Finite Group Databasebeta

← #33149070

Subgroups — #33149070

Order 18688.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988882C211yesmaximal, characteristic, complemented
2C2×C4×#11924015C211yesmaximal, characteristic
3#22613781C211yesmaximal, characteristic
4#22611455C211yesmaximal, characteristic
5#22611413C211yesmaximal, characteristic, complemented
6#22611316C211yesmaximal, characteristic
7#22611264C211yesmaximal, characteristic, complemented
8#22610661C211yesmaximal, characteristic, complemented
9#22610490C211yesmaximal, characteristic, complemented
10#22610286C211yesmaximal, characteristic
11#22610286C211yesmaximal, characteristic
12#22610199C211yesmaximal, characteristic, complemented
13#22610185C211yesmaximal, characteristic, complemented
14#22610177C211yesmaximal, characteristic
15C73×#2351C211yesmaximal, characteristic, complemented
16C22×#11924015C2211yescharacteristic
17C2×#14556591C2211yescharacteristic
18C2×#14556569—21no—
19C2×C4×D292C2211yescharacteristic
20#17988882—24no—
21C4×#11924015—24no—
22C2×#14556524C2211yescharacteristic
23C2×#14556524C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556522C2211yescharacteristic
26C2×#14556522C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556515C2211yescharacteristic
29C2×#14556515C2211yescharacteristic
30C2×#14556515C2211yescharacteristic
31C2×#14556511C2211yescharacteristic
32C2×#14556511C2211yescharacteristic
33C2×#14556511C2211yescharacteristic
34C2×C4×#10596831C2211yescharacteristic
35#17988509—21no—
36#17988500—21no—
37#17988489C2211yescharacteristic
38#17988489C2211yescharacteristic
39#17988485C2211yescharacteristic
40#17988485C2211yescharacteristic
41#17988461—21no—
42#17988349C2211yescharacteristic
43#17988349C2211yescharacteristic
44#17988349C2211yescharacteristic
45#17988349C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988342C2211yescharacteristic
49#17988342C2211yescharacteristic
50C2×C4×Q8×C73C2211yescharacteristic
51C2×C4×Q8×C73C2211yescharacteristic
52C2×C4×Q8×C73C2211yescharacteristic
53#389×C73—21no—
54#389×C73—21no—
55#389×C73C2211yescharacteristic
56#389×C73C2211yescharacteristic
57#382×C73—21no—
58#382×C73C2211yescharacteristic
59C4×C4⋊C4×C73—21no—
60C4×C4⋊C4×C73C2211yescharacteristic
61C22×Dic146—21no—
62C22×D292C2311yescharacteristic
63C22×C4×D73—21no—
64C2×#11924015—22no—
65C2×#11924015—22no—
66C2×#11924015Q814yescomplemented
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71#14556591—22no—
72#14556591—41no—
73#14556569—44no—
74C4×D292—22no—
75C4×D292—41no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73C2311yescharacteristic
78C2×C4⋊Dic73C2311yescharacteristic
79C2×C4⋊Dic73C2311yescharacteristic
80C2×#11924007—21no—
81C2×#11924007—21no—
82C2×#11924005—21no—
83C2×#11924005—21no—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×#11924005—21no—
87#14556524—22no—
88#14556524—22no—
89#14556524—22no—
90#14556522—22no—
91#14556522—41no—
92C4⋊#10596831—22no—
93#14556515—22no—
94#14556515—22no—
95#14556515D414yes—
96#14556511—22no—
97#14556511—22no—
98#14556511—41no—
99C4×#10596831—22no—
100#14556498—21no—
101#14556498—21no—
102C22×Q8×C73C2311yescharacteristic
103C4×Q8×C73—22no—
104C4×Q8×C73—22no—
105C4×Q8×C73—22no—
106C4×Q8×C73—22no—
107C4×Q8×C73—41no—
108C4×Q8×C73—41no—
109C2×C4⋊C4×C73—21no—
110C2×C4⋊C4×C73—21no—
111C2×C4⋊C4×C73—21no—
