Finite Group Databasebeta

← #46839017

Subgroups — #46839017

Order 34688.

808 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576130C211yesmaximal, characteristic
2C2×#22576122C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D271C211yesmaximal, characteristic, complemented
4#31773799C211yesmaximal, characteristic, complemented
5#31773797C211yesmaximal, characteristic, complemented
6#31773790C211yesmaximal, characteristic, complemented
7#31773787C211yesmaximal, characteristic, complemented
8#31773786C211yesmaximal, characteristic, complemented
9#31773775C211yesmaximal, characteristic, complemented
10#31773775C211yesmaximal, characteristic, complemented
11#31773773C211yesmaximal, characteristic, complemented
12#31773771C211yesmaximal, characteristic, complemented
13#31773752C211yesmaximal, characteristic, complemented
14C4×(C2×D271)⋊C4C211yesmaximal, characteristic, complemented
15#388×C271C211yesmaximal, characteristic, complemented
16C24×D271C2211yescharacteristic
17C22×D1084C2211yescharacteristic
18C22×D1084C2211yescharacteristic
19C22×C22⋊D271C2211yescharacteristic
20C22×C4×D271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22C22×C4×D271C2211yescharacteristic, complemented
23#22576130C418yescomplemented
24#22576122—24no—
25C2×C4⋊Dic271C2211yescharacteristic, complemented
26C2×Dic271⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic271C2211yescharacteristic
28C2×C22⋊Dic271C2211yescharacteristic
29C2×(C2×D271)⋊C4C2211yescharacteristic
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×(C2×D271)⋊C4C2211yescharacteristic
35C2×(C2×D271)⋊C4C2211yescharacteristic
36C2×(C2×D271)⋊C4C2211yescharacteristic
37C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
40C22⋊C4×D271—24no—
41C2×C4×Dic271C2211yescharacteristic, complemented
42C2×C4×Dic271C2211yescharacteristic, complemented
43C271⋊#95C2211yescharacteristic, complemented
44C271⋊#95C2211yescharacteristic, complemented
45C271⋊#95C2211yescharacteristic, complemented
46C271⋊#95C2211yescharacteristic, complemented
47C2×C4⋊C4×C271C2211yescharacteristic, complemented
48C2×C22⋊C4×C271C2211yescharacteristic
49C2×C22⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C42×C271C2211yescharacteristic, complemented
52#95×C271C2211yescharacteristic, complemented
53#95×C271C2211yescharacteristic, complemented
54C23×D271—22no—
55C23×D271—22no—
56C23×D271—41no—
57C23×D271—41no—
58C23×D271—41no—
59C23×D271—41no—
60C23×D271C2×C414yescomplemented
61C23×D271C2311yescharacteristic
62C23×D271C2311yescharacteristic
63C23×D271C2311yescharacteristic
64C2×D1084—22no—
65C2×D1084—22no—
66C2×D1084—41no—
67C2×D1084—41no—
68C2×D1084C2×C414yes—
69C2×D1084C2×C414yescomplemented
70C2×C22⋊D271—41no—
71C2×C22⋊D271—41no—
72C2×C22⋊D271C2×C414yescomplemented
73C22×Dic271C2311yescharacteristic
74C22×Dic271C2311yescharacteristic
75C22×Dic271C2311yescharacteristic
76C22×Dic271C2311yescharacteristic
77C22×Dic271C2311yescharacteristic
78C2×C4×D271—22no—
79C2×C4×D271—22no—
80C2×C4×D271—41no—
81C2×C4×D271—41no—
82C2×C4×D271—41no—
83C2×C4×D271—41no—
84C2×C4×D271—41no—
85C2×C4×D271—41no—
86C2×C4×D271D414yescomplemented
87C4⋊Dic271—41no—
88Dic271⋊C4—41no—
89C22⋊Dic271—22no—
