Finite Group Databasebeta

← #46839904

Subgroups — #46839904

Order 34688.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576132C211yesmaximal, characteristic, complemented
2C2×#22576132C211yesmaximal, characteristic, complemented
3C2×#22576129C211yesmaximal, characteristic, complemented
4C2×D4×Dic271C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic271C211yesmaximal, characteristic, complemented
6#31773820C211yesmaximal, characteristic, complemented
7#31773813C211yesmaximal, characteristic, complemented
8#31773811C211yesmaximal, characteristic, complemented
9#31773805C211yesmaximal, characteristic, complemented
10#31773797C211yesmaximal, characteristic, complemented
11#31773791C211yesmaximal, characteristic, complemented
12#31773777C211yesmaximal, characteristic, complemented
13#31773774C211yesmaximal, characteristic, complemented
14#31773769C211yesmaximal, characteristic, complemented
15C2×#120×C271C211yesmaximal, characteristic, complemented
16C22×C22⋊D271C2211yescharacteristic
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic, complemented
20C23×Dic271C2211yescharacteristic
21C23×Dic271C2211yescharacteristic
22C23×Dic271C2211yescharacteristic, complemented
23#22576132—24no—
24#22576132—24no—
25#22576129—24no—
26D4×Dic271—24no—
27C2×C4⋊Dic271C2211yescharacteristic, complemented
28C2×Dic271⋊C4C2211yescharacteristic, complemented
29C2×Dic271⋊C4C2211yescharacteristic, complemented
30C2×Dic271⋊C4C2211yescharacteristic, complemented
31C2×C22⋊Dic271—24no—
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C22⋊Dic271C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×C22⋊Dic271C2211yescharacteristic
38C2×C22⋊Dic271C2211yescharacteristic
39C2×C22⋊Dic271C2211yescharacteristic, complemented
40C2×C22⋊Dic271C2211yescharacteristic, complemented
41C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
44C2×C4×Dic271C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C271⋊#95C2211yescharacteristic, complemented
48C271⋊#95C2211yescharacteristic, complemented
49C25×C271C2211yescharacteristic
50C22×D4×C271C2211yescharacteristic
51C22×D4×C271C2211yescharacteristic
52C22×D4×C271C2211yescharacteristic
53#120×C271—24no—
54C2×C22⋊C4×C271C2211yescharacteristic
55C2×C22⋊C4×C271C2211yescharacteristic
56C2×C22⋊C4×C271C2211yescharacteristic
57C23×D271C2311yescharacteristic
58C2×C22⋊D271—22no—
59C2×C22⋊D271—22no—
60C2×C22⋊D271—22no—
61C2×C22⋊D271—22no—
62C2×C22⋊D271—22no—
63C2×C22⋊D271—22no—
64C2×C22⋊D271—22no—
65C2×C22⋊D271—22no—
66C2×C22⋊D271—22no—
67C2×C22⋊D271—41no—
68C2×C22⋊D271—41no—
69C2×C22⋊D271—41no—
70C22×Dic271—22no—
71C22×Dic271—22no—
72C22×Dic271—22no—
73C22×Dic271—22no—
74C22×Dic271—22no—
75C22×Dic271—22no—
76C22×Dic271—41no—
77C22×Dic271—41no—
78C22×Dic271—41no—
79C22×Dic271C2311yescharacteristic
80C22×Dic271C2311yescharacteristic
81C22×Dic271C2311yescharacteristic
82C22×Dic271C2311yescharacteristic
83C22×Dic271C2311yescharacteristic
84C22×Dic271C2311yescharacteristic
85C22×Dic271C2311yescharacteristic
86C4⋊Dic271—22no—
87Dic271⋊C4—22no—
88Dic271⋊C4—22no—
89Dic271⋊C4—41no—
90C22⋊Dic271—22no—
91C22⋊Dic271—22no—
92C22⋊Dic271—22no—
93C22⋊Dic271—22no—
94C22⋊Dic271—22no—
95C22⋊Dic271—22no—
96C22⋊Dic271—22no—
97C22⋊Dic271D414yescomplemented