112C2×C4⋊C4×C73—21no—
113C2×C4⋊C4×C73—21no—
114C2×C4⋊C4×C73D411yescharacteristic
115C2×C4⋊C4×C73D411yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C4⋊C4×C73C2311yescharacteristic
119C2×C42×C73—21no—
120C2×C42×C73—21no—
121C2×C42×C73—21no—
122C2×C42×C73C2311yescharacteristic
123C2×C42×C73C2311yescharacteristic
124C2×C42×C73C2311yescharacteristic
125#95×C73—21no—
126#95×C73—21no—
127#95×C73—21no—
128#95×C73—21no—
129#95×C73D411yescharacteristic
130#95×C73D411yescharacteristic
131C23×D73—21no—
132C2×Dic146—42no—
133C2×Dic146—42no—
134C2×Dic146—81no—
135C2×D292—21no—
136C2×D292—21no—
137C2×D292—21no—
138C2×D292—21no—
139C2×D292—21no—
140C2×D292C2×Q812yes—
141C22×Dic73—21no—
142C22×Dic73—21no—
143C22×Dic73—21no—
144C2×C4×D73—42no—
145C2×C4×D73—42no—
146C2×C4×D73—81no—
147#11924015—28no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831—21no—
158C2×#10596831C2×D412yes—
159C2×#10596831C2×Q812yes—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—42no—
166C4⋊Dic73C2×D412yes—
167#11924007—42no—
168#11924007—42no—
169#11924005—42no—
170#11924005—42no—
171#11924005—42no—
172#11924005—42no—
173#11924005—81no—
174C2×Q8×C73—21no—
175C2×Q8×C73—21no—
176C2×Q8×C73—21no—
177C2×Q8×C73C2×D412yes—
178C2×Q8×C73C2×Q812yes—
179C2×Q8×C73#4112yes—
180C22×C4×C73—21no—
181C22×C4×C73—21no—
182C22×C4×C73—21no—
183C22×C4×C73—21no—
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×D411yescharacteristic
187C22×C4×C73C2×D411yescharacteristic
188C22×C4×C73C2×D411yescharacteristic
189C22×C4×C73C2×D411yescharacteristic
190C22×C4×C73C2411yescharacteristic
191C4⋊C4×C73—24no—
192C4⋊C4×C73—22no—
193C4⋊C4×C73—22no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—21no—
196C4⋊C4×C73—21no—
197C4⋊C4×C73—21no—
198C4⋊C4×C73—21no—
199C4⋊C4×C73—42no—
200C4⋊C4×C73—42no—
201C4⋊C4×C73—42no—
202C4⋊C4×C73—42no—
203C4⋊C4×C73#4112yes—
204C42×C73—24no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—21no—
210C42×C73—21no—
211C42×C73—42no—
212C42×C73—42no—
213C23×C73C22×D411yescharacteristic
214C22×D73—22no—
215C22×D73—41no—
216C22×D73—41no—
217C22×D73—41no—
218C22×D73—41no—
219C22×D73—41no—
220C22×D73—41no—
221Q8×C73—22no—
222Q8×C73—22no—
223Q8×C73—22no—
224Q8×C73C22⋊Q814yescomplemented
225C2×C4×C73—22no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—22no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—41no—
253C2×C4×C73#13512yes—
254C2×C4×C73AGL(1,ℤ/8)12yes—
255C2×C4×C73C22×D411yescharacteristic
256C2×C4×C73C22×Q811yescharacteristic
257C2×C4×C73C2×#4111yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×C4×C73H(5,𝔽2)11yescharacteristic
260C2×C4×C732−1+411yescharacteristic
261C2×Dic73—22no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—41no—
266C2×Dic73—41no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—41no—
270D292—22no—
271D292—22no—
272D292—41no—
273D292—41no—
274C4×D73—84no—
275Dic146—84no—
276#10596831—22no—
277#10596831—22no—
278#10596831—41no—
279#10596831—41no—
280C22×C73#52511yescharacteristic
281C22×C73D4211yescharacteristic
282C22×C73D4×Q811yescharacteristic
283C22×C73C2×#13511yescharacteristic
284C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C73#57711yescharacteristic
286C22×C73#57811yescharacteristic
287C2×D73—42no—
288C2×D73—42no—
289C2×D73—42no—
290C2×D73—81no—
291C2×D73—81no—
292C2×D73—81no—
293C2×D73—81no—
294C4×C73—24no—
295C4×C73—22no—
296C4×C73—22no—