90C22⋊Dic271—41no—
91(C2×D271)⋊C4—22no—
92(C2×D271)⋊C4—22no—
93(C2×D271)⋊C4—22no—
94(C2×D271)⋊C4—22no—
95(C2×D271)⋊C4—22no—
96(C2×D271)⋊C4—41no—
97(C2×D271)⋊C4—41no—
98(C2×D271)⋊C4—41no—
99(C2×D271)⋊C4—41no—
100(C2×D271)⋊C4C2×C414yescomplemented
101(C2×D271)⋊C4C2×C414yescomplemented
102C4×Dic271—22no—
103C4×Dic271—41no—
104C24×C271C2311yescharacteristic
105C22×C4×C271C2311yescharacteristic
106C22×C4×C271C2311yescharacteristic
107C22×C4×C271C2311yescharacteristic
108C22×C4×C271C2311yescharacteristic
109C22×C4×C271C2311yescharacteristic
110C22×C4×C271C2311yescharacteristic
111C4⋊C4×C271—22no—
112C22⋊C4×C271—22no—
113C22⋊C4×C271—41no—
114C22⋊C4×C271C2×C414yescomplemented
115C42×C271—41no—
116C22×D271—21no—
117C22×D271—21no—
118C22×D271—21no—
119C22×D271—21no—
120C22×D271—21no—
121C22×D271—21no—
122C22×D271—21no—
123C22×D271—21no—
124C22×D271—21no—
125C22×D271—21no—
126C22×D271—21no—
127C22×D271—21no—
128C22×D271—21no—
129C22×D271—21no—
130C22×D271—21no—
131C22×D271—21no—
132C22×D271—28no—
133C22×D271—44no—
134C22×D271—44no—
135C22×D271—44no—
136C22×D271—44no—
137C22×D271—44no—
138C22×D271—44no—
139C22×D271C22×C412yes—
140C22×D271C22×C412yes—
141C22×D271C22×C412yes—
142C22×D271C2×D412yes—
143C22×D271#4112yes—
144D1084—28no—
145D1084—44no—
146C22⋊D271—44no—
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—21no—
158C2×Dic271—21no—
159C2×Dic271—21no—
160C2×Dic271C22×C412yes—
161C2×Dic271C2×D412yes—
162C4×D271—44no—
163C4×D271—44no—
164C4×D271—44no—
165C23×C271—21no—
166C23×C271—21no—
167C23×C271—21no—
168C23×C271—21no—
169C23×C271—21no—
170C23×C271—21no—
171C23×C271C22×C412yes—
172C23×C271C2411yescharacteristic
173C2×C4×C271—21no—
174C2×C4×C271—21no—
175C2×C4×C271—21no—
176C2×C4×C271—21no—
177C2×C4×C271—21no—
178C2×C4×C271—21no—
179C2×C4×C271—21no—
180C2×C4×C271—21no—
181C2×C4×C271—21no—
182C2×C4×C271—21no—
183C2×C4×C271—21no—
184C2×C4×C271—21no—
185C2×C4×C271—21no—
186C2×C4×C271—21no—
187C2×C4×C271—21no—
188C2×C4×C271C22×C412yes—
189C2×C4×C271C22×C412yes—
190C2×C4×C271C2×D412yes—
191C22×C271—22no—
192C22×C271—22no—
193C22×C271—22no—
194C22×C271—41no—
195C22×C271—41no—
196C22×C271—41no—
197C22×C271—41no—
198C22×C271C23×C411yescharacteristic
199C22×C271C22×D411yescharacteristic
200C22×C271C2×#4111yescharacteristic
201C22×C271H(5,𝔽2)11yescharacteristic
202C22×C271H(5,𝔽2)11yescharacteristic
203C22×C271H(5,𝔽2)11yescharacteristic
204C22×C271H(5,𝔽2)11yescharacteristic
205C2×D271—22no—
206C2×D271—22no—
207C2×D271—22no—
208C2×D271—22no—
209C2×D271—22no—
210C2×D271—22no—
211C2×D271—22no—
212C2×D271—22no—
213C2×D271—22no—
214C2×D271—22no—
215C2×D271—22no—
216C2×D271—22no—
217C2×D271—416no—
218C2×D271—41no—
219C2×D271—41no—
220C2×D271—41no—
221C2×D271—41no—
222C2×D271—41no—
223C2×D271—41no—
224C2×D271—41no—
225C2×D271—41no—
226C2×D271C4×D414yes—
227C4×C271—22no—
228C4×C271—22no—
229C4×C271—22no—
230C4×C271—41no—
231C4×C271—41no—
232C4×C271—41no—
233Dic271—22no—
234Dic271—22no—
235Dic271—41no—
236Dic271—41no—
237Dic271—41no—
238C2×C271—42no—
239C2×C271C2×C4×D411yescharacteristic
240C2×C271#51811yescharacteristic
241C2×C271#51811yescharacteristic
242C2×C271#53411yescharacteristic
243C2×C271#53511yescharacteristic
