Finite Group Databasebeta

← #46839900

Subgroups — #46839900

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×#22576134C211yesmaximal, characteristic, complemented
2C2×#22576133C211yesmaximal, characteristic, complemented
3C2×#22576129C211yesmaximal, characteristic, complemented
4C2×#22576129C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊D271C211yesmaximal, characteristic, complemented
6#31773820C211yesmaximal, characteristic, complemented
7#31773812C211yesmaximal, characteristic, complemented
8#31773812C211yesmaximal, characteristic, complemented
9#31773806C211yesmaximal, characteristic, complemented
10#31773797C211yesmaximal, characteristic, complemented
11#31773790C211yesmaximal, characteristic, complemented
12#31773776C211yesmaximal, characteristic, complemented
13#31773773C211yesmaximal, characteristic, complemented
14#31773770C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C271C211yesmaximal, characteristic, complemented
16C22×D1084C2211yescharacteristic, complemented
17C22×C22⋊D271C2211yescharacteristic
18C22×C22⋊D271C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C22⋊D271C2211yescharacteristic
21C22×C22⋊D271C2211yescharacteristic
22C22×C22⋊D271C2211yescharacteristic
23C22×C4×D271C2211yescharacteristic, complemented
24#22576134—24no—
25#22576133—24no—
26#22576129—24no—
27#22576129—24no—
28C4×C22⋊D271—24no—
29C2×C4⋊Dic271C2211yescharacteristic, complemented
30C2×Dic271⋊C4C2211yescharacteristic, complemented
31C2×Dic271⋊C4C2211yescharacteristic, complemented
32C2×Dic271⋊C4C2211yescharacteristic, complemented
33C2×C22⋊Dic271C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×C22⋊Dic271C2211yescharacteristic
38C2×C22⋊Dic271C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D271)⋊C4C2211yescharacteristic, 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
50C23×C4×C271C2211yescharacteristic
51C23×C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271—24no—
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C22⋊C4×C271C2211yescharacteristic
55C2×C22⋊C4×C271C2211yescharacteristic
56C2×C22⋊C4×C271C2211yescharacteristic
57C23×D271C2311yescharacteristic
58C23×D271C2311yescharacteristic
59C2×D1084—22no—
60C2×D1084—22no—
61C2×D1084—41no—
62C2×C22⋊D271—22no—
63C2×C22⋊D271—22no—
64C2×C22⋊D271—22no—
65C2×C22⋊D271—22no—
66C2×C22⋊D271—22no—
67C2×C22⋊D271—22no—
68C2×C22⋊D271—22no—
69C2×C22⋊D271—22no—
70C2×C22⋊D271—22no—
71C2×C22⋊D271—22no—
72C2×C22⋊D271—22no—
73C2×C22⋊D271—22no—
74C2×C22⋊D271—22no—
75C2×C22⋊D271—41no—
76C2×C22⋊D271—41no—
77C2×C22⋊D271—41no—
78C2×C22⋊D271—41no—
79C2×C22⋊D271D414yescomplemented
80C22×Dic271C2311yescharacteristic
81C22×Dic271C2311yescharacteristic
82C22×Dic271C2311yescharacteristic
83C22×Dic271C2311yescharacteristic
84C22×Dic271C2311yescharacteristic
85C22×Dic271C2311yescharacteristic
86C2×C4×D271—22no—
87C2×C4×D271—22no—
88C2×C4×D271—41no—
89C4⋊Dic271—22no—
90Dic271⋊C4—22no—
91Dic271⋊C4—22no—
92Dic271⋊C4—41no—
93C22⋊Dic271—22no—
94C22⋊Dic271—22no—
95C22⋊Dic271—22no—
96C22⋊Dic271—22no—
97C22⋊Dic271—22no—
98C22⋊Dic271D414yescomplemented
99(C2×D271)⋊C4—22no—
100(C2×D271)⋊C4—22no—
101(C2×D271)⋊C4—22no—
102(C2×D271)⋊C4—22no—
103(C2×D271)⋊C4—41no—
104C4×Dic271—22no—
105C24×C271—22no—
106C24×C271—22no—
107C24×C271—22no—
108C24×C271—22no—
109C24×C271—22no—
