Finite Group Databasebeta

← #44487207

Subgroups — #44487207

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×#21439928C211yesmaximal, characteristic
2C2×(C22⋊D239)⋊C4C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D239C211yesmaximal, characteristic, complemented
4#29504672C211yesmaximal, characteristic, complemented
5#29504666C211yesmaximal, characteristic, complemented
6#29504666C211yesmaximal, characteristic, complemented
7#29504663C211yesmaximal, characteristic, complemented
8#29504662C211yesmaximal, characteristic, complemented
9#29504652C211yesmaximal, characteristic, complemented
10#29504649C211yesmaximal, characteristic, complemented
11#29504648C211yesmaximal, characteristic, complemented
12#29504643C211yesmaximal, characteristic, complemented
13#29504627C211yesmaximal, characteristic, complemented
14C4×(C2×D239)⋊C4C211yesmaximal, characteristic, complemented
15#390×C239C211yesmaximal, characteristic, complemented
16C24×D239C2211yescharacteristic
17C22×C22⋊D239C2211yescharacteristic
18C22×C22⋊D239C2211yescharacteristic
19C23×Dic239C2211yescharacteristic, complemented
20C22×C4×D239C2211yescharacteristic
21C22×C4×D239C2211yescharacteristic
22#21439928C418yescomplemented
23(C22⋊D239)⋊C4—24no—
24C2×Dic239⋊C4C2211yescharacteristic, complemented
25C2×Dic239⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic239C2211yescharacteristic
27C2×C22⋊Dic239C2211yescharacteristic
28C2×C22⋊Dic239C2211yescharacteristic
29C2×C22⋊Dic239C2211yescharacteristic, complemented
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
38C22⋊C4×D239—24no—
39C2×C4×Dic239C2211yescharacteristic, complemented
40C2×C4×Dic239C2211yescharacteristic, complemented
41C239⋊#95C2211yescharacteristic, complemented
42C239⋊#95C2211yescharacteristic, complemented
43C239⋊#95C2211yescharacteristic, complemented
44C239⋊#95C2211yescharacteristic, complemented
45C239⋊#95C2211yescharacteristic, complemented
46C239⋊#95C2211yescharacteristic, complemented
47C22×D4×C239C2211yescharacteristic
48C2×C4⋊C4×C239C2211yescharacteristic, complemented
49C2×C22⋊C4×C239C2211yescharacteristic
50C2×C22⋊C4×C239C2211yescharacteristic
51C2×C22⋊C4×C239C2211yescharacteristic, complemented
52C2×C22⋊C4×C239C2211yescharacteristic, complemented
53C2×C42×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
63C2×C22⋊D239—22no—
64C2×C22⋊D239—22no—
65C2×C22⋊D239—41no—
66C2×C22⋊D239—41no—
67C2×C22⋊D239C2×C414yes—
68C2×C22⋊D239C2×C414yescomplemented
69C22×Dic239—22no—
70C22×Dic239—41no—
71C22×Dic239—41no—
72C22×Dic239C2311yescharacteristic
73C22×Dic239C2311yescharacteristic
74C22×Dic239C2311yescharacteristic
75C22×Dic239C2311yescharacteristic
76C22×Dic239C2311yescharacteristic
77C22×Dic239C2311yescharacteristic
78C2×C4×D239—22no—
79C2×C4×D239—41no—
80C2×C4×D239—41no—
81C2×C4×D239—41no—
82C2×C4×D239—41no—
83C2×C4×D239D414yescomplemented
84Dic239⋊C4—22no—
85Dic239⋊C4—41no—
86C22⋊Dic239—22no—
87C22⋊Dic239—41no—
88C22⋊Dic239—41no—
89C22⋊Dic239C2×C414yescomplemented
90(C2×D239)⋊C4—22no—
91(C2×D239)⋊C4—22no—
92(C2×D239)⋊C4—22no—
93(C2×D239)⋊C4—22no—
94(C2×D239)⋊C4—41no—
95(C2×D239)⋊C4—41no—
96(C2×D239)⋊C4C2×C414yescomplemented
97(C2×D239)⋊C4C2×C414yescomplemented
98C4×Dic239—22no—
99C4×Dic239—41no—
100C24×C239C2311yescharacteristic
101C24×C239C2311yescharacteristic
102C2×D4×C239—41no—
103C2×D4×C239—41no—
104C2×D4×C239C2×C414yescomplemented
105C22×C4×C239C2311yescharacteristic
106C22×C4×C239C2311yescharacteristic
107C22×C4×C239C2311yescharacteristic
108C22×C4×C239C2311yescharacteristic
109C22×C4×C239C2311yescharacteristic
110C4⋊C4×C239—41no—