98C22⋊Dic271D414yescomplemented
99(C2×D271)⋊C4—22no—
100(C2×D271)⋊C4—22no—
101(C2×D271)⋊C4—41no—
102C4×Dic271—22no—
103C24×C271—22no—
104C24×C271—22no—
105C24×C271—22no—
106C24×C271—22no—
107C24×C271—22no—
108C24×C271—41no—
109C24×C271—41no—
110C24×C271C2311yescharacteristic
111C24×C271C2311yescharacteristic
112C24×C271C2311yescharacteristic
113C24×C271C2311yescharacteristic
114C2×D4×C271—22no—
115C2×D4×C271—22no—
116C2×D4×C271—22no—
117C2×D4×C271—22no—
118C2×D4×C271—22no—
119C2×D4×C271—22no—
120C2×D4×C271—41no—
121C2×D4×C271—41no—
122C2×D4×C271D414yescomplemented
123C22×C4×C271C2311yescharacteristic
124C22×C4×C271C2311yescharacteristic
125C22×C4×C271C2311yescharacteristic
126C22⋊C4×C271—22no—
127C22⋊C4×C271—22no—
128C22⋊C4×C271—22no—
129C22×D271—21no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—21no—
133C22×D271—21no—
134C22×D271—21no—
135C22×D271—21no—
136C22⋊D271—44no—
137C22⋊D271—44no—
138C22⋊D271—44no—
139C22⋊D271—44no—
140C2×Dic271—21no—
141C2×Dic271—21no—
142C2×Dic271—21no—
143C2×Dic271—21no—
144C2×Dic271—21no—
145C2×Dic271—21no—
146C2×Dic271—21no—
147C2×Dic271—21no—
148C2×Dic271—21no—
149C2×Dic271—21no—
150C2×Dic271—21no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—21no—
154C2×Dic271—21no—
155C2×Dic271—21no—
156C2×Dic271—21no—
157C2×Dic271—44no—
158C2×Dic271—44no—
159C2×Dic271—44no—
160C2×Dic271C2×D412yes—
161C2×Dic271C2×D412yes—
162C2×Dic271C2×D412yes—
163C2×Dic271C2×D412yes—
164C23×C271—21no—
165C23×C271—21no—
166C23×C271—21no—
167C23×C271—21no—
168C23×C271—28no—
169C23×C271—21no—
170C23×C271—21no—
171C23×C271—21no—
172C23×C271—21no—
173C23×C271—21no—
174C23×C271—21no—
175C23×C271—21no—
176C23×C271—21no—
177C23×C271—21no—
178C23×C271—21no—
179C23×C271—21no—
180C23×C271—21no—
181C23×C271—21no—
182C23×C271—21no—
183C23×C271—21no—
184C23×C271—21no—
185C23×C271—21no—
186C23×C271—21no—
187C23×C271—44no—
188C23×C271—44no—
189C23×C271—44no—
190C23×C271—44no—
191C23×C271—44no—
192C23×C271—44no—
193C23×C271C2×D412yes—
194C23×C271C2×D412yes—
195C23×C271C2×D412yes—
196C23×C271C2×D412yes—
197C23×C271#4112yes—
198C23×C271#4112yes—
199C23×C271C2411yescharacteristic
200D4×C271—28no—
201D4×C271—44no—
202D4×C271—44no—
203C2×C4×C271—21no—
204C2×C4×C271—21no—
205C2×C4×C271—21no—
206C2×C4×C271—21no—
207C2×C4×C271—21no—
208C2×C4×C271—21no—
209C2×C4×C271—21no—
210C2×C4×C271—21no—
211C2×C4×C271C2×D412yes—
212C22×C271—22no—
213C22×C271—22no—
214C22×C271—22no—
215C22×C271—22no—
216C22×C271—22no—
217C22×C271—22no—
218C22×C271—22no—
219C22×C271—22no—
220C22×C271—22no—
221C22×C271—22no—
222C22×C271—22no—
223C22×C271—22no—
224C22×C271—22no—
225C22×C271—22no—
226C22×C271—22no—
227C22×C271—416no—
228C22×C271—41no—
229C22×C271—41no—
230C22×C271—41no—
231C22×C271—41no—
232C22×C271—41no—
233C22×C271—41no—
234C22×C271—41no—
235C22×C271—41no—
236C22×C271—41no—
237C22×C271—41no—
238C22×C271—41no—
239C22×C271—41no—
240C22×C271#12114yescomplemented
241C22×C271C22×D411yescharacteristic
242C22×C271C22×D411yescharacteristic
243C22×C271C22×D411yescharacteristic
244C22×C271C2×#4111yescharacteristic
245C22×C271C2×#4111yescharacteristic
246C22×C271H(5,𝔽2)11yescharacteristic
247C22×C271H(5,𝔽2)11yescharacteristic
248C2×D271—41no—
249C2×D271—41no—
250C2×D271—41no—
