Finite Group Databasebeta

← #33108463

Subgroups — #33108463

Order 18688.

954 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988807C211yesmaximal, characteristic
2C2×#17988691C211yesmaximal, characteristic, complemented
3#22613004C211yesmaximal, characteristic
4Q8×#11924007C211yesmaximal, characteristic, complemented
5#22610746C211yesmaximal, characteristic
6#22610691C211yesmaximal, characteristic, complemented
7#22610625C211yesmaximal, characteristic
8#22610607C211yesmaximal, characteristic, complemented
9#22610288C211yesmaximal, characteristic
10#22610272C211yesmaximal, characteristic, complemented
11#22610196C211yesmaximal, characteristic, complemented
12#22610192C211yesmaximal, characteristic
13#22609805C211yesmaximal, characteristic, complemented
14#22609763C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C73C211yesmaximal, characteristic
16C22×#11924018C2211yescharacteristic
17C2×#14556578C2211yescharacteristic
18C2×Q8×Dic73C2211yescharacteristic
19C2×Q8×Dic73C2211yescharacteristic
20C2×#14556556C2211yescharacteristic
21#17988807C418yescomplemented
22#17988691—24no—
23C2×#14556522C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556514C2211yescharacteristic
29C2×#14556512C2211yescharacteristic
30C2×#14556512C2211yescharacteristic
31C2×#14556501C2211yescharacteristic
32C2×#14556501C2211yescharacteristic
33#17988493C2211yescharacteristic
34#17988490C2211yescharacteristic
35#17988485—21no—
36#17988485C2211yescharacteristic
37#17988483—21no—
38#17988483C2211yescharacteristic
39#17988481—21no—
40#17988475—21no—
41#17988460—21no—
42#17988456—21no—
43#17988456C2211yescharacteristic
44C4⋊C4×Dic73C2211yescharacteristic
45C4×#11924007—21no—
46C4×#11924007C2211yescharacteristic
47C4×C22⋊Dic73—21no—
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988342C2211yescharacteristic
53#17988341C2211yescharacteristic
54C22×SD16×C73C2211yescharacteristic
55C2×C4×Q8×C73C2211yescharacteristic
56C2×C4×D4×C73C2211yescharacteristic
57C4×SD16×C73—24no—
58C2×#106×C73C2211yescharacteristic
59C2×#103×C73C2211yescharacteristic
60C2×#102×C73C2211yescharacteristic
61C2×C4×C8×C73C2211yescharacteristic
62C22×Dic146C2311yescharacteristic
63C2×#11924018—22no—
64C2×#11924018—22no—
65C2×#11924018C2×C414yescomplemented
66C22×#10596831C2311yescharacteristic
67C22×#10596831C2311yescharacteristic
68#14556578—22no—
69#14556578C2×C414yes—
70Q8×Dic73—22no—
71Q8×Dic73—22no—
72Q8×Dic73—41no—
73Q8×Dic73C2×C414yescomplemented
74#14556556—22no—
75#14556556—41no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73—21no—
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×#11924007—21no—
82C2×#11924007—21no—
83C2×#11924007—21no—
84C2×#11924007—21no—
85C2×#11924007—21no—
86C2×#11924007D411yescharacteristic
87C2×#11924007D411yescharacteristic
88C2×C22⋊Dic73—21no—
89C2×C22⋊Dic73—21no—
90C2×C22⋊Dic73—21no—
91C2×C22⋊Dic73—21no—
92C2×C4×Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95C2×C4×Dic73—21no—
96C2×C4×Dic73C2311yescharacteristic
97C2×C4×Dic73C2311yescharacteristic
98#14556522—22no—
99#14556520C2×C414yescomplemented
100#14556516—22no—
101#14556516C2×C414yescomplemented
102#14556515—22no—
103#14556514—41no—
104#14556512—41no—
105#14556512C2×C414yes—
106#14556501—22no—
107#14556501—22no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112#14556498—21no—
113#14556498D411yescharacteristic
114#14556498D411yescharacteristic
115C22×Q8×C73C2311yescharacteristic
116C22×D4×C73C2311yescharacteristic
117C23×C4×C73—21no—
118C2×SD16×C73—22no—
119C2×SD16×C73—22no—
120C2×SD16×C73C2×C414yes—
121C22×C8×C73C2311yescharacteristic