297C4×C73—41no—
298C4×C73—42no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73—41no—
302C4×C73—81no—
303C4×C73—81no—
304C4×C73C2×C22⋊Q811yescharacteristic
305C4×C73#53911yescharacteristic
306C4×C73#54711yescharacteristic
307C4×C73#55411yescharacteristic
308Dic73—42no—
309Dic73—81no—
310Dic73—81no—
311#26377—731nomaximal, Sylow
312C2×C73#244911yescharacteristic
313C2×C73#310611yescharacteristic
314C2×C73#312111yescharacteristic
315C2×C73#326511yescharacteristic
316C2×C73#328511yescharacteristic
317C2×C73#338111yescharacteristic
318C2×C73#338611yescharacteristic
319D73—82no—
320C2×#477—731no—
321C2×C4×SD16—731no—
322#2452—731no—
323#2351D7311yescharacteristic, complemented
324#2092—731no—
325#2085—731no—
326#2067—731no—
327#2058—731no—
328#1942—731no—
329#1916—731no—
330#1880—731no—
331#1880—731no—
332#1865—731no—
333#1863—731no—
334#1862—731no—
335C73#2637711yescharacteristic, complemented, Sylow
336C22×SD16—731no—
337C2×C4⋊Q8—731no—
338C2×C22⋊Q8—1461no—
339C2×C4×Q8C2×D7311yescharacteristic
340C2×C4×Q8C2×D7311yescharacteristic
341C2×C4×Q8C2×D7311yescharacteristic
342C2×C4×D4—731no—
343#477—1464no—
344C4×SD16—1464no—
345C2×#107—731no—
346C2×#107—731no—
347C2×#107—731no—
348C2×#106—731no—
349C2×#106—731no—
350C2×C4⋊C8—731no—
351C2×#103—731no—
352C2×#103—731no—
353C2×#103—731no—
354C2×#102—731no—
355C2×#102—731no—
356C2×#102—731no—
357C2×C4×C8—731no—
358#396—1461no—
359#393—1461no—
360#390—731no—
361#390—731no—
362#389—21no—
363#389—21no—
364#389—731no—
365#389—731no—
366#389C2×D7311yescharacteristic
367#389C2×D7311yescharacteristic
368#385—1461no—
369#382—21no—
370#382C2×D7311yescharacteristic
371C4×C4⋊C4—21no—
372C4×C4⋊C4C2×D7311yescharacteristic
373#340—731no—
374#340—731no—
375#340—731no—
376#340—731no—
377#336—731no—
378#336—731no—
379#336—731no—
380#336—731no—
381C22×Q8—1461no—
382C22×Q8C22×D7311yescharacteristic
383C22×D4—731no—
384C23×C4—1461no—
385C2×SD16—734no—
386C2×SD16—1462no—
387C2×SD16—1462no—
388C22×C8—731no—
389C22×C8—731no—
390C22×C8—731no—
391C22×C8—731no—
392C4⋊Q8—2921no—
393C4⋊Q8—1462no—
394C22⋊Q8—2924no—
395C4×Q8—22no—
396C4×Q8—22no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8—41no—
400C4×Q8—41no—
401C4×D4—2921no—
402C4×D4—1462no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—1461no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4C22⋊D7311yescharacteristic
415C2×C4⋊C4C22⋊D7311yescharacteristic
416C2×C4⋊C4C22×D7311yescharacteristic
417C2×C4⋊C4C22×D7311yescharacteristic
418C2×C4⋊C4C22×D7311yescharacteristic
419C2×C22⋊C4—1461no—
420C2×C22⋊C4—1461no—
421C2×C22⋊C4—1461no—
422C2×C22⋊C4—1461no—
423C2×C22⋊C4—1461no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—21no—
427C2×C42C22×D7311yescharacteristic
428C2×C42C22×D7311yescharacteristic
429C2×C42C22×D7311yescharacteristic
430#107—1462no—
431#107—1462no—
432#107—1462no—
433#106—2921no—
434#106—1462no—
435C4⋊C8—1462no—
436#103—734no—
437#103—1462no—
438#103—1462no—
439#102—2921no—
440#102—1462no—
441#102—1462no—
442C4×C8—1462no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—1461no—
448#95—1461no—
449#95C22⋊D7311yescharacteristic
450#95C22⋊D7311yescharacteristic
451C24—1461no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—2922no—
456C2×Q8—2922no—
457C2×Q8—5841no—
458C2×Q8D4×D7312yes—
459C2×Q8Q8×D7312yes—
460C2×Q8#1192402812yes—
461C2×D4—732no—
462C2×D4—1461no—
