Finite Group Databasebeta

← #46839902

Subgroups — #46839902

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×#22576133C211yesmaximal, characteristic, complemented
2C2×#22576132C211yesmaximal, characteristic, complemented
3C2×#22576130C211yesmaximal, characteristic, complemented
4C2×#22576128C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊D271C211yesmaximal, characteristic, complemented
6C22×(C2×D271)⋊C4C211yesmaximal, characteristic, complemented
7#31773821C211yesmaximal, characteristic, complemented
8#31773815C211yesmaximal, characteristic, complemented
9#31773812C211yesmaximal, characteristic, complemented
10#31773806C211yesmaximal, characteristic, complemented
11#31773797C211yesmaximal, characteristic, complemented
12#31773786C211yesmaximal, characteristic, complemented
13#31773775C211yesmaximal, characteristic, complemented
14#31773769C211yesmaximal, characteristic, complemented
15#386×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C22×D1084C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C22⋊D271C2211yescharacteristic
21C22×C22⋊D271C2211yescharacteristic
22C22×C22⋊D271C2211yescharacteristic, complemented
23C23×Dic271C2211yescharacteristic
24C22×C4×D271C2211yescharacteristic, complemented
25#22576133—24no—
26#22576132—24no—
27#22576130—24no—
28#22576128—24no—
29C4×C22⋊D271—24no—
30C2×Dic271⋊C4C2211yescharacteristic, complemented
31C2×Dic271⋊C4C2211yescharacteristic, complemented
32C2×Dic271⋊C4C2211yescharacteristic, complemented
33C2×Dic271⋊C4C2211yescharacteristic, complemented
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic, complemented
36C2×(C2×D271)⋊C4—24no—
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×(C2×D271)⋊C4C2211yescharacteristic
43C2×(C2×D271)⋊C4C2211yescharacteristic
44C2×(C2×D271)⋊C4C2211yescharacteristic
45C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
47C2×C4×Dic271C2211yescharacteristic, complemented
48C271⋊#95C2211yescharacteristic, complemented
49C271⋊#95C2211yescharacteristic, complemented
50C22×D4×C271C2211yescharacteristic
51C23×C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic, complemented
54C2×C22⋊C4×C271C2211yescharacteristic, complemented
55#95×C271C2211yescharacteristic, complemented
56#95×C271C2211yescharacteristic, complemented
57C23×D271—22no—
58C23×D271—22no—
59C23×D271—22no—
60C23×D271—22no—
61C23×D271—22no—
62C23×D271—41no—
63C23×D271—41no—
64C23×D271C2311yescharacteristic
65C23×D271C2311yescharacteristic
66C23×D271C2311yescharacteristic
67C2×D1084—22no—
68C2×D1084—22no—
69C2×D1084—22no—
70C2×D1084—22no—
71C2×D1084—41no—
72C2×D1084—41no—
73C2×C22⋊D271—22no—
74C2×C22⋊D271—22no—
75C2×C22⋊D271—22no—
76C2×C22⋊D271—22no—
77C2×C22⋊D271—22no—
78C2×C22⋊D271—22no—
79C2×C22⋊D271—22no—
80C2×C22⋊D271—22no—
81C2×C22⋊D271—41no—
82C2×C22⋊D271—41no—
83C2×C22⋊D271—41no—
84C2×C22⋊D271D414yescomplemented
85C22×Dic271—22no—
86C22×Dic271—22no—
87C22×Dic271—41no—
88C22×Dic271C2311yescharacteristic
89C22×Dic271C2311yescharacteristic
90C22×Dic271C2311yescharacteristic
91C22×Dic271C2311yescharacteristic