122C22×C8×C73C2311yescharacteristic
123C4×Q8×C73—22no—
124C4×Q8×C73—41no—
125C4×D4×C73—22no—
126C4×D4×C73—41no—
127C2×C4⋊C4×C73—21no—
128C2×C4⋊C4×C73C2311yescharacteristic
129C2×C4⋊C4×C73C2311yescharacteristic
130C2×C22⋊C4×C73—21no—
131C2×C42×C73—21no—
132C2×C42×C73C2311yescharacteristic
133#106×C73—22no—
134#103×C73D414yes—
135#102×C73—22no—
136C4×C8×C73—22no—
137C2×Dic146—21no—
138C2×Dic146—21no—
139C2×Dic146—21no—
140C2×Dic146—21no—
141C2×Dic146—21no—
142C2×Dic146C22×C412yes—
143C22×Dic73—21no—
144C22×Dic73—21no—
145C22×Dic73—21no—
146C22×Dic73—21no—
147C22×Dic73C2×D411yescharacteristic
148C22×Dic73C2×D411yescharacteristic
149C22×Dic73C2×D411yescharacteristic
150C22×Dic73C2×D411yescharacteristic
151#11924018—28no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831C22×C412yes—
158C4⋊Dic73—21no—
159C4⋊Dic73—24no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—42no—
165C4⋊Dic73C22×C412yes—
166#11924007—22no—
167#11924007—41no—
168#11924007—42no—
169#11924007—42no—
170#11924007—81no—
171#11924007—81no—
172#11924007—81no—
173C22⋊Dic73—24no—
174C22⋊Dic73—24no—
175C22⋊Dic73—42no—
176C22⋊Dic73—81no—
177C4×Dic73—21no—
178C4×Dic73—21no—
179C4×Dic73—24no—
180C4×Dic73—42no—
181C4×Dic73—42no—
182C4×Dic73—81no—
183C4×Dic73C22×C412yes—
184C4×Dic73#4112yes—
185C24×C73—21no—
186C2×Q8×C73—21no—
187C2×Q8×C73—21no—
188C2×Q8×C73—21no—
189C2×Q8×C73C22×C412yes—
190C2×Q8×C73C2×D412yes—
191C2×Q8×C73#4112yes—
192C2×D4×C73—21no—
193C2×D4×C73—21no—
194C2×D4×C73—21no—
195C2×D4×C73—21no—
196C2×D4×C73—21no—
197C2×D4×C73C22×C412yes—
198C22×C4×C73—21no—
199C22×C4×C73—21no—
200C22×C4×C73—21no—
201C22×C4×C73—42no—
202C22×C4×C73—42no—
203C22×C4×C73—81no—
204C22×C4×C73C2×D411yescharacteristic
205C22×C4×C73C2×D411yescharacteristic
206C22×C4×C73C2411yescharacteristic
207SD16×C73—28no—
208C2×C8×C73—21no—
209C2×C8×C73—21no—
210C2×C8×C73—21no—
211C2×C8×C73—21no—
212C2×C8×C73C22×C412yes—
213C2×C8×C73C2×D412yes—
214C4⋊C4×C73—21no—
215C4⋊C4×C73—21no—
216C4⋊C4×C73—21no—
217C4⋊C4×C73—42no—
218C4⋊C4×C73C2×D412yes—
219C22⋊C4×C73—42no—
220C42×C73—21no—
221C42×C73—21no—
222C42×C73—42no—
223C23×C73—22no—
224C23×C73—41no—
225C23×C73—41no—
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×C73C4×D414yes—
235D4×C73—22no—
236D4×C73—22no—
237D4×C73—41no—
238D4×C73—41no—
239C2×C4×C73—22no—
240C2×C4×C73—21no—
241C2×C4×C73—22no—
242C2×C4×C73—22no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—22no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73—84no—
254C2×C4×C73#13712yes—
255C2×C4×C73C23×C411yescharacteristic
256C2×C4×C73C22×D411yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73H(5,𝔽2)11yescharacteristic
260C2×C4×C732−1+411yescharacteristic
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—22no—
266C2×Dic73—22no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73—21no—
272C2×Dic73—21no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—41no—
277C2×Dic73—41no—
278C2×Dic73—41no—
279C2×Dic73—41no—
280C2×Dic73—41no—
281C2×Dic73—41no—
282C2×Dic73—41no—
283C2×Dic73C4×D412yes—
284C2×Dic73C4×D412yes—
285Dic146—22no—
286Dic146—22no—
287Dic146—41no—
288Dic146—41no—
289C8×C73—22no—
290C8×C73—22no—
291#10596831—22no—
292#10596831—41no—
293C22×C73—42no—
294C22×C73—42no—
295C22×C73—42no—
296C22×C73—81no—
297C22×C73—81no—
298C22×C73—81no—
299C22×C73—81no—
300C22×C73C2×C4×D411yescharacteristic
301C22×C73D4211yescharacteristic
302C22×C73#54811yescharacteristic
303C22×C73C2×#13711yescharacteristic
304C22×C73#57511yescharacteristic
305C22×C73#57711yescharacteristic
306C22×C73#57711yescharacteristic
307C4×C73—24no—