463C2×D4—1461no—
464C2×D4—1461no—
465C2×D4—1461no—
466C2×D4—1461no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—2922no—
472C22×C4—2922no—
473C22×C4—5841no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4C2×C22⋊D7311yescharacteristic
478C22×C4C2×C22⋊D7311yescharacteristic
479C22×C4C2×C22⋊D7311yescharacteristic
480C22×C4C2×C22⋊D7311yescharacteristic
481C22×C4C2×C22⋊D7311yescharacteristic
482C22×C4C2×C22⋊D7311yescharacteristic
483C22×C4C23×D7311yescharacteristic
484SD16—1468no—
485C2×C8—732no—
486C2×C8—732no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—2922no—
510C4⋊C4—2922no—
511C4⋊C4—2922no—
512C4⋊C4—732no—
513C4⋊C4—1461no—
514C4⋊C4—1461no—
515C4⋊C4—1461no—
516C4⋊C4—1461no—
517C4⋊C4—1461no—
518C4⋊C4#1192402812yes—
519C22⋊C4—2922no—
520C22⋊C4—2922no—
521C22⋊C4—2922no—
522C22⋊C4—2922no—
523C22⋊C4—5841no—
524C42—24no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—42no—
532C42—42no—
533C23—2921no—
534C23—2921no—
535C23—2921no—
536C23—2921no—
537C23—2921no—
538C23—2921no—
539C23—1462no—
540C23C22×C22⋊D7311yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—22no—
544Q8—5844no—
545Q8#1455656914yescomplemented
546D4—2921no—
547D4—2921no—
548D4—1462no—
549D4—1462no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
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—5844no—
587C2×C4—1462no—
588C2×C4#1455661112yes—
589C2×C4#1455661612yes—
590C2×C4C2×D4×D7311yescharacteristic
591C2×C4C2×Q8×D7311yescharacteristic
592C2×C4C2×#1192402811yescharacteristic
593C2×C4C2×#1192402811yescharacteristic
594C2×C4#1455663611yescharacteristic
595C2×C4#1455663911yescharacteristic
596C8—2921no—
597C8—2921no—
598C8—1462no—
599C8—1462no—
600C22—2922no—
601C22—2922no—
602C22—2922no—
603C22—5841no—
604C22—5841no—
605C22—5841no—
606C22—5841no—
607C22#1798909811yescharacteristic
608C22D4×C22⋊D7311yescharacteristic
609C22Q8×C22⋊D7311yescharacteristic
610C22C2×#1455661111yescharacteristic
611C22C2×#1455661611yescharacteristic
612C22#1798939511yescharacteristic
613C22#1798940211yescharacteristic
614C4—22no—
615C4—24no—
616C4—22no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—2922no—
625C4—5841no—
626C4—5841no—
627C4C2×#1455656911yescharacteristic
628C4#1798918411yescharacteristic
629C4#1798923811yescharacteristic
630C4#1798929211yescharacteristic
631C2—5842no—
632C2#2261374011yescharacteristic
633C2#2261999111yescharacteristic
634C2#2262016011yescharacteristic
635C2#2262142711yescharacteristic
636C2#2262166811yescharacteristic
637C2#2262280211yescharacteristic
638C2#2262284811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#2351C2
socleC23×C73C22×D4
prime radicalC22×D292C23
hypercenter#2351D73
hypocenterC73#26377
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C2×#17988882C2semidirect
5. #22611413C2semidirect
7. #22611264C2semidirect
8. #22610661C2semidirect
9. #22610490C2semidirect
12. #22610199C2semidirect
13. #22610185C2semidirect
15. C73×#2351C2semidirect
66. C2×#11924015Q8semidirect
224. Q8×C73C22⋊Q8semidirect
323. #2351D73semidirect
335. C73#26377semidirect
545. Q8#14556569semidirect

Maximal subgroups

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

subgroupindex
C2×#179888822
C2×C4×#119240152
#226137812
#226114552
#226114132
#226113162
#226112642
#226106612
#226104902
#226102862
#226101992
#226101852
#226101772
C73×#23512
#2637773