Finite Group Databasebeta

← #44487201

Subgroups — #44487201

Order 30592.

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×#21439930C211yesmaximal, characteristic
2C2×#21439922C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D239C211yesmaximal, characteristic, complemented
4#29504674C211yesmaximal, characteristic, complemented
5#29504672C211yesmaximal, characteristic, complemented
6#29504665C211yesmaximal, characteristic, complemented
7#29504662C211yesmaximal, characteristic, complemented
8#29504661C211yesmaximal, characteristic, complemented
9#29504650C211yesmaximal, characteristic, complemented
10#29504650C211yesmaximal, characteristic, complemented
11#29504648C211yesmaximal, characteristic, complemented
12#29504646C211yesmaximal, characteristic, complemented
13#29504627C211yesmaximal, characteristic, complemented
14C4×(C2×D239)⋊C4C211yesmaximal, characteristic, complemented
15#388×C239C211yesmaximal, characteristic, complemented
16C24×D239C2211yescharacteristic
17C22×D956C2211yescharacteristic
18C22×D956C2211yescharacteristic
19C22×C22⋊D239C2211yescharacteristic
20C22×C4×D239C2211yescharacteristic
21C22×C4×D239C2211yescharacteristic
22C22×C4×D239C2211yescharacteristic, complemented
23#21439930C418yescomplemented
24#21439922—24no—
25C2×C4⋊Dic239C2211yescharacteristic, complemented
26C2×Dic239⋊C4C2211yescharacteristic, complemented
27C2×C22⋊Dic239C2211yescharacteristic
28C2×C22⋊Dic239C2211yescharacteristic
29C2×(C2×D239)⋊C4C2211yescharacteristic
30C2×(C2×D239)⋊C4C2211yescharacteristic
31C2×(C2×D239)⋊C4C2211yescharacteristic
32C2×(C2×D239)⋊C4C2211yescharacteristic
33C2×(C2×D239)⋊C4C2211yescharacteristic
34C2×(C2×D239)⋊C4C2211yescharacteristic
35C2×(C2×D239)⋊C4C2211yescharacteristic
36C2×(C2×D239)⋊C4C2211yescharacteristic
37C2×(C2×D239)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D239)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D239)⋊C4C2211yescharacteristic, complemented
40C22⋊C4×D239—24no—
41C2×C4×Dic239C2211yescharacteristic, complemented
42C2×C4×Dic239C2211yescharacteristic, complemented
43C239⋊#95C2211yescharacteristic, complemented
44C239⋊#95C2211yescharacteristic, complemented
45C239⋊#95C2211yescharacteristic, complemented
46C239⋊#95C2211yescharacteristic, complemented
47C2×C4⋊C4×C239C2211yescharacteristic, complemented
48C2×C22⋊C4×C239C2211yescharacteristic
49C2×C22⋊C4×C239C2211yescharacteristic
50C2×C22⋊C4×C239C2211yescharacteristic
51C2×C42×C239C2211yescharacteristic, complemented
52#95×C239C2211yescharacteristic, complemented
53#95×C239C2211yescharacteristic, complemented
54C23×D239—22no—
55C23×D239—22no—
56C23×D239—41no—
57C23×D239—41no—
58C23×D239—41no—
59C23×D239—41no—
60C23×D239C2×C414yescomplemented
61C23×D239C2311yescharacteristic
62C23×D239C2311yescharacteristic
63C23×D239C2311yescharacteristic
64C2×D956—22no—
65C2×D956—22no—
66C2×D956—41no—
67C2×D956—41no—
68C2×D956C2×C414yes—
69C2×D956C2×C414yescomplemented
70C2×C22⋊D239—41no—
71C2×C22⋊D239—41no—
72C2×C22⋊D239C2×C414yescomplemented
73C22×Dic239C2311yescharacteristic
74C22×Dic239C2311yescharacteristic
75C22×Dic239C2311yescharacteristic
76C22×Dic239C2311yescharacteristic
77C22×Dic239C2311yescharacteristic
78C2×C4×D239—22no—
79C2×C4×D239—22no—
80C2×C4×D239—41no—
81C2×C4×D239—41no—
82C2×C4×D239—41no—
83C2×C4×D239—41no—