111C22⋊C4×C239—22no—
112C22⋊C4×C239—22no—
113C22⋊C4×C239—41no—
114C22⋊C4×C239—41no—
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—28no—
127C22×D239—44no—
128C22×D239—44no—
129C22×D239—44no—
130C22×D239—44no—
131C22×D239—44no—
132C22×D239—44no—
133C22×D239C22×C412yes—
134C22×D239C22×C412yes—
135C22×D239C2×D412yes—
136C22×D239#4112yes—
137C22⋊D239—28no—
138C22⋊D239—44no—
139C2×Dic239—21no—
140C2×Dic239—21no—
141C2×Dic239—21no—
142C2×Dic239—21no—
143C2×Dic239—21no—
144C2×Dic239—21no—
145C2×Dic239—21no—
146C2×Dic239—21no—
147C2×Dic239—21no—
148C2×Dic239—21no—
149C2×Dic239—21no—
150C2×Dic239—21no—
151C2×Dic239—21no—
152C2×Dic239—21no—
153C2×Dic239—21no—
154C2×Dic239—44no—
155C2×Dic239C22×C412yes—
156C2×Dic239C22×C412yes—
157C2×Dic239C2×D412yes—
158C4×D239—44no—
159C4×D239—44no—
160C23×C239—21no—
161C23×C239—21no—
162C23×C239—21no—
163C23×C239—21no—
164C23×C239—21no—
165C23×C239—21no—
166C23×C239—21no—
167C23×C239—21no—
168C23×C239—21no—
169C23×C239—21no—
170C23×C239—21no—
171C23×C239—21no—
172C23×C239C22×C412yes—
173C23×C239C22×C412yes—
174C23×C239C2411yescharacteristic
175D4×C239—44no—
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×C239—21no—
189C2×C4×C239C22×C412yes—
190C2×C4×C239C2×D412yes—
191C22×C239—22no—
192C22×C239—22no—
193C22×C239—22no—
194C22×C239—22no—
195C22×C239—22no—
196C22×C239—22no—
197C22×C239—41no—
198C22×C239—41no—
199C22×C239—41no—
200C22×C239—41no—
201C22×C239—41no—
202C22×C239—41no—
203C22×C239—41no—
204C22×C239—41no—
205C22×C239C23×C411yescharacteristic
206C22×C239C22×D411yescharacteristic
207C22×C239C2×#4111yescharacteristic
208C22×C239H(5,𝔽2)11yescharacteristic
209C22×C239H(5,𝔽2)11yescharacteristic
210C22×C239H(5,𝔽2)11yescharacteristic
211C22×C239H(5,𝔽2)11yescharacteristic
212C2×D239—22no—
213C2×D239—22no—
214C2×D239—22no—
215C2×D239—22no—
216C2×D239—22no—
217C2×D239—22no—
218C2×D239—22no—
219C2×D239—22no—
220C2×D239—22no—
221C2×D239—416no—
222C2×D239—41no—
223C2×D239—41no—
224C2×D239—41no—
225C2×D239—41no—
226C2×D239C4×D414yes—
227C4×C239—22no—
228C4×C239—22no—
229C4×C239—41no—
230C4×C239—41no—
231C4×C239—41no—
232Dic239—22no—
233Dic239—22no—
234Dic239—22no—
235Dic239—41no—
236Dic239—41no—
237Dic239—41no—
238C2×C239—42no—
239C2×C239—42no—
240C2×C239C2×C4×D411yescharacteristic
241C2×C239#51811yescharacteristic
242C2×C239#51811yescharacteristic
243C2×C239#53411yescharacteristic
244C2×C239#53511yescharacteristic
245C2×C239#53811yescharacteristic
246C2×C239#53811yescharacteristic
247D239—24no—
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#390D23911yescharacteristic, complemented
256#388—2391no—
257#388—2391no—
258#388—2391no—
259#388—2391no—
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×D4C2×D23911yescharacteristic
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⋊C4C2×D23911yescharacteristic
292C2×C22⋊C4C2×D23911yescharacteristic
293C2×C22⋊C4C2×D23911yescharacteristic, complemented
294C2×C22⋊C4C2×D23911yescharacteristic, complemented
295C2×C42—2391no—
296C2×C42—2391no—
297C2×C42C2×D23911yescharacteristic, complemented
298#95—2391no—
299#95—2391no—
300#95—2391no—
301#95—2391no—
302#95—2391no—
303#95—2391no—
304C24—9561no—
305C24—9561no—
306C24—9561no—
307C24—9561no—
308C24—4782no—
309C24—4782no—
310C24—2394no—
311C24—2391no—
312C24—2391no—
313C24C22×D23911yescharacteristic
314C24C22×D23911yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—9561no—
318C2×D4—9561no—
319C2×D4—4782no—
320C2×D4—4782no—
321C2×D4—2394no—
322C2×D4—2394no—
323C2×D4C4×D23914yescomplemented
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×C4—2391no—