110C24×C271—41no—
111C24×C271—41no—
112C24×C271C2311yescharacteristic
113C24×C271C2311yescharacteristic
114C24×C271C2311yescharacteristic
115C22×C4×C271—22no—
116C22×C4×C271—22no—
117C22×C4×C271—22no—
118C22×C4×C271—22no—
119C22×C4×C271—41no—
120C22×C4×C271—41no—
121C22×C4×C271C2311yescharacteristic
122C22×C4×C271C2311yescharacteristic
123C22×C4×C271C2311yescharacteristic
124C22×C4×C271C2311yescharacteristic
125C22⋊C4×C271—22no—
126C22⋊C4×C271—22no—
127C22⋊C4×C271—22no—
128C22⋊C4×C271D414yescomplemented
129C22×D271—21no—
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×D271C2×D412yes—
143D1084—44no—
144C22⋊D271—28no—
145C22⋊D271—44no—
146C22⋊D271—44no—
147C22⋊D271—44no—
148C22⋊D271—44no—
149C22⋊D271—44no—
150C2×Dic271—21no—
151C2×Dic271—21no—
152C2×Dic271—21no—
153C2×Dic271—21no—
154C2×Dic271—21no—
155C2×Dic271—21no—
156C2×Dic271—21no—
157C2×Dic271—21no—
158C2×Dic271—21no—
159C2×Dic271—21no—
160C2×Dic271—21no—
161C2×Dic271—21no—
162C2×Dic271—21no—
163C2×Dic271—21no—
164C2×Dic271—21no—
165C2×Dic271C2×D412yes—
166C2×Dic271C2×D412yes—
167C2×Dic271C2×D412yes—
168C4×D271—44no—
169C23×C271—21no—
170C23×C271—21no—
171C23×C271—21no—
172C23×C271—21no—
173C23×C271—28no—
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—44no—
187C23×C271—44no—
188C23×C271—44no—
189C23×C271—44no—
190C23×C271—44no—
191C23×C271—44no—
192C23×C271C2×D412yes—
193C23×C271C2×D412yes—
194C23×C271C2×D412yes—
195C23×C271#4112yes—
196C23×C271#4112yes—
197C23×C271C2411yescharacteristic
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—44no—
209C2×C4×C271—44no—
210C2×C4×C271C2×D412yes—
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—416no—
225C22×C271—41no—
226C22×C271—41no—
227C22×C271—41no—
228C22×C271—41no—
229C22×C271—41no—
230C22×C271—41no—
231C22×C271—41no—
232C22×C271—41no—
233C22×C271#12114yescomplemented
234C22×C271C22×D411yescharacteristic
235C22×C271C22×D411yescharacteristic
236C22×C271C22×D411yescharacteristic
237C22×C271C2×#4111yescharacteristic
238C22×C271C2×#4111yescharacteristic
239C22×C271H(5,𝔽2)11yescharacteristic
240C22×C271H(5,𝔽2)11yescharacteristic
241C2×D271—22no—
242C2×D271—22no—
243C2×D271—22no—
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—22no—
257C4×C271—41no—
258C4×C271—41no—
259Dic271—22no—
260Dic271—22no—
261Dic271—22no—
262Dic271—41no—
263Dic271—41no—
264Dic271—41no—
265C2×C271—24no—
266C2×C271—42no—
267C2×C271—42no—
268C2×C271C2×#12111yescharacteristic
269C2×C271#52511yescharacteristic
270C2×C271#53811yescharacteristic
271C2×C271D4211yescharacteristic
272C2×C271D4211yescharacteristic
273C2×C271#54611yescharacteristic
274C2×C271#54611yescharacteristic
275D271—42no—
276D271—81no—
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⋊C4D27111yescharacteristic, complemented
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×D4—2711no—
300C22×D4—2711no—
301C22×D4—2711no—
302C23×C4—2711no—
303C23×C4C2×D27111yescharacteristic
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—24no—
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⋊C4C2×D27111yescharacteristic
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—
343C24—2711no—
344C24C22×D27111yescharacteristic
345C24C22×D27111yescharacteristic
346C24C22×D27111yescharacteristic
347C2×D4—2714no—
348C2×D4—5422no—
349C2×D4—5422no—