84C2×C4×D239—41no—
85C2×C4×D239—41no—
86C2×C4×D239D414yescomplemented
87C4⋊Dic239—41no—
88Dic239⋊C4—41no—
89C22⋊Dic239—22no—
90C22⋊Dic239—41no—
91(C2×D239)⋊C4—22no—
92(C2×D239)⋊C4—22no—
93(C2×D239)⋊C4—22no—
94(C2×D239)⋊C4—22no—
95(C2×D239)⋊C4—22no—
96(C2×D239)⋊C4—41no—
97(C2×D239)⋊C4—41no—
98(C2×D239)⋊C4—41no—
99(C2×D239)⋊C4—41no—
100(C2×D239)⋊C4C2×C414yescomplemented
101(C2×D239)⋊C4C2×C414yescomplemented
102C4×Dic239—22no—
103C4×Dic239—41no—
104C24×C239C2311yescharacteristic
105C22×C4×C239C2311yescharacteristic
106C22×C4×C239C2311yescharacteristic
107C22×C4×C239C2311yescharacteristic
108C22×C4×C239C2311yescharacteristic
109C22×C4×C239C2311yescharacteristic
110C22×C4×C239C2311yescharacteristic
111C4⋊C4×C239—22no—
112C22⋊C4×C239—22no—
113C22⋊C4×C239—41no—
114C22⋊C4×C239C2×C414yescomplemented
115C42×C239—41no—
116C22×D239—21no—
117C22×D239—21no—
118C22×D239—21no—
119C22×D239—21no—
120C22×D239—21no—
121C22×D239—21no—
122C22×D239—21no—
123C22×D239—21no—
124C22×D239—21no—
125C22×D239—21no—
126C22×D239—21no—
127C22×D239—21no—
128C22×D239—21no—
129C22×D239—21no—
130C22×D239—21no—
131C22×D239—21no—
132C22×D239—28no—
133C22×D239—44no—
134C22×D239—44no—
135C22×D239—44no—
136C22×D239—44no—
137C22×D239—44no—
138C22×D239—44no—
139C22×D239C22×C412yes—
140C22×D239C22×C412yes—
141C22×D239C22×C412yes—
142C22×D239C2×D412yes—
143C22×D239#4112yes—
144D956—28no—
145D956—44no—
146C22⋊D239—44no—
147C2×Dic239—21no—
148C2×Dic239—21no—
149C2×Dic239—21no—
150C2×Dic239—21no—
151C2×Dic239—21no—
152C2×Dic239—21no—
153C2×Dic239—21no—
154C2×Dic239—21no—
155C2×Dic239—21no—
156C2×Dic239—21no—
157C2×Dic239—21no—
158C2×Dic239—21no—
159C2×Dic239—21no—
160C2×Dic239C22×C412yes—
161C2×Dic239C2×D412yes—
162C4×D239—44no—
163C4×D239—44no—
164C4×D239—44no—
165C23×C239—21no—
166C23×C239—21no—
167C23×C239—21no—
168C23×C239—21no—
169C23×C239—21no—
170C23×C239—21no—
171C23×C239C22×C412yes—
172C23×C239C2411yescharacteristic
173C2×C4×C239—21no—
174C2×C4×C239—21no—
175C2×C4×C239—21no—
176C2×C4×C239—21no—
177C2×C4×C239—21no—
178C2×C4×C239—21no—
179C2×C4×C239—21no—
180C2×C4×C239—21no—
181C2×C4×C239—21no—
182C2×C4×C239—21no—
183C2×C4×C239—21no—
184C2×C4×C239—21no—
185C2×C4×C239—21no—
186C2×C4×C239—21no—
187C2×C4×C239—21no—
188C2×C4×C239C22×C412yes—
189C2×C4×C239C22×C412yes—
190C2×C4×C239C2×D412yes—
191C22×C239—22no—
192C22×C239—22no—
193C22×C239—22no—
194C22×C239—41no—
195C22×C239—41no—
196C22×C239—41no—
197C22×C239—41no—
198C22×C239C23×C411yescharacteristic
199C22×C239C22×D411yescharacteristic
200C22×C239C2×#4111yescharacteristic
201C22×C239H(5,𝔽2)11yescharacteristic
202C22×C239H(5,𝔽2)11yescharacteristic
203C22×C239H(5,𝔽2)11yescharacteristic
204C22×C239H(5,𝔽2)11yescharacteristic
205C2×D239—22no—
206C2×D239—22no—
207C2×D239—22no—
208C2×D239—22no—
209C2×D239—22no—
210C2×D239—22no—
211C2×D239—22no—
212C2×D239—22no—
213C2×D239—22no—
214C2×D239—22no—
215C2×D239—22no—
216C2×D239—22no—
217C2×D239—416no—
218C2×D239—41no—
219C2×D239—41no—
220C2×D239—41no—
221C2×D239—41no—
222C2×D239—41no—
223C2×D239—41no—
224C2×D239—41no—
225C2×D239—41no—
226C2×D239C4×D414yes—
227C4×C239—22no—
228C4×C239—22no—
229C4×C239—22no—
230C4×C239—41no—
231C4×C239—41no—
232C4×C239—41no—
233Dic239—22no—
234Dic239—22no—
235Dic239—41no—
236Dic239—41no—
237Dic239—41no—
238C2×C239—42no—