92C22×Dic271C2311yescharacteristic
93C2×C4×D271—22no—
94C2×C4×D271—22no—
95C2×C4×D271—41no—
96Dic271⋊C4—22no—
97Dic271⋊C4—22no—
98Dic271⋊C4—22no—
99Dic271⋊C4—41no—
100C22⋊Dic271—22no—
101C22⋊Dic271D414yescomplemented
102(C2×D271)⋊C4—22no—
103(C2×D271)⋊C4—22no—
104(C2×D271)⋊C4—22no—
105(C2×D271)⋊C4—22no—
106(C2×D271)⋊C4—22no—
107(C2×D271)⋊C4—22no—
108(C2×D271)⋊C4—22no—
109(C2×D271)⋊C4—22no—
110(C2×D271)⋊C4—22no—
111(C2×D271)⋊C4D414yescomplemented
112C4×Dic271—22no—
113C24×C271C2311yescharacteristic
114C24×C271C2311yescharacteristic
115C2×D4×C271—22no—
116C2×D4×C271—22no—
117C2×D4×C271—22no—
118C22×C4×C271—22no—
119C22×C4×C271—22no—
120C22×C4×C271—41no—
121C22×C4×C271C2311yescharacteristic
122C22×C4×C271C2311yescharacteristic
123C22×C4×C271C2311yescharacteristic
124C22×C4×C271C2311yescharacteristic
125C22×C4×C271C2311yescharacteristic
126C22⋊C4×C271—22no—
127C22⋊C4×C271—22no—
128C22⋊C4×C271—41no—
129C22×D271—28no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—21no—
133C22×D271—21no—
134C22×D271—21no—
135C22×D271—21no—
136C22×D271—21no—
137C22×D271—21no—
138C22×D271—21no—
139C22×D271—21no—
140C22×D271—21no—
141C22×D271—21no—
142C22×D271—21no—
143C22×D271—21no—
144C22×D271—21no—
145C22×D271—21no—
146C22×D271—21no—
147C22×D271—21no—
148C22×D271—44no—
149C22×D271—44no—
150C22×D271—44no—
151C22×D271—44no—
152C22×D271—44no—
153C22×D271—44no—
154C22×D271C2×D412yes—
155C22×D271C2×D412yes—
156C22×D271#4112yes—
157D1084—44no—
158D1084—44no—
159C22⋊D271—28no—
160C22⋊D271—44no—
161C22⋊D271—44no—
162C22⋊D271—44no—
163C2×Dic271—21no—
164C2×Dic271—21no—
165C2×Dic271—21no—
166C2×Dic271—21no—
167C2×Dic271—21no—
168C2×Dic271—21no—
169C2×Dic271—21no—
170C2×Dic271—21no—
171C2×Dic271—21no—
172C2×Dic271—21no—
173C2×Dic271—21no—
174C2×Dic271—44no—
175C2×Dic271C2×D412yes—
176C2×Dic271C2×D412yes—
177C2×Dic271C2×D412yes—
178C2×Dic271C2×D412yes—
179C4×D271—44no—
180C23×C271—21no—
181C23×C271—21no—
182C23×C271—21no—
183C23×C271—21no—
184C23×C271—21no—
185C23×C271—21no—
186C23×C271—21no—
187C23×C271—21no—
188C23×C271—21no—
189C23×C271—21no—
190C23×C271—21no—
191C23×C271C2×D412yes—
192C23×C271C2×D412yes—
193C23×C271#4112yes—
194C23×C271C2411yescharacteristic
195D4×C271—44no—
196C2×C4×C271—21no—
197C2×C4×C271—21no—
198C2×C4×C271—21no—
199C2×C4×C271—21no—
200C2×C4×C271—21no—
201C2×C4×C271—21no—
202C2×C4×C271—21no—
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—44no—
211C2×C4×C271C2×D412yes—
212C22×C271—22no—
213C22×C271—22no—
214C22×C271—22no—
215C22×C271—22no—
216C22×C271—22no—
217C22×C271—22no—
218C22×C271—41no—
219C22×C271—41no—
220C22×C271—41no—
221C22×C271—41no—
222C22×C271—41no—
223C22×C271—41no—
224C22×C271—41no—
225C22×C271#12114yescomplemented
226C22×C271C22×D411yescharacteristic
227C22×C271C22×D411yescharacteristic
228C22×C271C22×D411yescharacteristic
229C22×C271C2×#4111yescharacteristic
230C22×C271C2×#4111yescharacteristic
231C22×C271H(5,𝔽2)11yescharacteristic
232C22×C271H(5,𝔽2)11yescharacteristic
233C2×D271—22no—
234C2×D271—22no—
235C2×D271—22no—
236C2×D271—22no—