350C2×D4—5422no—
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—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—2711no—
375C22×C4—2711no—
376C22×C4—2711no—
377C22×C4—2711no—
378C22×C4—2711no—
379C22×C4—2711no—
380C22×C4—5422no—
381C22×C4—5422no—
382C22×C4—10841no—
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—22no—
394C22⋊C4—2714no—
395C22⋊C4—5422no—
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—10841no—
405C22⋊C4C22⋊D27114yescomplemented
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—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—2712no—
431C23—5421no—
432C23—5421no—
433C23—5421no—
434C23—5421no—
435C23—5421no—
436C23—5421no—
437C23—5421no—
438C23—5421no—
439C23—5421no—
440C23—5421no—
441C23—5421no—
442C23—5421no—
443C23—5421no—
444C23D4×D27112yes—
445C23D4×D27112yes—
446C23C2×C22⋊D27112yes—
447C23#1686465412yes—
448C23#1686465412yes—
449C23C23×D27111yescharacteristic
450D4—5428no—
451D4—10844no—
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—44no—
468C2×C4—44no—
469C2×C4—2712no—
470C2×C4—2712no—
471C2×C4—2712no—
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—5421no—
487C2×C4—10844no—
488C2×C4C2×C22⋊D27112yes—
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—416no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—5422no—
512C22—5422no—
513C22—5422no—
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
528C22C22×C22⋊D27111yescharacteristic
529C22C2×#1686465411yescharacteristic
530C22C2×#1686465411yescharacteristic
531C22#2257620511yescharacteristic
532C22#2257620511yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—5422no—
538C4—5422no—
539C4—5422no—
540C4—10841no—
541C4—10841no—
542C4—10841no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—10842no—
547C2—21681no—
548C2C2×#2257613311yescharacteristic
549C2#3177440811yescharacteristic
550C2#3177448511yescharacteristic
551C2D4×C22⋊D27111yescharacteristic
552C2D4×C22⋊D27111yescharacteristic
553C2#3177453711yescharacteristic
554C2#3177453711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC22×C22⋊C4×C271C2
hypercenterC22×C22⋊C4D271
hypocenterC271#2424
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#22576134C2semidirect
2. C2×#22576133C2semidirect
3. C2×#22576129C2semidirect
4. C2×#22576129C2semidirect
5. C2×C4×C22⋊D271C2semidirect
6. #31773820C2semidirect
7. #31773812C2semidirect
8. #31773812C2semidirect
9. #31773806C2semidirect
10. #31773797C2semidirect
11. #31773790C2semidirect
12. #31773776C2semidirect
13. #31773773C2semidirect
14. #31773770C2semidirect
15. C22×C22⋊C4×C271C2semidirect
16. C22×D1084C22semidirect
23. C22×C4×D271C22semidirect
29. C2×C4⋊Dic271C22semidirect
30. C2×Dic271⋊C4C22semidirect
31. C2×Dic271⋊C4C22semidirect
32. C2×Dic271⋊C4C22semidirect
39. C2×(C2×D271)⋊C4C22semidirect
40. C2×(C2×D271)⋊C4C22semidirect
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
79. C2×C22⋊D271D4semidirect
98. C22⋊Dic271D4semidirect
128. C22⋊C4×C271D4semidirect
233. C22×C271#121semidirect
277. C271#2424semidirect
284. C22×C22⋊C4D271semidirect
405. C22⋊C4C22⋊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×#225761342
C2×#225761332
C2×#225761292
C2×C4×C22⋊D2712
#317738202
#317738122
#317738062
#317737972
#317737902
#317737762
#317737732
#317737702
C22×C22⋊C4×C2712
#2424271