308C4×C73—22no—
309C4×C73—42no—
310C4×C73—41no—
311C4×C73—81no—
312C4×C73C2×C4×D411yescharacteristic
313C4×C73#52011yescharacteristic
314C4×C73#54711yescharacteristic
315C4×C73#55311yescharacteristic
316Dic73—22no—
317Dic73—22no—
318Dic73—42no—
319Dic73—42no—
320Dic73—41no—
321Dic73—41no—
322Dic73—81no—
323Dic73—81no—
324#23013—731nomaximal, Sylow
325C2×C73—82no—
326C2×C73#236111yescharacteristic
327C2×C73C4×#13711yescharacteristic
328C2×C73#295211yescharacteristic
329C2×C73#326611yescharacteristic
330C2×C73#327111yescharacteristic
331C2×C73#336911yescharacteristic
332C2×C73#337111yescharacteristic
333C2×#464—731no—
334C2×#441—731no—
335C2×C4×SD16D7311yescharacteristic
336#2367—731no—
337Q8×C4⋊C4—731no—
338#1959—731no—
339#1948—731no—
340#1936—731no—
341#1933—731no—
342#1880—731no—
343#1878—731no—
344#1865—731no—
345#1864—731no—
346#1771—731no—
347#1763—731no—
348C73#2301311yescharacteristic, complemented, Sylow
349C22×Q16—731no—
350C22×SD16C2×D7311yescharacteristic
351C2×#124—731no—
352C2×C4×Q8—731no—
353C2×C4×Q8—731no—
354C2×C4×Q8—731no—
355C2×C4×Q8C2×D7311yescharacteristic
356C2×C4×D4C2×D7311yescharacteristic
357#464—738no—
358#441—1464no—
359C4×SD16—24no—
360C2×#106—731no—
361C2×#106C2×D7311yescharacteristic
362C2×C4⋊C8—731no—
363C2×#103—731no—
364C2×#103—731no—
365C2×#103—731no—
366C2×#103—731no—
367C2×#103C2×D7311yescharacteristic
368C2×#102—731no—
369C2×#102—731no—
370C2×#102C2×D7311yescharacteristic
371C2×#97—731no—
372C2×#97—731no—
373C2×C4×C8C2×D7311yescharacteristic
374#391—731no—
375#390—731no—
376#389—731no—
377#389—731no—
378#389—1461no—
379#389—1461no—
380#388—1461no—
381#388—1461no—
382#385—1461no—
383#384—731no—
384#384—1461no—
385C4×C4⋊C4—731no—
386C4×C4⋊C4—731no—
387C4×C4⋊C4—1461no—
388C4×C22⋊C4—1461no—
389#340—731no—
390#340—731no—
391#340—731no—
392#340—731no—
393#336—731no—
394#336—731no—
395C22×Q8—731no—
396C22×Q8C22×D7311yescharacteristic
397C22×D4C22×D7311yescharacteristic
398C23×C4—21no—
399C2×Q16—734no—
400C2×Q16—1462no—
401C2×Q16—1462no—
402C2×SD16—22no—
403C2×SD16—22no—
404C2×SD16C4×D7314yes—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8C22×D7311yescharacteristic
408C22×C8C22×D7311yescharacteristic
409#124—734no—
410#124—1462no—
411C4×Q8—22no—
412C4×Q8—41no—
413C4×Q8—2921no—
414C4×Q8—2921no—
415C4×Q8—734no—
416C4×Q8—1462no—
417C4×Q8—1462no—
418C4×Q8—1462no—
419C4×D4—22no—
420C4×D4—41no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—731no—
423C2×C4⋊C4—731no—
424C2×C4⋊C4—731no—
425C2×C4⋊C4—731no—
426C2×C4⋊C4—731no—
427C2×C4⋊C4—1461no—
428C2×C4⋊C4—1461no—
429C2×C4⋊C4—1461no—
430C2×C4⋊C4—1461no—
431C2×C4⋊C4—1461no—
432C2×C4⋊C4—1461no—
433C2×C4⋊C4—1461no—
434C2×C4⋊C4C22×D7311yescharacteristic
435C2×C4⋊C4C22×D7311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1461no—
438C2×C22⋊C4—1461no—
439C2×C22⋊C4—1461no—
440C2×C22⋊C4—1461no—
441C2×C42—21no—
442C2×C42—731no—
443C2×C42—731no—
444C2×C42—1461no—
445C2×C42—1461no—
446C2×C42—1461no—
447C2×C42—1461no—
448C2×C42C22×D7311yescharacteristic
449#106—22no—
450#106—1462no—
451C4⋊C8—734no—
452#103—2921no—
453#103—734no—
454#103—1462no—
455#103—1462no—
456#103C22⋊D7314yes—
457#102—22no—
458#102—2921no—
459#102—734no—
460#97—1462no—
461#97—1462no—
462C4×C8—22no—
463#95—731no—
464#95—731no—
465#95—1461no—
466#95—1461no—
467#95—1461no—
468#95—1461no—
469#95—1461no—
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—732no—
475C2×Q8—1461no—
476C2×Q8—1461no—
477C2×Q8—1461no—
478C2×Q8—1461no—
479C2×Q8—1461no—
480C2×Q8C2×C4×D7312yes—
481C2×Q8C2×C22⋊D7312yes—