239C2×C239C2×C4×D411yescharacteristic
240C2×C239#51811yescharacteristic
241C2×C239#51811yescharacteristic
242C2×C239#53411yescharacteristic
243C2×C239#53511yescharacteristic
244C2×C239#53811yescharacteristic
245C2×C239#53811yescharacteristic
246D239—24no—
247D239—42no—
248D239—42no—
249C239#232111yescharacteristic, complemented, Sylow
250#2321—2391nomaximal, Sylow
251C2×#120—2391no—
252C2×C4×D4—2391no—
253C22×C22⋊C4—2391no—
254#390—2391no—
255#390—2391no—
256#388—2391no—
257#388—2391no—
258#388—2391no—
259#388D23911yescharacteristic, complemented
260#386—2391no—
261#386—2391no—
262#386—2391no—
263#385—2391no—
264#379—2391no—
265C4×C22⋊C4—2391no—
266C25—2391no—
267C22×D4—2391no—
268C22×D4—2391no—
269C22×D4—2391no—
270C23×C4—2391no—
271C23×C4—2391no—
272C23×C4—2391no—
273#120—2398no—
274C4×D4—4784no—
275C2×C4⋊C4—2391no—
276C2×C4⋊C4—2391no—
277C2×C4⋊C4C2×D23911yescharacteristic, complemented
278C2×C22⋊C4—4784no—
279C2×C22⋊C4—2391no—
280C2×C22⋊C4—2391no—
281C2×C22⋊C4—2391no—
282C2×C22⋊C4—2391no—
283C2×C22⋊C4—2391no—
284C2×C22⋊C4—2391no—
285C2×C22⋊C4—2391no—
286C2×C22⋊C4—2391no—
287C2×C22⋊C4—2391no—
288C2×C22⋊C4—2391no—
289C2×C22⋊C4—2391no—
290C2×C22⋊C4—2391no—
291C2×C22⋊C4—2391no—
292C2×C22⋊C4C2×D23911yescharacteristic
293C2×C22⋊C4C2×D23911yescharacteristic
294C2×C22⋊C4C2×D23911yescharacteristic
295C2×C42—2391no—
296C2×C42—2391no—
297C2×C42C2×D23911yescharacteristic, complemented
298#95—2391no—
299#95—2391no—
300#95—2391no—
301#95—2391no—
302#95C2×D23911yescharacteristic, complemented
303#95C2×D23911yescharacteristic, complemented
304C24—9561no—
305C24—9561no—
306C24—9561no—
307C24—9561no—
308C24—4782no—
309C24—4782no—
310C24—2394no—
311C24—2391no—
312C24—2391no—
313C24—2391no—
314C24C22×D23911yescharacteristic
315C2×D4—9561no—
316C2×D4—9561no—
317C2×D4—9561no—
318C2×D4—9561no—
319C2×D4—4782no—
320C2×D4—4782no—
321C2×D4—2394no—
322C2×D4—2394no—
323C2×D4—2394no—
324C22×C4—9561no—
325C22×C4—9561no—
326C22×C4—9561no—
327C22×C4—9561no—
328C22×C4—9561no—
329C22×C4—9561no—
330C22×C4—4782no—
331C22×C4—4782no—
332C22×C4—2394no—
333C22×C4—2391no—
334C22×C4—2391no—
335C22×C4—2391no—
336C22×C4—2391no—
337C22×C4—2391no—
338C22×C4C22×D23911yescharacteristic
339C22×C4C22×D23911yescharacteristic
340C22×C4C22×D23911yescharacteristic
341C22×C4C22×D23911yescharacteristic
342C22×C4C22×D23911yescharacteristic
343C22×C4C22×D23911yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—9561no—
346C4⋊C4—9561no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—9561no—
350C22⋊C4—9561no—
351C22⋊C4—9561no—
352C22⋊C4—9561no—
353C22⋊C4—9561no—
354C22⋊C4—4782no—
355C22⋊C4—4782no—
356C22⋊C4—4782no—
357C22⋊C4—4782no—
358C22⋊C4—4782no—
359C22⋊C4—4782no—
360C22⋊C4—2394no—
361C22⋊C4—2394no—
362C22⋊C4C4×D23914yescomplemented
363C42—41no—
364C42—9561no—
365C42—4782no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—9564no—
373C23—9564no—
374C23—9564no—
375C23—9564no—
376C23—9564no—
377C23—9564no—
378C23—4788no—
379C23—4781no—
380C23—4781no—
381C23—4781no—
382C23—4781no—
383C23—4781no—
384C23—4781no—
385C23—4781no—
386C23—4781no—
387C23—4781no—
388C23—4781no—
389C23—4781no—
390C23—4781no—
391C23—4781no—
392C23—4781no—
393C23—4781no—
394C23—4781no—
395C23—2392no—
396C23—2392no—
397C23—2392no—
398C23—2392no—
399C23—2392no—
400C23C2×C4×D23912yes—