237C2×D271—22no—
238C2×D271—22no—
239C2×D271—22no—
240C2×D271—22no—
241C2×D271—22no—
242C2×D271—416no—
243C2×D271—41no—
244C2×D271—41no—
245C2×D271—41no—
246C2×D271—41no—
247C2×D271—41no—
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—
258C4×C271—41no—
259C4×C271—41no—
260Dic271—22no—
261Dic271—22no—
262Dic271—22no—
263Dic271—22no—
264Dic271—41no—
265C2×C271—24no—
266C2×C271—81no—
267C2×C271C2×#12111yescharacteristic
268C2×C271#52511yescharacteristic
269C2×C271#53811yescharacteristic
270C2×C271D4211yescharacteristic
271C2×C271D4211yescharacteristic
272C2×C271#54611yescharacteristic
273C2×C271#54611yescharacteristic
274D271—42no—
275D271—42no—
276D271—42no—
277C271#242411yescharacteristic, complemented, Sylow
278#2424—2711nomaximal, Sylow
279C2×#121—2711no—
280C2×#121—2711no—
281C2×#120—2711no—
282C2×#120—2711no—
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#386D27111yescharacteristic, complemented
293#385—2711no—
294C25—2711no—
295C22×D4—2711no—
296C22×D4—2711no—
297C22×D4—2711no—
298C22×D4—2711no—
299C22×D4—2711no—
300C22×D4—2711no—
301C22×D4C2×D27111yescharacteristic
302C23×C4—2711no—
303C23×C4—2711no—
304C23×C4C2×D27111yescharacteristic
305#121—5424no—
306#121—5424no—
307#120—5424no—
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, complemented
329C2×C22⋊C4C2×D27111yescharacteristic, complemented
330C2×C42—2711no—
331#95—2711no—
332#95—2711no—
333#95C2×D27111yescharacteristic, complemented
334#95C2×D27111yescharacteristic, complemented
335C24—2711no—
336C24—2711no—
337C24—2711no—
338C24—5422no—
339C24—5422no—
340C24—5422no—
341C24—5422no—
342C24—5422no—
343C24—10841no—
344C24—10841no—
345C24C22×D27111yescharacteristic
346C24C22×D27111yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—2714no—
351C2×D4—5422no—
352C2×D4—5422no—
353C2×D4—5422no—
354C2×D4—5422no—
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—10841no—
364C2×D4—10841no—
365C2×D4—10841no—
366C2×D4—10841no—
367C2×D4—10841no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—2711no—
372C22×C4—2711no—
373C22×C4—2711no—
374C22×C4—2711no—
375C22×C4—2711no—
376C22×C4—5422no—
377C22×C4—5422no—
378C22×C4—5422no—
379C22×C4—5422no—
380C22×C4—10841no—
381C22×C4—10841no—
382C22×C4C22×D27111yescharacteristic
383C22×C4C22×D27111yescharacteristic
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—41no—
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—5422no—
406C42—5422no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—2712no—
419C23—2712no—
420C23—2712no—
421C23—5428no—
422C23—5421no—
423C23—5421no—
424C23—5421no—
425C23—5421no—
426C23—5421no—
427C23—5421no—
428C23—5421no—
429C23—5421no—
430C23—5421no—
431C23—5421no—
432C23—5421no—
433C23—5421no—
434C23—5421no—
435C23—5421no—
436C23—5421no—
437C23—5421no—
438C23—5421no—
439C23—5421no—
440C23—10844no—
441C23—10844no—
442C23—10844no—
443C23—10844no—
444C23—10844no—
445C23—10844no—
446C23D4×D27112yes—
447C23D4×D27112yes—
448C23#1686465412yes—
449C23C23×D27111yescharacteristic
450D4—44no—
451D4—5428no—
452D4—10844no—
