Finite Group Databasebeta

← #49123917

Subgroups — #49123917

Order 36224.

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×#22583578C211yesmaximal, characteristic
2C2×C4×C22⋊D283C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D283C211yesmaximal, characteristic, complemented
4#31826368C211yesmaximal, characteristic, complemented
5#31826365C211yesmaximal, characteristic, complemented
6#31826357C211yesmaximal, characteristic, complemented
7#31826354C211yesmaximal, characteristic, complemented
8#31826353C211yesmaximal, characteristic, complemented
9#31826353C211yesmaximal, characteristic, complemented
10#31826344C211yesmaximal, characteristic, complemented
11#31826341C211yesmaximal, characteristic, complemented
12#31826339C211yesmaximal, characteristic, complemented
13#31826320C211yesmaximal, characteristic, complemented
14#31826312C211yesmaximal, characteristic, complemented
15#386×C283C211yesmaximal, characteristic, complemented
16C24×D283C2211yescharacteristic
17C22×C22⋊D283C2211yescharacteristic
18C22×C22⋊D283C2211yescharacteristic
19C22×C4×D283C2211yescharacteristic
20C22×C4×D283C2211yescharacteristic
21#22583578C418yescomplemented
22C4×C22⋊D283—24no—
23C2×C4⋊Dic283C2211yescharacteristic, complemented
24C2×Dic283⋊C4C2211yescharacteristic, complemented
25C2×Dic283⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic283C2211yescharacteristic
27C2×C22⋊Dic283C2211yescharacteristic
28C2×C22⋊Dic283C2211yescharacteristic
29C2×C22⋊Dic283C2211yescharacteristic, complemented
30C2×C22⋊Dic283C2211yescharacteristic, complemented
31C2×(C2×D283)⋊C4C2211yescharacteristic
32C2×(C2×D283)⋊C4C2211yescharacteristic
33C2×(C2×D283)⋊C4C2211yescharacteristic
34C2×(C2×D283)⋊C4C2211yescharacteristic
35C2×(C2×D283)⋊C4C2211yescharacteristic
36C2×(C2×D283)⋊C4C2211yescharacteristic
37C2×(C2×D283)⋊C4C2211yescharacteristic
38C2×(C2×D283)⋊C4C2211yescharacteristic
39C22⋊C4×D283—24no—
40C2×C4×Dic283C2211yescharacteristic, complemented
41C2×C4×Dic283C2211yescharacteristic, complemented
42C2×C4×Dic283C2211yescharacteristic, complemented
43C283⋊#95C2211yescharacteristic, complemented
44C283⋊#95C2211yescharacteristic, complemented
45C283⋊#95C2211yescharacteristic, complemented
46C283⋊#95C2211yescharacteristic, complemented
47C22×D4×C283C2211yescharacteristic
48C23×C4×C283C2211yescharacteristic, complemented
49C2×C22⋊C4×C283C2211yescharacteristic
50C2×C22⋊C4×C283C2211yescharacteristic
51C2×C22⋊C4×C283C2211yescharacteristic, complemented
52#95×C283C2211yescharacteristic, complemented
53#95×C283C2211yescharacteristic, complemented
54C23×D283—22no—
55C23×D283—22no—
56C23×D283—41no—
57C23×D283—41no—
58C23×D283—41no—
59C23×D283—41no—
60C23×D283C2×C414yescomplemented
61C23×D283C2311yescharacteristic
62C23×D283C2311yescharacteristic
63C2×C22⋊D283—22no—
64C2×C22⋊D283—22no—
65C2×C22⋊D283—41no—
66C2×C22⋊D283—41no—
67C2×C22⋊D283C2×C414yes—
68C2×C22⋊D283C2×C414yescomplemented
69C22×Dic283C2311yescharacteristic