244C2×C271#53811yescharacteristic
245C2×C271#53811yescharacteristic
246D271—24no—
247D271—42no—
248D271—42no—
249C271#232111yescharacteristic, complemented, Sylow
250#2321—2711nomaximal, Sylow
251C2×#120—2711no—
252C2×C4×D4—2711no—
253C22×C22⋊C4—2711no—
254#390—2711no—
255#390—2711no—
256#388—2711no—
257#388—2711no—
258#388—2711no—
259#388D27111yescharacteristic, complemented
260#386—2711no—
261#386—2711no—
262#386—2711no—
263#385—2711no—
264#379—2711no—
265C4×C22⋊C4—2711no—
266C25—2711no—
267C22×D4—2711no—
268C22×D4—2711no—
269C22×D4—2711no—
270C23×C4—2711no—
271C23×C4—2711no—
272C23×C4—2711no—
273#120—2718no—
274C4×D4—5424no—
275C2×C4⋊C4—2711no—
276C2×C4⋊C4—2711no—
277C2×C4⋊C4C2×D27111yescharacteristic, complemented
278C2×C22⋊C4—2711no—
279C2×C22⋊C4—2711no—
280C2×C22⋊C4—2711no—
281C2×C22⋊C4—2711no—
282C2×C22⋊C4—2711no—
283C2×C22⋊C4—2711no—
284C2×C22⋊C4—2711no—
285C2×C22⋊C4—2711no—
286C2×C22⋊C4—2711no—
287C2×C22⋊C4—2711no—
288C2×C22⋊C4—2711no—
289C2×C22⋊C4—2711no—
290C2×C22⋊C4—2711no—
291C2×C22⋊C4—5424no—
292C2×C22⋊C4C2×D27111yescharacteristic
293C2×C22⋊C4C2×D27111yescharacteristic
294C2×C22⋊C4C2×D27111yescharacteristic
295C2×C42—2711no—
296C2×C42—2711no—
297C2×C42C2×D27111yescharacteristic, complemented
298#95—2711no—
299#95—2711no—
300#95—2711no—
301#95—2711no—
302#95C2×D27111yescharacteristic, complemented
303#95C2×D27111yescharacteristic, complemented
304C24—2711no—
305C24—2711no—
306C24—2711no—
307C24—2714no—
308C24—5422no—
309C24—5422no—
310C24—10841no—
311C24—10841no—
312C24—10841no—
313C24—10841no—
314C24C22×D27111yescharacteristic
315C2×D4—2714no—
316C2×D4—2714no—
317C2×D4—2714no—
318C2×D4—5422no—
319C2×D4—5422no—
320C2×D4—10841no—
321C2×D4—10841no—
322C2×D4—10841no—
323C2×D4—10841no—
324C22×C4—2711no—
325C22×C4—2711no—
326C22×C4—2714no—
327C22×C4—2711no—
328C22×C4—2711no—
329C22×C4—2711no—
330C22×C4—5422no—
331C22×C4—5422no—
332C22×C4—10841no—
333C22×C4—10841no—
334C22×C4—10841no—
335C22×C4—10841no—
336C22×C4—10841no—
337C22×C4—10841no—
338C22×C4C22×D27111yescharacteristic
339C22×C4C22×D27111yescharacteristic
340C22×C4C22×D27111yescharacteristic
341C22×C4C22×D27111yescharacteristic
342C22×C4C22×D27111yescharacteristic
343C22×C4C22×D27111yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—10841no—
346C4⋊C4—10841no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—2714no—
350C22⋊C4—2714no—
351C22⋊C4—5422no—
352C22⋊C4—5422no—
353C22⋊C4—5422no—
354C22⋊C4—5422no—
355C22⋊C4—5422no—
356C22⋊C4—5422no—
357C22⋊C4—10841no—
358C22⋊C4—10841no—
359C22⋊C4—10841no—
360C22⋊C4—10841no—
361C22⋊C4—10841no—
362C22⋊C4C4×D27114yescomplemented
363C42—41no—
364C42—5422no—
365C42—10841no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—2712no—
373C23—2712no—
374C23—2712no—
375C23—2712no—
376C23—2712no—
377C23—5428no—
378C23—5421no—
379C23—5421no—
380C23—5421no—
381C23—5421no—
382C23—5421no—
383C23—5421no—
384C23—5421no—
385C23—5421no—
386C23—5421no—
387C23—5421no—
388C23—5421no—
389C23—5421no—
390C23—5421no—
391C23—5421no—
392C23—5421no—
393C23—5421no—
394C23—10844no—
395C23—10844no—
396C23—10844no—
397C23—10844no—
398C23—10844no—
399C23—10844no—
400C23C2×C4×D27112yes—
401C23C23×D27111yescharacteristic
