Finite Group Databasebeta

← #46839905

Subgroups — #46839905

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×#22576135C211yesmaximal, characteristic, complemented
2C2×#22576133C211yesmaximal, characteristic, complemented
3C2×#22576130C211yesmaximal, characteristic, complemented
4C2×#22576130C211yesmaximal, characteristic, complemented
5C2×C4×D1084C211yesmaximal, characteristic, complemented
6C2×C22⋊C4×D271C211yesmaximal, characteristic, complemented
7#31773819C211yesmaximal, characteristic, complemented
8#31773813C211yesmaximal, characteristic, complemented
9#31773813C211yesmaximal, characteristic, complemented
10#31773807C211yesmaximal, characteristic, complemented
11#31773799C211yesmaximal, characteristic, complemented
12#31773775C211yesmaximal, characteristic, complemented
13#31773775C211yesmaximal, characteristic, complemented
14#31773771C211yesmaximal, characteristic, complemented
15#388×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C22×D1084C2211yescharacteristic
19C22×D1084C2211yescharacteristic
20C22×D1084C2211yescharacteristic
21C22×D1084C2211yescharacteristic, complemented
22C22×C22⋊D271C2211yescharacteristic
23C22×C22⋊D271C2211yescharacteristic
24C22×C4×D271C2211yescharacteristic
25C22×C4×D271C2211yescharacteristic
26C22×C4×D271C2211yescharacteristic, complemented
27#22576135—24no—
28#22576133—24no—
29#22576130—24no—
30#22576130—24no—
31C4×D1084—24no—
32C2×C4⋊Dic271C2211yescharacteristic, complemented
33C2×C4⋊Dic271C2211yescharacteristic, complemented
34C2×Dic271⋊C4C2211yescharacteristic, complemented
35C2×C22⋊Dic271C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic
38C2×(C2×D271)⋊C4C2211yescharacteristic
39C2×(C2×D271)⋊C4C2211yescharacteristic
40C2×(C2×D271)⋊C4C2211yescharacteristic
41C2×(C2×D271)⋊C4C2211yescharacteristic
42C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
47C22⋊C4×D271—24no—
48C271⋊#95C2211yescharacteristic, complemented
49C271⋊#95C2211yescharacteristic, complemented
50C2×C4⋊C4×C271C2211yescharacteristic, complemented
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54C2×C42×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
67C23×D271C2311yescharacteristic
68C2×D1084—22no—
69C2×D1084—22no—
70C2×D1084—22no—
71C2×D1084—22no—
72C2×D1084—22no—
73C2×D1084—22no—
74C2×D1084—22no—
75C2×D1084—22no—
76C2×D1084—22no—
77C2×D1084—22no—
78C2×D1084—22no—
79C2×D1084—41no—
80C2×D1084—41no—
81C2×D1084—41no—
82C2×D1084D414yescomplemented
83C2×C22⋊D271—22no—
84C2×C22⋊D271—22no—
85C2×C22⋊D271—22no—
86C2×C22⋊D271—22no—
87C2×C22⋊D271—41no—
88C2×C22⋊D271—41no—
89C22×Dic271C2311yescharacteristic
90C22×Dic271C2311yescharacteristic
91C22×Dic271C2311yescharacteristic
92C22×Dic271C2311yescharacteristic
93C2×C4×D271—22no—
94C2×C4×D271—22no—
95C2×C4×D271—22no—
96C2×C4×D271—22no—
97C2×C4×D271—22no—
98C2×C4×D271—22no—
99C2×C4×D271—41no—
100C2×C4×D271—41no—
101C2×C4×D271—41no—
102C4⋊Dic271—22no—
103C4⋊Dic271—41no—
104Dic271⋊C4—22no—
105C22⋊Dic271—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)⋊C4—22no—
112(C2×D271)⋊C4—22no—
113(C2×D271)⋊C4—22no—
114(C2×D271)⋊C4—22no—
115(C2×D271)⋊C4—41no—
116(C2×D271)⋊C4D414yescomplemented
117C24×C271C2311yescharacteristic
118C22×C4×C271C2311yescharacteristic
119C22×C4×C271C2311yescharacteristic
120C22×C4×C271C2311yescharacteristic
121C22×C4×C271C2311yescharacteristic
122C22×C4×C271C2311yescharacteristic
123C22×C4×C271C2311yescharacteristic
124C4⋊C4×C271—22no—
125C22⋊C4×C271—22no—
126C22⋊C4×C271—22no—
127C22⋊C4×C271D414yescomplemented
128C42×C271—22no—
129C22×D271—21no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—21no—
