C4×C73C2411yescharacteristic
203SD16×C73C4⋊C4116yescomplemented
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73C22×C412yes—
207C2×C8×C73C22×C412yes—
208C2×C8×C73C2×D412yes—
209C2×C8×C73C2×Q812yes—
210C4⋊C4×C73—24no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—21no—
213C4⋊C4×C73C22×C412yes—
214C4⋊C4×C73C22×C412yes—
215C22⋊C4×C73—24no—
216C42×C73—24no—
217C42×C73C22×C412yes—
218C42×C73#4112yes—
219C23×C73—22no—
220C23×C73—22no—
221C23×C73—22no—
222C23×C73—41no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—41no—
226C23×C73C22×D411yescharacteristic
227Q8×C73—22no—
228Q8×C73—22no—
229Q8×C73—22no—
230Q8×C73C2×C4⋊C414yes—
231D4×C73—22no—
232D4×C73—22no—
233D4×C73—22no—
234D4×C73C2×C4⋊C414yes—
235C2×C4×C73—22no—
236C2×C4×C73—22no—
237C2×C4×C73—22no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—22no—
243C2×C4×C73—22no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—84no—
249C2×C4×C73C4×D412yes—
250C2×C4×C73C4×D412yes—
251C2×C4×C73C23×C411yescharacteristic
252C2×C4×C73C22×D411yescharacteristic
253C2×C4×C73C22×Q811yescharacteristic
254C2×C4×C73C2×#4111yescharacteristic
255C2×C4×C73H(5,𝔽2)11yescharacteristic
256C2×C4×C732−1+411yescharacteristic
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270C2×Dic73—41no—
271C2×Dic73—41no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—84no—
281C2×Dic73—84no—
282C2×Dic73#13712yes—
283C8×C73—22no—
284C8×C73C2×C4⋊C414yes—
285#10596831—41no—
286#10596831—41no—
287C22×C73—24no—
288C22×C73—42no—
289C22×C73—42no—
290C22×C73—42no—
291C22×C73—42no—
292C22×C73—42no—
293C22×C73—42no—
294C22×C73C2×C4×D411yescharacteristic
295C22×C73D4211yescharacteristic
296C22×C73D4×Q811yescharacteristic
297C22×C73C2×#13711yescharacteristic
298C22×C73#57511yescharacteristic
299C22×C73#57611yescharacteristic
300C22×C73#57711yescharacteristic
301C4×C73—24no—
302C4×C73—22no—
303C4×C73—22no—
304C4×C73—42no—
305C4×C73—42no—
306C4×C73C22×C4⋊C411yescharacteristic
307C4×C73#52011yescharacteristic
308C4×C73#54711yescharacteristic
309C4×C73#55411yescharacteristic
310Dic73—22no—
311Dic73—41no—
312Dic73—41no—
313Dic73—41no—
314Dic73—81no—
315Dic73—81no—
316Dic73—81no—
317Dic73—81no—
318#22678—731nomaximal, Sylow
319C2×C73—44no—
320C2×C73D4×C4⋊C411yescharacteristic
321C2×C73C4×#13711yescharacteristic
322C2×C73#294611yescharacteristic
323C2×C73#326611yescharacteristic
324C2×C73#326811yescharacteristic
325C2×C73#338511yescharacteristic
326C2×C73#338811yescharacteristic
327C2×#440—731no—
328C2×#440D7311yescharacteristic
329C2×C4×SD16—731no—
330Q8×C4⋊C4—731no—
331#2349—731no—
332#1943—731no—
333#1942—731no—
334#1865—731no—
335#1864—731no—
336#1863—731no—
337#1862—731no—
338#1847—731no—
339#1805—731no—
340#1802—731no—
341#1771—731no—
342C73#2267811yescharacteristic, complemented, Sylow
343C22×SD16C2×D7311yescharacteristic
344C2×C4×Q8—731no—
345C2×C4×Q8—731no—
346C2×C4×Q8C2×D7311yescharacteristic
347C2×C4×D4—731no—
348C2×C4×D4—731no—
349C2×C4×D4C2×D7311yescharacteristic
350#440—1464no—
351#440Dic7318yescomplemented
352C4×SD16—1464no—
353C2×#107—731no—
354C2×#107—731no—
355C2×#107C2×D7311yescharacteristic