339C22×C4C22×D23911yescharacteristic
340C22×C4C22×D23911yescharacteristic
341C22×C4C22×D23911yescharacteristic
342C22×C4C22×D23911yescharacteristic
343C22×C4C22×D23911yescharacteristic
344C4⋊C4—41no—
345C4⋊C4—9561no—
346C4⋊C4—4782no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—9561no—
352C22⋊C4—9561no—
353C22⋊C4—9561no—
354C22⋊C4—9561no—
355C22⋊C4—4782no—
356C22⋊C4—4782no—
357C22⋊C4—4782no—
358C22⋊C4—4782no—
359C22⋊C4—4782no—
360C22⋊C4—2394no—
361C22⋊C4—2394no—
362C22⋊C4—2394no—
363C42—41no—
364C42—9561no—
365C42—4782no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—9564no—
379C23—9564no—
380C23—9564no—
381C23—9564no—
382C23—9564no—
383C23—9564no—
384C23—4788no—
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—
399C23C2×C4×D23912yes—
400C23C2×C4×D23912yes—
401C23C23×D23911yescharacteristic
402D4—44no—
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—9564no—
419C2×C4—9564no—
420C2×C4—9564no—
421C2×C4—4781no—
422C2×C4—4781no—
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×C4—2392no—
439C2×C4C2×C4×D23912yes—
440C2×C4D4×D23912yes—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—95616no—
456C22—9561no—
457C22—9561no—
458C22—9561no—
459C22—9561no—
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#2144000511yescharacteristic
475C22#2144000511yescharacteristic
476C22#2144000611yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—9561no—
483C4—9561no—
484C4—9561no—
485C4—4782no—
486C4—4782no—
487C4—4782no—
488C2—42no—
489C2—42no—
490C2—9562no—
491C2—4784no—
492C2C4×D4×D23911yescharacteristic
493C2#2950524411yescharacteristic
494C2#2950524511yescharacteristic
495C2#2950533011yescharacteristic
496C2#2950533511yescharacteristic
497C2#2950535411yescharacteristic
498C2#2950535611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D239
commutatorC22×C239C23×C4
Fitting#390×C239C2
socle, cosocleC23×C239C24
prime radicalC2×#21439928C2
hypercenter#390D239
hypocenterC239#2321
LCSG ▷ C22×C239 ▷ C239
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D239)⋊C4C2semidirect
3. C2×C22⋊C4×D239C2semidirect
4. #29504672C2semidirect
5. #29504666C2semidirect
6. #29504666C2semidirect
7. #29504663C2semidirect
8. #29504662C2semidirect
9. #29504652C2semidirect
10. #29504649C2semidirect
11. #29504648C2semidirect
12. #29504643C2semidirect
13. #29504627C2semidirect
14. C4×(C2×D239)⋊C4C2semidirect
15. #390×C239C2semidirect
19. C23×Dic239C22semidirect
22. #21439928C4semidirect
24. C2×Dic239⋊C4C22semidirect
25. C2×Dic239⋊C4C22semidirect
29. C2×C22⋊Dic239C22semidirect
39. C2×C4×Dic239C22semidirect
40. C2×C4×Dic239C22semidirect
41. C239⋊#95C22semidirect
42. C239⋊#95C22semidirect
43. C239⋊#95C22semidirect
44. C239⋊#95C22semidirect
45. C239⋊#95C22semidirect
46. C239⋊#95C22semidirect
48. C2×C4⋊C4×C239C22semidirect
51. C2×C22⋊C4×C239C22semidirect
52. C2×C22⋊C4×C239C22semidirect
53. C2×C42×C239C22semidirect
60. C23×D239C2×C4semidirect
68. C2×C22⋊D239C2×C4semidirect
83. C2×C4×D239D4semidirect
89. C22⋊Dic239C2×C4semidirect
96. (C2×D239)⋊C4C2×C4semidirect
97. (C2×D239)⋊C4C2×C4semidirect
104. C2×D4×C239C2×C4semidirect
249. C239#2321semidirect
255. #390D239semidirect
277. C2×C4⋊C4C2×D239semidirect
293. C2×C22⋊C4C2×D239semidirect
294. C2×C22⋊C4C2×D239semidirect
297. C2×C42C2×D239semidirect
323. C2×D4C4×D239semidirect

Maximal subgroups

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

subgroupindex
C2×#214399282
C2×(C22⋊D239)⋊C42
C2×C22⋊C4×D2392
#295046722
#295046662
#295046632
#295046622
#295046522
#295046492
#295046482
#295046432
#295046272
C4×(C2×D239)⋊C42
#390×C2392
#2321239