133C22×D271—28no—
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—21no—
149C22×D271—21no—
150C22×D271—21no—
151C22×D271—21no—
152C22×D271—21no—
153C22×D271—44no—
154C22×D271—44no—
155C22×D271—44no—
156C22×D271—44no—
157C22×D271—44no—
158C22×D271—44no—
159C22×D271C2×D412yes—
160C22×D271C2×D412yes—
161C22×D271C2×D412yes—
162C22×D271#4112yes—
163C22×D271#4112yes—
164D1084—28no—
165D1084—44no—
166D1084—44no—
167D1084—44no—
168D1084—44no—
169C22⋊D271—44no—
170C22⋊D271—44no—
171C2×Dic271—21no—
172C2×Dic271—21no—
173C2×Dic271—21no—
174C2×Dic271—21no—
175C2×Dic271—21no—
176C2×Dic271—21no—
177C2×Dic271—21no—
178C2×Dic271—21no—
179C2×Dic271—21no—
180C2×Dic271—21no—
181C2×Dic271—21no—
182C2×Dic271C2×D412yes—
183C4×D271—44no—
184C4×D271—44no—
185C4×D271—44no—
186C23×C271—21no—
187C23×C271—21no—
188C23×C271—21no—
189C23×C271—21no—
190C23×C271—21no—
191C23×C271—21no—
192C23×C271C2×D412yes—
193C23×C271C2411yescharacteristic
194C2×C4×C271—21no—
195C2×C4×C271—21no—
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×C271C2×D412yes—
209C2×C4×C271C2×D412yes—
210C2×C4×C271C2×D412yes—
211C2×C4×C271C2×D412yes—
212C22×C271—22no—
213C22×C271—22no—
214C22×C271—22no—
215C22×C271—41no—
216C22×C271—41no—
217C22×C271—41no—
218C22×C271—41no—
219C22×C271C22×D411yescharacteristic
220C22×C271C22×D411yescharacteristic
221C22×C271C22×D411yescharacteristic
222C22×C271C2×#4111yescharacteristic
223C22×C271C2×#4111yescharacteristic
224C22×C271H(5,𝔽2)11yescharacteristic
225C22×C271H(5,𝔽2)11yescharacteristic
226C2×D271—22no—
227C2×D271—22no—
228C2×D271—22no—
229C2×D271—22no—
230C2×D271—22no—
231C2×D271—22no—
232C2×D271—22no—
233C2×D271—22no—
234C2×D271—22no—
235C2×D271—22no—
236C2×D271—22no—
237C2×D271—22no—
238C2×D271—416no—
239C2×D271—41no—
240C2×D271—41no—
241C2×D271—41no—
242C2×D271—41no—
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#12114yescomplemented
255C4×C271—22no—
256C4×C271—22no—
257C4×C271—22no—
258C4×C271—22no—
259C4×C271—41no—
260C4×C271—41no—
261Dic271—22no—
262Dic271—41no—
263Dic271—41no—
264Dic271—41no—
265C2×C271—42no—
266C2×C271C2×#12111yescharacteristic
267C2×C271#52511yescharacteristic
268C2×C271#53811yescharacteristic
269C2×C271D4211yescharacteristic
270C2×C271D4211yescharacteristic
271C2×C271#54611yescharacteristic
272C2×C271#54611yescharacteristic
273D271—24no—
274D271—42no—
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⋊C4—2711no—
285#396—2711no—
286#394—2711no—
287#394—2711no—
288#392—2711no—
289#390—2711no—
290#388D27111yescharacteristic, complemented
291#386—2711no—
292#386—2711no—
293#385—2711no—
294C25—2711no—
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×C4—2711no—
304C23×C4—2711no—
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⋊C4C2×D27111yescharacteristic, complemented
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×C42C2×D27111yescharacteristic, complemented
331#95—2711no—
332#95—2711no—
333#95C2×D27111yescharacteristic, complemented
334#95C2×D27111yescharacteristic, complemented
335C24—2711no—
336C24—2711no—
337C24—2711no—
338C24—2711no—
339C24—5422no—
340C24—5422no—
341C24—5422no—
342C24—5422no—
343C24—5422no—
344C24—10841no—
345C24—10841no—
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—2711no—
369C22×C4—2711no—
370C22×C4—2711no—
371C22×C4—2711no—
372C22×C4—5422no—
373C22×C4—5422no—
374C22×C4—5422no—
375C22×C4—5422no—
376C22×C4—5422no—
377C22×C4—5422no—
378C22×C4—10841no—
379C22×C4—10841no—
380C22×C4—10841no—
381C22×C4C22×D27111yescharacteristic
382C22×C4C22×D27111yescharacteristic
383C22×C4C22×D27111yescharacteristic
384C22×C4C22×D27111yescharacteristic