70C22×Dic283C2311yescharacteristic
71C22×Dic283C2311yescharacteristic
72C22×Dic283C2311yescharacteristic
73C22×Dic283C2311yescharacteristic
74C22×Dic283C2311yescharacteristic
75C2×C4×D283—22no—
76C2×C4×D283—41no—
77C2×C4×D283—41no—
78C2×C4×D283—41no—
79C2×C4×D283—41no—
80C2×C4×D283D414yescomplemented
81C4⋊Dic283—41no—
82Dic283⋊C4—22no—
83Dic283⋊C4—41no—
84C22⋊Dic283—22no—
85C22⋊Dic283—22no—
86C22⋊Dic283—41no—
87C22⋊Dic283—41no—
88C22⋊Dic283C2×C414yescomplemented
89(C2×D283)⋊C4—22no—
90(C2×D283)⋊C4—22no—
91(C2×D283)⋊C4—22no—
92(C2×D283)⋊C4—22no—
93(C2×D283)⋊C4—41no—
94(C2×D283)⋊C4—41no—
95(C2×D283)⋊C4C2×C414yescomplemented
96(C2×D283)⋊C4C2×C414yescomplemented
97C4×Dic283—22no—
98C4×Dic283—41no—
99C4×Dic283—41no—
100C24×C283C2311yescharacteristic
101C24×C283C2311yescharacteristic
102C2×D4×C283—41no—
103C2×D4×C283—41no—
104C2×D4×C283C2×C414yescomplemented
105C22×C4×C283—22no—
106C22×C4×C283—41no—
107C22×C4×C283—41no—
108C22×C4×C283C2311yescharacteristic
109C22×C4×C283C2311yescharacteristic
110C22×C4×C283C2311yescharacteristic
111C22×C4×C283C2311yescharacteristic
112C22×C4×C283C2311yescharacteristic
113C22⋊C4×C283—22no—
114C22⋊C4×C283—41no—
115C22⋊C4×C283—41no—
116C22×D283—21no—
117C22×D283—21no—
118C22×D283—21no—
119C22×D283—21no—
120C22×D283—21no—
121C22×D283—21no—
122C22×D283—21no—
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—21no—
126C22×D283—28no—
127C22×D283—44no—
128C22×D283—44no—
129C22×D283—44no—
130C22×D283—44no—
131C22×D283—44no—
132C22×D283—44no—
133C22×D283C22×C412yes—
134C22×D283C22×C412yes—
135C22×D283C2×D412yes—
136C22×D283#4112yes—
137C22⋊D283—28no—
138C22⋊D283—44no—
139C2×Dic283—21no—
140C2×Dic283—21no—
141C2×Dic283—21no—
142C2×Dic283—21no—
143C2×Dic283—21no—
144C2×Dic283—21no—
145C2×Dic283—21no—
146C2×Dic283—21no—
147C2×Dic283—21no—
148C2×Dic283—21no—
149C2×Dic283—21no—
150C2×Dic283—21no—
151C2×Dic283—21no—
152C2×Dic283—21no—
153C2×Dic283—21no—
154C2×Dic283C22×C412yes—
155C2×Dic283C22×C412yes—
156C2×Dic283C2×D412yes—
157C4×D283—44no—
158C4×D283—44no—
159C23×C283—21no—
160C23×C283—21no—
161C23×C283—21no—
162C23×C283—21no—
163C23×C283—21no—
164C23×C283—21no—
165C23×C283—21no—
166C23×C283—21no—
167C23×C283—21no—
168C23×C283—21no—
169C23×C283—21no—
170C23×C283—21no—
171C23×C283C22×C412yes—
172C23×C283C22×C412yes—
173C23×C283C2411yescharacteristic
174D4×C283—44no—
175C2×C4×C283—21no—
176C2×C4×C283—21no—
177C2×C4×C283—21no—
178C2×C4×C283—21no—
179C2×C4×C283—21no—
180C2×C4×C283—21no—
181C2×C4×C283—21no—
182C2×C4×C283—21no—
183C2×C4×C283—21no—
184C2×C4×C283—21no—
185C2×C4×C283—21no—
186C2×C4×C283—21no—
187C2×C4×C283—21no—
188C2×C4×C283—44no—
189C2×C4×C283C22×C412yes—
190C2×C4×C283C2×D412yes—
191C22×C283—22no—
192C22×C283—22no—
193C22×C283—22no—
194C22×C283—22no—
195C22×C283—22no—
196C22×C283—22no—
197C22×C283—41no—
198C22×C283—41no—
199C22×C283—41no—
200C22×C283—41no—
201C22×C283—41no—
202C22×C283—41no—
203C22×C283—41no—
204C22×C283—41no—