401C23C23×D23911yescharacteristic
402D4—9564no—
403D4—9564no—
404D4—4788no—
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—9564no—
421C2×C4—9564no—
422C2×C4—9564no—
423C2×C4—4781no—
424C2×C4—4781no—
425C2×C4—4781no—
426C2×C4—4781no—
427C2×C4—4781no—
428C2×C4—4781no—
429C2×C4—4781no—
430C2×C4—4781no—
431C2×C4—4781no—
432C2×C4—4781no—
433C2×C4—4781no—
434C2×C4—4781no—
435C2×C4—4781no—
436C2×C4—2392no—
437C2×C4—2392no—
438C2×C4C2×C4×D23912yes—
439C2×C4C2×C4×D23912yes—
440C2×C4D4×D23912yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—95616no—
449C22—9561no—
450C22—9561no—
451C22—9561no—
452C22—9561no—
453C22—9561no—
454C22—9561no—
455C22—9561no—
456C22—9561no—
457C22—4782no—
458C22—4782no—
459C22—4782no—
460C22—4782no—
461C22—4782no—
462C22—4782no—
463C22—4782no—
464C22—4782no—
465C22—4782no—
466C22—4782no—
467C22—4782no—
468C22—4782no—
469C22—2394no—
470C22C22×C4×D23911yescharacteristic
471C22C2×D4×D23911yescharacteristic
472C22#41×D23911yescharacteristic
473C22#2144000511yescharacteristic
474C22#2144000611yescharacteristic
475C22#2144000611yescharacteristic
476C22#2144000611yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—41no—
482C4—41no—
483C4—9561no—
484C4—9561no—
485C4—9561no—
486C4—4782no—
487C4—4782no—
488C2—42no—
489C2—9562no—
490C2—9562no—
491C2—4784no—
492C2C4×D4×D23911yescharacteristic
493C2#2950524411yescharacteristic
494C2#2950524611yescharacteristic
495C2#2950533011yescharacteristic
496C2#2950534011yescharacteristic
497C2#2950535811yescharacteristic
498C2#2950536111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutatorC22×C239C23×C4
Fitting#388×C239C2
socle, cosocleC23×C239C24
prime radicalC2×#21439930C2
hypercenter#388D239
hypocenterC239#2321
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#21439922C2semidirect
3. C2×C22⋊C4×D239C2semidirect
4. #29504674C2semidirect
5. #29504672C2semidirect
6. #29504665C2semidirect
7. #29504662C2semidirect
8. #29504661C2semidirect
9. #29504650C2semidirect
10. #29504650C2semidirect
11. #29504648C2semidirect
12. #29504646C2semidirect
13. #29504627C2semidirect
14. C4×(C2×D239)⋊C4C2semidirect
15. #388×C239C2semidirect
22. C22×C4×D239C22semidirect
23. #21439930C4semidirect
25. C2×C4⋊Dic239C22semidirect
26. C2×Dic239⋊C4C22semidirect
37. C2×(C2×D239)⋊C4C22semidirect
38. C2×(C2×D239)⋊C4C22semidirect
39. C2×(C2×D239)⋊C4C22semidirect
41. C2×C4×Dic239C22semidirect
42. C2×C4×Dic239C22semidirect
43. C239⋊#95C22semidirect
44. C239⋊#95C22semidirect
45. C239⋊#95C22semidirect
46. C239⋊#95C22semidirect
47. C2×C4⋊C4×C239C22semidirect
51. C2×C42×C239C22semidirect
52. #95×C239C22semidirect
53. #95×C239C22semidirect
60. C23×D239C2×C4semidirect
69. C2×D956C2×C4semidirect
72. C2×C22⋊D239C2×C4semidirect
86. C2×C4×D239D4semidirect
100. (C2×D239)⋊C4C2×C4semidirect
101. (C2×D239)⋊C4C2×C4semidirect
114. C22⋊C4×C239C2×C4semidirect
249. C239#2321semidirect
259. #388D239semidirect
277. C2×C4⋊C4C2×D239semidirect
297. C2×C42C2×D239semidirect
302. #95C2×D239semidirect
303. #95C2×D239semidirect
362. C22⋊C4C4×D239semidirect

Maximal subgroups

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

subgroupindex
C2×#214399302
C2×#214399222
C2×C22⋊C4×D2392
#295046742
#295046722
#295046652
#295046622
#295046612
#295046502
#295046482
#295046462
#295046272
C4×(C2×D239)⋊C42
#388×C2392
#2321239