385C22×C4C22×D27111yescharacteristic
386C22×C4C22×D27111yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—5422no—
389C4⋊C4—5422no—
390C4⋊C4—10841no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—2714no—
394C22⋊C4—5422no—
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⋊C4D108414yescomplemented
406C42—22no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—2712no—
414C23—2712no—
415C23—2712no—
416C23—2712no—
417C23—2712no—
418C23—5428no—
419C23—5421no—
420C23—5421no—
421C23—5421no—
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—5421no—
441C23—5421no—
442C23—10844no—
443C23—10844no—
444C23—10844no—
445C23—10844no—
446C23—10844no—
447C23—10844no—
448C23C2×D108412yes—
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—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
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—10844no—
484C2×C4—10844no—
485C2×C4—10844no—
486C2×C4D4×D27112yes—
487C2×C4D4×D27112yes—
488C2×C4C2×D108412yes—
489C2×C4C2×D108412yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—2714no—
498C22—5422no—
499C22—5422no—
500C22—5422no—
501C22—5422no—
502C22—5422no—
503C22—5422no—
504C22—5422no—
505C22—5422no—
506C22—5422no—
507C22—5422no—
508C22—5422no—
509C22—5422no—
510C22—108416no—
511C22—10841no—
512C22—10841no—
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—10841no—
526C22C2×D4×D27111yescharacteristic
527C22C2×D4×D27111yescharacteristic
528C22C22×D108411yescharacteristic
529C22#41×D27111yescharacteristic
530C22#41×D27111yescharacteristic
531C22#2257620611yescharacteristic
532C22#2257620611yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—41no—
539C4—5422no—
540C4—10841no—
541C4—10841no—
542C4—10841no—
543C2—42no—
544C2—5424no—
545C2—10842no—
546C2—10842no—
547C2—21681no—
548C2#121×D27111yescharacteristic
549C2#3177440911yescharacteristic
550C2#3177448611yescharacteristic
551C2D4×D108411yescharacteristic
552C2D4×D108411yescharacteristic
553C2#3177453911yescharacteristic
554C2#3177454011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#388×C271C2
hypercenter#388D271
hypocenterC271#2424
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#22576135C2semidirect
2. C2×#22576133C2semidirect
3. C2×#22576130C2semidirect
4. C2×#22576130C2semidirect
5. C2×C4×D1084C2semidirect
6. C2×C22⋊C4×D271C2semidirect
7. #31773819C2semidirect
8. #31773813C2semidirect
9. #31773813C2semidirect
10. #31773807C2semidirect
11. #31773799C2semidirect
12. #31773775C2semidirect
13. #31773775C2semidirect
14. #31773771C2semidirect
15. #388×C271C2semidirect
21. C22×D1084C22semidirect
26. C22×C4×D271C22semidirect
32. C2×C4⋊Dic271C22semidirect
33. C2×C4⋊Dic271C22semidirect
34. C2×Dic271⋊C4C22semidirect
42. C2×(C2×D271)⋊C4C22semidirect
43. C2×(C2×D271)⋊C4C22semidirect
44. C2×(C2×D271)⋊C4C22semidirect
45. C2×(C2×D271)⋊C4C22semidirect
46. C2×(C2×D271)⋊C4C22semidirect
48. C271⋊#95C22semidirect
49. C271⋊#95C22semidirect
50. C2×C4⋊C4×C271C22semidirect
54. C2×C42×C271C22semidirect
55. #95×C271C22semidirect
56. #95×C271C22semidirect
82. C2×D1084D4semidirect
116. (C2×D271)⋊C4D4semidirect
127. C22⋊C4×C271D4semidirect
254. C2×D271#121semidirect
277. C271#2424semidirect
290. #388D271semidirect
313. C2×C4⋊C4C2×D271semidirect
330. C2×C42C2×D271semidirect
333. #95C2×D271semidirect
334. #95C2×D271semidirect
405. C22⋊C4D1084semidirect

Maximal subgroups

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

subgroupindex
C2×#225761352
C2×#225761332
C2×#225761302
C2×C4×D10842
C2×C22⋊C4×D2712
#317738192
#317738132
#317738072
#317737992
#317737752
#317737712
#388×C2712
#2424271