356C2×#106—731no—
357C2×#106—731no—
358C2×C4⋊C8—731no—
359C2×#103—731no—
360C2×#103—731no—
361C2×#103C2×D7311yescharacteristic
362C2×#102—731no—
363C2×#102—731no—
364C2×#102C2×D7311yescharacteristic
365C2×#97—731no—
366C2×#97C2×D7311yescharacteristic
367C2×C4×C8—731no—
368#389—731no—
369#389—731no—
370#389—731no—
371#389—731no—
372#389—1461no—
373#389—1461no—
374#385—1461no—
375#385—1461no—
376#384—731no—
377#384—1461no—
378#382—731no—
379#380—1461no—
380C4×C4⋊C4—731no—
381C4×C4⋊C4—1461no—
382C2×#95—1461no—
383#340—731no—
384#340—731no—
385#340—731no—
386#340—731no—
387#336—731no—
388#336—731no—
389C22×Q8C22×D7311yescharacteristic
390C22×D4C22×D7311yescharacteristic
391C23×C4—21no—
392C23×C4—1461no—
393C23×C4—1461no—
394C2×SD16Dic14614yes—
395C2×SD16D29214yes—
396C2×SD16C2×Dic7314yes—
397C22×C8—731no—
398C22×C8—731no—
399C22×C8C22×D7311yescharacteristic
400C22×C8C22×D7311yescharacteristic
401C4×Q8—22no—
402C4×Q8—2921no—
403C4×Q8—2921no—
404C4×Q8—1462no—
405C4×Q8—1462no—
406C4×Q8C2×Dic7314yescomplemented
407C4×D4—22no—
408C4×D4—2921no—
409C4×D4—2921no—
410C4×D4—1462no—
411C4×D4—1462no—
412C4×D4C2×Dic7314yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4—731no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—1461no—
421C2×C4⋊C4—1461no—
422C2×C4⋊C4—1461no—
423C2×C4⋊C4—1461no—
424C2×C4⋊C4—1461no—
425C2×C4⋊C4—1461no—
426C2×C4⋊C4—1461no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4C22×D7311yescharacteristic
429C2×C4⋊C4C22×D7311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1461no—
432C2×C22⋊C4—1461no—
433C2×C42—21no—
434C2×C42—731no—
435C2×C42—731no—
436C2×C42—1461no—
437C2×C42—1461no—
438C2×C42C22×D7311yescharacteristic
439#107—2921no—
440#107—1462no—
441#107C2×Dic7314yescomplemented
442#106—2921no—
443#106—1462no—
444C4⋊C8—1462no—
445#103—1462no—
446#103—1462no—
447#103C2×Dic7314yescomplemented
448#102—1462no—
449#102—1462no—
450#102C2×Dic7314yes—
451#97—1462no—
452#97C2×Dic7314yescomplemented
453C4×C8—1462no—
454#95—731no—
455#95—731no—
456#95—1461no—
457#95—1461no—
458#95—1461no—
459#95—1461no—
460#95—1461no—
461#95—1461no—
462C24—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C22×Dic7312yes—
467C2×Q8C2×D29212yes—
468C2×Q8C2×Dic14612yes—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C22×Dic7312yes—
473C2×D4C2×D29212yes—
474C2×D4C2×Dic14612yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—24no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—2922no—
482C22×C4—2922no—
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—5841no—
486C22×C4—5841no—
487C22×C4—731no—
488C22×C4—731no—
489C22×C4—731no—
490C22×C4—731no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4—1461no—
495C22×C4D4×D7311yescharacteristic
496C22×C4D4×D7311yescharacteristic
497C22×C4C23×D7311yescharacteristic
498SD16C4⋊Dic73116yescomplemented
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8C22×Dic7312yes—
508C2×C8C22×Dic7312yes—
509C2×C8C2×D29212yes—
510C2×C8C2×Dic14612yes—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—2922no—
515C4⋊C4—2922no—
516C4⋊C4—2922no—
517C4⋊C4—2921no—
518C4⋊C4—5841no—
519C4⋊C4—5841no—