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—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—44no—
472C2×C4—2712no—
473C2×C4—2712no—
474C2×C4—2712no—
475C2×C4—2712no—
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—5421no—
487C2×C4—10844no—
488C2×C4—10844no—
489C2×C4D4×D27112yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—5422no—
504C22—5422no—
505C22—5422no—
506C22—5422no—
507C22—5422no—
508C22—5422no—
509C22—5422no—
510C22—5422no—
511C22—5422no—
512C22—108416no—
513C22—10841no—
514C22—10841no—
515C22—10841no—
516C22—10841no—
517C22—10841no—
518C22—10841no—
519C22—10841no—
520C22—10841no—
521C22—10841no—
522C22—10841no—
523C22—10841no—
524C22—10841no—
525C22#2257613314yescomplemented
526C22C2×D4×D27111yescharacteristic
527C22C2×D4×D27111yescharacteristic
528C22C2×D4×D27111yescharacteristic
529C22#41×D27111yescharacteristic
530C22C2×#1686465411yescharacteristic
531C22#2257620511yescharacteristic
532C22#2257620611yescharacteristic
533C4—22no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—5422no—
539C4—5422no—
540C4—5422no—
541C4—5422no—
542C4—10841no—
543C2—24no—
544C2—81no—
545C2—10842no—
546C2—10842no—
547C2—10842no—
548C2C2×#2257613311yescharacteristic
549C2#3177440611yescharacteristic
550C2#3177448011yescharacteristic
551C2#3177452911yescharacteristic
552C2#3177453111yescharacteristic
553C2#3177453611yescharacteristic
554C2#3177453911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#386×C271C2
hypercenter#386D271
hypocenterC271#2424
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#22576133C2semidirect
2. C2×#22576132C2semidirect
3. C2×#22576130C2semidirect
4. C2×#22576128C2semidirect
5. C2×C4×C22⋊D271C2semidirect
6. C22×(C2×D271)⋊C4C2semidirect
7. #31773821C2semidirect
8. #31773815C2semidirect
9. #31773812C2semidirect
10. #31773806C2semidirect
11. #31773797C2semidirect
12. #31773786C2semidirect
13. #31773775C2semidirect
14. #31773769C2semidirect
15. #386×C271C2semidirect
22. C22×C22⋊D271C22semidirect
24. C22×C4×D271C22semidirect
30. C2×Dic271⋊C4C22semidirect
31. C2×Dic271⋊C4C22semidirect
32. C2×Dic271⋊C4C22semidirect
33. C2×Dic271⋊C4C22semidirect
35. C2×C22⋊Dic271C22semidirect
45. C2×(C2×D271)⋊C4C22semidirect
46. C2×(C2×D271)⋊C4C22semidirect
47. C2×C4×Dic271C22semidirect
48. C271⋊#95C22semidirect
49. C271⋊#95C22semidirect
53. C2×C22⋊C4×C271C22semidirect
54. C2×C22⋊C4×C271C22semidirect
55. #95×C271C22semidirect
56. #95×C271C22semidirect
84. C2×C22⋊D271D4semidirect
101. C22⋊Dic271D4semidirect
111. (C2×D271)⋊C4D4semidirect
225. C22×C271#121semidirect
277. C271#2424semidirect
292. #386D271semidirect
328. C2×C22⋊C4C2×D271semidirect
329. C2×C22⋊C4C2×D271semidirect
333. #95C2×D271semidirect
334. #95C2×D271semidirect
525. C22#22576133semidirect

Maximal subgroups

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

subgroupindex
C2×#225761332
C2×#225761322
C2×#225761302
C2×#225761282
C2×C4×C22⋊D2712
C22×(C2×D271)⋊C42
#317738212
#317738152
#317738122
#317738062
#317737972
#317737862
#317737752
#317737692
#386×C2712
#2424271