205C22×C283C23×C411yescharacteristic
206C22×C283C22×D411yescharacteristic
207C22×C283C2×#4111yescharacteristic
208C22×C283H(5,𝔽2)11yescharacteristic
209C22×C283H(5,𝔽2)11yescharacteristic
210C22×C283H(5,𝔽2)11yescharacteristic
211C22×C283H(5,𝔽2)11yescharacteristic
212C2×D283—22no—
213C2×D283—22no—
214C2×D283—22no—
215C2×D283—22no—
216C2×D283—22no—
217C2×D283—22no—
218C2×D283—22no—
219C2×D283—22no—
220C2×D283—22no—
221C2×D283—416no—
222C2×D283—41no—
223C2×D283—41no—
224C2×D283—41no—
225C2×D283—41no—
226C2×D283C4×D414yes—
227C4×C283—22no—
228C4×C283—22no—
229C4×C283—41no—
230C4×C283—41no—
231C4×C283—41no—
232Dic283—22no—
233Dic283—22no—
234Dic283—22no—
235Dic283—41no—
236Dic283—41no—
237Dic283—41no—
238C2×C283—42no—
239C2×C283—42no—
240C2×C283C2×C4×D411yescharacteristic
241C2×C283#51811yescharacteristic
242C2×C283#51811yescharacteristic
243C2×C283#53411yescharacteristic
244C2×C283#53511yescharacteristic
245C2×C283#53811yescharacteristic
246C2×C283#53811yescharacteristic
247D283—24no—
248D283—42no—
249C283#232111yescharacteristic, complemented, Sylow
250#2321—2831nomaximal, Sylow
251C2×#120—2831no—
252C2×C4×D4—2831no—
253C22×C22⋊C4—2831no—
254#390—2831no—
255#390—2831no—
256#388—2831no—
257#388—2831no—
258#388—2831no—
259#388—2831no—
260#386—2831no—
261#386—2831no—
262#386D28311yescharacteristic, complemented
263#385—2831no—
264#379—2831no—
265C4×C22⋊C4—2831no—
266C25—2831no—
267C22×D4—2831no—
268C22×D4—2831no—
269C22×D4C2×D28311yescharacteristic
270C23×C4—2831no—
271C23×C4—2831no—
272C23×C4C2×D28311yescharacteristic, complemented
273#120—2838no—
274C4×D4—5664no—
275C2×C4⋊C4—2831no—
276C2×C4⋊C4—2831no—
277C2×C4⋊C4—2831no—
278C2×C22⋊C4—2831no—
279C2×C22⋊C4—2831no—
280C2×C22⋊C4—2831no—
281C2×C22⋊C4—2831no—
282C2×C22⋊C4—2831no—
283C2×C22⋊C4—2831no—
284C2×C22⋊C4—2831no—
285C2×C22⋊C4—2831no—
286C2×C22⋊C4—2831no—
287C2×C22⋊C4—2831no—
288C2×C22⋊C4—2831no—
289C2×C22⋊C4—2831no—
290C2×C22⋊C4—2831no—
291C2×C22⋊C4—5664no—
292C2×C22⋊C4C2×D28311yescharacteristic
293C2×C22⋊C4C2×D28311yescharacteristic
294C2×C22⋊C4C2×D28311yescharacteristic, complemented
295C2×C42—2831no—
296C2×C42—2831no—
297C2×C42—2831no—
298#95—2831no—
299#95—2831no—
300#95—2831no—
301#95—2831no—
302#95C2×D28311yescharacteristic, complemented
303#95C2×D28311yescharacteristic, complemented
304C24—2831no—
305C24—2831no—
306C24—2834no—
307C24—5662no—
308C24—5662no—
309C24—11321no—
310C24—11321no—
311C24—11321no—
312C24—11321no—
313C24C22×D28311yescharacteristic
314C24C22×D28311yescharacteristic
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—2834no—
318C2×D4—2834no—
319C2×D4—5662no—
320C2×D4—5662no—
321C2×D4—11321no—
322C2×D4—11321no—
323C2×D4C4×D28314yescomplemented
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—2831no—
328C22×C4—2831no—
329C22×C4—2831no—
330C22×C4—2834no—
331C22×C4—2831no—
332C22×C4—2831no—
333C22×C4—2831no—
334C22×C4—5662no—
335C22×C4—11321no—
336C22×C4—11321no—
337C22×C4—11321no—
338C22×C4—11321no—