520C4⋊C4—5841no—
521C4⋊C4—5841no—
522C4⋊C4—5841no—
523C4⋊C4—1462no—
524C4⋊C4—1461no—
525C4⋊C4—1461no—
526C4⋊C4—1461no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4—1461no—
532C4⋊C4C22×Dic7312yes—
533C4⋊C4C22×Dic7312yes—
534C22⋊C4—24no—
535C22⋊C4—2922no—
536C22⋊C4—2922no—
537C42—24no—
538C42—2922no—
539C42—2922no—
540C42—1461no—
541C42—1461no—
542C42—1461no—
543C42—1461no—
544C42C22×Dic7312yes—
545C42#1192402712yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C2×D4×D7311yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C2×C4⋊Dic7314yes—
558D4—22no—
559D4—22no—
560D4—22no—
561D4C2×C4⋊Dic7314yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—2921no—
577C2×C4—2921no—
578C2×C4—2921no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—5844no—
589C2×C4—5844no—
590C2×C4—732no—
591C2×C4—1461no—
592C2×C4—1461no—
593C2×C4—1461no—
594C2×C4—1461no—
595C2×C4—1461no—
596C2×C4—1461no—
597C2×C4—1461no—
598C2×C4—1461no—
599C2×C4—1461no—
600C2×C4—1461no—
601C2×C4—1461no—
602C2×C4D4×Dic7312yes—
603C2×C4D4×Dic7312yes—
604C2×C4C23×Dic7311yescharacteristic
605C2×C4C22×D29211yescharacteristic
606C2×C4C22×Dic14611yescharacteristic
607C2×C4C2×#1192402711yescharacteristic
608C2×C4#1455663711yescharacteristic
609C2×C4#1455663811yescharacteristic
610C8—22no—
611C8—2921no—
612C8—2921no—
613C8C2×C4⋊Dic7314yes—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C2×D4×Dic7311yescharacteristic
622C22D4×D29211yescharacteristic
623C22D4×Dic14611yescharacteristic
624C22#137×D7311yescharacteristic
625C22#1798938611yescharacteristic
626C22#1798939111yescharacteristic
627C22#1798939711yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—2921no—
634C4—2921no—
635C4—2921no—
636C4—5841no—
637C4—5841no—
638C4—5841no—
639C4—5841no—
640C4—1462no—
641C4C22×C4⋊Dic7311yescharacteristic
642C4#1798907011yescharacteristic
643C4#1798924111yescharacteristic
644C4#1798929411yescharacteristic
645C2—44no—
646C2D4×C4⋊Dic7311yescharacteristic
647C2#137×Dic7311yescharacteristic
648C2#2261876411yescharacteristic
649C2#2262144311yescharacteristic
650C2#2262147311yescharacteristic
651C2#2262284211yescharacteristic
652C2#2262287711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#440×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC2×#440D73
hypocenterC73#22678
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. Q8×C4⋊Dic73C2semidirect
5. #22610670C2semidirect
6. #22610665C2semidirect
7. #22610201C2semidirect
9. #22610187C2semidirect
11. #22610099C2semidirect
12. #22609946C2semidirect
14. #22609803C2semidirect
57. #440×C73C4semidirect
122. C4×Q8×C73C2×C4semidirect
131. #107×C73C2×C4semidirect
132. #103×C73C2×C4semidirect
134. #97×C73C2×C4semidirect
203. SD16×C73C4⋊C4semidirect
342. C73#22678semidirect
351. #440Dic73semidirect
406. C4×Q8C2×Dic73semidirect
441. #107C2×Dic73semidirect
447. #103C2×Dic73semidirect
452. #97C2×Dic73semidirect
498. SD16C4⋊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
Q8×C4⋊Dic732
#226128412
#226106702
#226106652
#226102012
#226101942
#226101872
#226101802
#226100992
#226099462
#226099312
#226098032
C2×#440×C732
#2267873