402D4—5428no—
403D4—10844no—
404D4—10844no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—2712no—
421C2×C4—2712no—
422C2×C4—5421no—
423C2×C4—5421no—
424C2×C4—5421no—
425C2×C4—5421no—
426C2×C4—5421no—
427C2×C4—5421no—
428C2×C4—5421no—
429C2×C4—5421no—
430C2×C4—5421no—
431C2×C4—5421no—
432C2×C4—5421no—
433C2×C4—5421no—
434C2×C4—5421no—
435C2×C4—10844no—
436C2×C4—10844no—
437C2×C4—10844no—
438C2×C4C2×C4×D27112yes—
439C2×C4C2×C4×D27112yes—
440C2×C4D4×D27112yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—2714no—
449C22—5422no—
450C22—5422no—
451C22—5422no—
452C22—5422no—
453C22—5422no—
454C22—5422no—
455C22—5422no—
456C22—5422no—
457C22—5422no—
458C22—5422no—
459C22—5422no—
460C22—5422no—
461C22—108416no—
462C22—10841no—
463C22—10841no—
464C22—10841no—
465C22—10841no—
466C22—10841no—
467C22—10841no—
468C22—10841no—
469C22—10841no—
470C22C22×C4×D27111yescharacteristic
471C22C2×D4×D27111yescharacteristic
472C22#41×D27111yescharacteristic
473C22#2257620511yescharacteristic
474C22#2257620611yescharacteristic
475C22#2257620611yescharacteristic
476C22#2257620611yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—5422no—
484C4—5422no—
485C4—10841no—
486C4—10841no—
487C4—10841no—
488C2—42no—
489C2—5424no—
490C2—10842no—
491C2—10842no—
492C2C4×D4×D27111yescharacteristic
493C2#3177436911yescharacteristic
494C2#3177437111yescharacteristic
495C2#3177445511yescharacteristic
496C2#3177446511yescharacteristic
497C2#3177448311yescharacteristic
498C2#3177448611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#388×C271C2
socle, cosocleC23×C271C24
prime radicalC2×#22576130C2
hypercenter#388D271
hypocenterC271#2321
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#22576122C2semidirect
3. C2×C22⋊C4×D271C2semidirect
4. #31773799C2semidirect
5. #31773797C2semidirect
6. #31773790C2semidirect
7. #31773787C2semidirect
8. #31773786C2semidirect
9. #31773775C2semidirect
10. #31773775C2semidirect
11. #31773773C2semidirect
12. #31773771C2semidirect
13. #31773752C2semidirect
14. C4×(C2×D271)⋊C4C2semidirect
15. #388×C271C2semidirect
22. C22×C4×D271C22semidirect
23. #22576130C4semidirect
25. C2×C4⋊Dic271C22semidirect
26. C2×Dic271⋊C4C22semidirect
37. C2×(C2×D271)⋊C4C22semidirect
38. C2×(C2×D271)⋊C4C22semidirect
39. C2×(C2×D271)⋊C4C22semidirect
41. C2×C4×Dic271C22semidirect
42. C2×C4×Dic271C22semidirect
43. C271⋊#95C22semidirect
44. C271⋊#95C22semidirect
45. C271⋊#95C22semidirect
46. C271⋊#95C22semidirect
47. C2×C4⋊C4×C271C22semidirect
51. C2×C42×C271C22semidirect
52. #95×C271C22semidirect
53. #95×C271C22semidirect
60. C23×D271C2×C4semidirect
69. C2×D1084C2×C4semidirect
72. C2×C22⋊D271C2×C4semidirect
86. C2×C4×D271D4semidirect
100. (C2×D271)⋊C4C2×C4semidirect
101. (C2×D271)⋊C4C2×C4semidirect
114. C22⋊C4×C271C2×C4semidirect
249. C271#2321semidirect
259. #388D271semidirect
277. C2×C4⋊C4C2×D271semidirect
297. C2×C42C2×D271semidirect
302. #95C2×D271semidirect
303. #95C2×D271semidirect
362. C22⋊C4C4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761302
C2×#225761222
C2×C22⋊C4×D2712
#317737992
#317737972
#317737902
#317737872
#317737862
#317737752
#317737732
#317737712
#317737522
C4×(C2×D271)⋊C42
#388×C2712
#2321271