251C2×D271—41no—
252C2×D271—41no—
253C2×D271—41no—
254C2×D271—41no—
255C4×C271—22no—
256C4×C271—41no—
257C4×C271—41no—
258Dic271—22no—
259Dic271—22no—
260Dic271—22no—
261Dic271—22no—
262Dic271—41no—
263Dic271—41no—
264Dic271—41no—
265C2×C271—24no—
266C2×C271—42no—
267C2×C271—42no—
268C2×C271—42no—
269C2×C271C2×#12111yescharacteristic
270C2×C271#52511yescharacteristic
271C2×C271#53811yescharacteristic
272C2×C271D4211yescharacteristic
273C2×C271D4211yescharacteristic
274C2×C271#54611yescharacteristic
275C2×C271#54611yescharacteristic
276D271—81no—
277C271#242411yescharacteristic, complemented, Sylow
278#2424—2711nomaximal, Sylow
279C2×#121—2711no—
280C2×#121—2711no—
281C2×#120—2711no—
282C2×#120D27111yescharacteristic, complemented
283C2×C4×D4—2711no—
284C22×C22⋊C4—2711no—
285#396—2711no—
286#394—2711no—
287#394—2711no—
288#392—2711no—
289#390—2711no—
290#388—2711no—
291#386—2711no—
292#386—2711no—
293#385—2711no—
294C25C2×D27111yescharacteristic
295C22×D4—2711no—
296C22×D4—2711no—
297C22×D4—2711no—
298C22×D4—2711no—
299C22×D4C2×D27111yescharacteristic
300C22×D4C2×D27111yescharacteristic
301C22×D4C2×D27111yescharacteristic
302C23×C4—2711no—
303C23×C4—2711no—
304C23×C4—2711no—
305#121—5424no—
306#121—5424no—
307#120—24no—
308#120—5424no—
309C4×D4—5424no—
310C2×C4⋊C4—2711no—
311C2×C4⋊C4—2711no—
312C2×C4⋊C4—2711no—
313C2×C4⋊C4—2711no—
314C2×C22⋊C4—2711no—
315C2×C22⋊C4—2711no—
316C2×C22⋊C4—2711no—
317C2×C22⋊C4—2711no—
318C2×C22⋊C4—2711no—
319C2×C22⋊C4—2711no—
320C2×C22⋊C4—2711no—
321C2×C22⋊C4—2711no—
322C2×C22⋊C4—2711no—
323C2×C22⋊C4—2711no—
324C2×C22⋊C4—2711no—
325C2×C22⋊C4—2711no—
326C2×C22⋊C4—5424no—
327C2×C22⋊C4C2×D27111yescharacteristic
328C2×C22⋊C4C2×D27111yescharacteristic
329C2×C22⋊C4C2×D27111yescharacteristic
330C2×C42—2711no—
331#95—2711no—
332#95—2711no—
333#95—2711no—
334#95—2711no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—2711no—
343C24C22×D27111yescharacteristic
344C24C22×D27111yescharacteristic
345C24C22×D27111yescharacteristic
346C24C22×D27111yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—5422no—
356C2×D4—5422no—
357C2×D4—5422no—
358C2×D4—5422no—
359C2×D4—5422no—
360C2×D4—5422no—
361C2×D4—5422no—
362C2×D4—5422no—
363C2×D4—5422no—
364C2×D4—10841no—
365C2×D4—10841no—
366C2×D4—10841no—
367C2×D4C22⋊D27114yescomplemented
368C22×C4—2711no—
369C22×C4—2711no—
370C22×C4—2711no—
371C22×C4—2711no—
372C22×C4—2711no—
373C22×C4—2711no—
374C22×C4—2711no—
375C22×C4—5422no—
376C22×C4—5422no—
377C22×C4—5422no—
378C22×C4—5422no—
379C22×C4—5422no—
380C22×C4—5422no—
381C22×C4—10841no—
382C22×C4—10841no—
383C22×C4—10841no—
384C22×C4C22×D27111yescharacteristic
385C22×C4C22×D27111yescharacteristic
386C22×C4C22×D27111yescharacteristic
387C4⋊C4—5422no—
388C4⋊C4—5422no—
389C4⋊C4—5422no—
390C4⋊C4—10841no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—2714no—
395C22⋊C4—2714no—
396C22⋊C4—5422no—
397C22⋊C4—5422no—
398C22⋊C4—5422no—
399C22⋊C4—5422no—
400C22⋊C4—5422no—
401C22⋊C4—5422no—
402C22⋊C4—5422no—
403C22⋊C4—5422no—
404C22⋊C4—5422no—
405C22⋊C4—10841no—
406C42—5422no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—28no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—5421no—
437C23—5421no—
438C23—5421no—
439C23—5421no—
440C23—5421no—
441C23—5421no—