482C2×Q8#1192402812yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D7312yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—81no—
495C22×C4—731no—
496C22×C4—731no—
497C22×C4—731no—
498C22×C4—731no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4D4×D7311yescharacteristic
504C22×C4D4×D7311yescharacteristic
505C22×C4C23×D7311yescharacteristic
506Q16—1468no—
507SD16—28no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—732no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8C2×C4×D7312yes—
519C2×C8C2×C22⋊D7312yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—2922no—
525C4⋊C4—2922no—
526C4⋊C4—2921no—
527C4⋊C4—2922no—
528C4⋊C4—5841no—
529C4⋊C4—5841no—
530C4⋊C4—5841no—
531C4⋊C4—732no—
532C4⋊C4—1464no—
533C4⋊C4—1461no—
534C4⋊C4—1461no—
535C4⋊C4—1461no—
536C4⋊C4—1461no—
537C4⋊C4—1462no—
538C4⋊C4—1461no—
539C4⋊C4C2×C22⋊D7312yes—
540C22⋊C4—42no—
541C22⋊C4—2922no—
542C22⋊C4—5841no—
543C22⋊C4—1464no—
544C22⋊C4—1464no—
545C42—21no—
546C42—21no—
547C42—42no—
548C42—2922no—
549C42—2922no—
550C42—5841no—
551C42—732no—
552C42—732no—
553C42—1464no—
554C42—1461no—
555C42—1461no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×D4×D7311yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—2921no—
568Q8—2921no—
569Q8—1462no—
570Q8—1462no—
571Q8C4×C22⋊D7314yes—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—2921no—
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—732no—
602C2×C4—732no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1462no—
610C2×C4—1462no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1461no—
614C2×C4—1461no—
615C2×C4#1455662012yes—
616C2×C4C22×C4×D7311yescharacteristic
617C2×C4C22×C22⋊D7311yescharacteristic
618C2×C4#41×D7311yescharacteristic
619C2×C4C2×#1192402811yescharacteristic
620C2×C4#1455663611yescharacteristic
621C2×C4#1455663811yescharacteristic
622C8—22no—
623C8—22no—
624C8—2921no—
625C8—1462no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C4×D4×D7311yescharacteristic
634C22D4×C22⋊D7311yescharacteristic
635C22#1798924611yescharacteristic
636C22C2×#1455662011yescharacteristic
637C22#1798938411yescharacteristic
638C22#1798939311yescharacteristic
639C22#1798939411yescharacteristic
640C4—22no—
641C4—24no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—2921no—
646C4—2921no—
647C4—2922no—
648C4—2922no—
649C4—5841no—
650C4—5841no—
651C4—1462no—
652C4—1462no—
653C4C2×C4×C22⋊D7311yescharacteristic
654C4#1798906611yescharacteristic
655C4#1798923611yescharacteristic
656C4#1798928611yescharacteristic
657C2—82no—
658C2#2261294811yescharacteristic
659C2C4×#1455662011yescharacteristic
660C2#2261880911yescharacteristic
661C2#2262144111yescharacteristic
662C2#2262151611yescharacteristic
663C2#2262266911yescharacteristic
664C2#2262269511yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×C4×SD16×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
cosocleC22×C4×C73C24
hypercenterC2×C4×SD16D73
hypocenterC73#23013
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#17988691C2semidirect
4. Q8×#11924007C2semidirect
6. #22610691C2semidirect
8. #22610607C2semidirect
10. #22610272C2semidirect
11. #22610196C2semidirect
13. #22609805C2semidirect
14. #22609763C2semidirect
21. #17988807C4semidirect
65. C2×#11924018C2×C4semidirect
73. Q8×Dic73C2×C4semidirect
99. #14556520C2×C4semidirect
101. #14556516C2×C4semidirect
348. C73#23013semidirect

Maximal subgroups

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

subgroupindex
C2×#179888072
C2×#179886912
#226130042
Q8×#119240072
#226107462
#226106912
#226106252
#226106072
#226102882
#226102722
#226101962
#226101922
#226098052
#226097632
C2×C4×SD16×C732
#2301373