442C23—5421no—
443C23D4×D27112yes—
444C23D4×D27112yes—
445C23C2×C22⋊D27112yes—
446C23C2×C22⋊D27112yes—
447C23#1686465312yes—
448C23#1686465312yes—
449C23C23×D27111yescharacteristic
450D4—28no—
451D4—44no—
452D4—44no—
453D4—10844no—
454D4—10844no—
455D4—10844no—
456D4—10844no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—2712no—
466C2×C4—2712no—
467C2×C4—2712no—
468C2×C4—2712no—
469C2×C4—5421no—
470C2×C4—5421no—
471C2×C4—5421no—
472C2×C4—5421no—
473C2×C4—5421no—
474C2×C4—5421no—
475C2×C4—5421no—
476C2×C4—5421no—
477C2×C4—5421no—
478C2×C4—5421no—
479C2×C4—5421no—
480C2×C4—5421no—
481C2×C4—5421no—
482C2×C4—5421no—
483C2×C4—5421no—
484C2×C4—5421no—
485C2×C4—5421no—
486C2×C4—10844no—
487C2×C4—10844no—
488C2×C4—10844no—
489C2×C4C2×C22⋊D27112yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—416no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—10841no—
519C22—10841no—
520C22—10841no—
521C22—10841no—
522C22—10841no—
523C22—10841no—
524C22—10841no—
525C22#2257613214yescomplemented
526C22C2×D4×D27111yescharacteristic
527C22C2×D4×D27111yescharacteristic
528C22C22×C22⋊D27111yescharacteristic
529C22C2×#1686465311yescharacteristic
530C22C2×#1686465311yescharacteristic
531C22#2257620511yescharacteristic
532C22#2257620511yescharacteristic
533C4—22no—
534C4—41no—
535C4—41no—
536C4—5422no—
537C4—5422no—
538C4—5422no—
539C4—5422no—
540C4—10841no—
541C4—10841no—
542C4—10841no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—42no—
547C2—21681no—
548C2C2×#2257613211yescharacteristic
549C2#3177440311yescharacteristic
550C2#3177448211yescharacteristic
551C2#3177452911yescharacteristic
552C2D4×C22⋊D27111yescharacteristic
553C2#3177453411yescharacteristic
554C2#3177454111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×#120×C271C2
hypercenterC2×#120D271
hypocenterC271#2424
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#22576132C2semidirect
2. C2×#22576132C2semidirect
3. C2×#22576129C2semidirect
4. C2×D4×Dic271C2semidirect
5. C22×C22⋊Dic271C2semidirect
6. #31773820C2semidirect
7. #31773813C2semidirect
8. #31773811C2semidirect
9. #31773805C2semidirect
10. #31773797C2semidirect
11. #31773791C2semidirect
12. #31773777C2semidirect
13. #31773774C2semidirect
14. #31773769C2semidirect
15. C2×#120×C271C2semidirect
19. C22×C22⋊D271C22semidirect
22. C23×Dic271C22semidirect
27. C2×C4⋊Dic271C22semidirect
28. C2×Dic271⋊C4C22semidirect
29. C2×Dic271⋊C4C22semidirect
30. C2×Dic271⋊C4C22semidirect
39. C2×C22⋊Dic271C22semidirect
40. C2×C22⋊Dic271C22semidirect
41. C2×(C2×D271)⋊C4C22semidirect
42. C2×(C2×D271)⋊C4C22semidirect
43. C2×(C2×D271)⋊C4C22semidirect
44. C2×C4×Dic271C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
47. C271⋊#95C22semidirect
48. C271⋊#95C22semidirect
97. C22⋊Dic271D4semidirect
98. C22⋊Dic271D4semidirect
122. C2×D4×C271D4semidirect
240. C22×C271#121semidirect
277. C271#2424semidirect
282. C2×#120D271semidirect
367. C2×D4C22⋊D271semidirect
525. C22#22576132semidirect

Maximal subgroups

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

subgroupindex
C2×#225761322
C2×#225761292
C2×D4×Dic2712
C22×C22⋊Dic2712
#317738202
#317738132
#317738112
#317738052
#317737972
#317737912
#317737772
#317737742
#317737692
C2×#120×C2712
#2424271