339C22×C4C22×D28311yescharacteristic
340C22×C4C22×D28311yescharacteristic
341C22×C4C22×D28311yescharacteristic
342C22×C4C22×D28311yescharacteristic
343C22×C4C22×D28311yescharacteristic
344C4⋊C4—5662no—
345C4⋊C4—11321no—
346C4⋊C4—11321no—
347C22⋊C4—22no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—2834no—
351C22⋊C4—2834no—
352C22⋊C4—2834no—
353C22⋊C4—5662no—
354C22⋊C4—5662no—
355C22⋊C4—5662no—
356C22⋊C4—5662no—
357C22⋊C4—5662no—
358C22⋊C4—5662no—
359C22⋊C4—11321no—
360C22⋊C4—11321no—
361C22⋊C4—11321no—
362C22⋊C4—11321no—
363C42—5662no—
364C42—11321no—
365C42—11321no—
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—2832no—
379C23—2832no—
380C23—2832no—
381C23—2832no—
382C23—5668no—
383C23—5661no—
384C23—5661no—
385C23—5661no—
386C23—5661no—
387C23—5661no—
388C23—5661no—
389C23—5661no—
390C23—5661no—
391C23—5661no—
392C23—5661no—
393C23—11324no—
394C23—11324no—
395C23—11324no—
396C23—11324no—
397C23—11324no—
398C23—11324no—
399C23C2×C4×D28312yes—
400C23C2×C4×D28312yes—
401C23C23×D28311yescharacteristic
402D4—44no—
403D4—5668no—
404D4—11324no—
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—44no—
419C2×C4—2832no—
420C2×C4—2832no—
421C2×C4—2832no—
422C2×C4—5661no—
423C2×C4—5661no—
424C2×C4—5661no—
425C2×C4—5661no—
426C2×C4—5661no—
427C2×C4—5661no—
428C2×C4—5661no—
429C2×C4—5661no—
430C2×C4—5661no—
431C2×C4—5661no—
432C2×C4—5661no—
433C2×C4—5661no—
434C2×C4—5661no—
435C2×C4—5661no—
436C2×C4—5661no—
437C2×C4—11324no—
438C2×C4—11324no—
439C2×C4C2×C4×D28312yes—
440C2×C4D4×D28312yes—
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—2834no—
456C22—5662no—
457C22—5662no—
458C22—5662no—
459C22—5662no—
460C22—5662no—
461C22—5662no—
462C22—5662no—
463C22—5662no—
464C22—5662no—
465C22—113216no—
466C22—11321no—
467C22—11321no—
468C22—11321no—
469C22—11321no—
470C22C22×C4×D28311yescharacteristic
471C22C2×D4×D28311yescharacteristic
472C22#41×D28311yescharacteristic
473C22#2258365511yescharacteristic
474C22#2258365511yescharacteristic
475C22#2258365511yescharacteristic
476C22#2258365611yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—5662no—
483C4—5662no—
484C4—5662no—
485C4—11321no—
486C4—11321no—
487C4—11321no—
488C2—42no—
489C2—42no—
490C2—5664no—
491C2—11322no—
492C2C4×D4×D28311yescharacteristic
493C2#3182693711yescharacteristic
494C2#3182693811yescharacteristic
495C2#3182702511yescharacteristic
496C2#3182703111yescharacteristic
497C2#3182704711yescharacteristic
498C2#3182705111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#386×C283C2
socle, cosocleC23×C283C24
prime radicalC2×#22583578C2
hypercenter#386D283
hypocenterC283#2321
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D283C2semidirect
3. C2×C22⋊C4×D283C2semidirect
4. #31826368C2semidirect
5. #31826365C2semidirect
6. #31826357C2semidirect
7. #31826354C2semidirect
8. #31826353C2semidirect
9. #31826353C2semidirect
10. #31826344C2semidirect
11. #31826341C2semidirect
12. #31826339C2semidirect
13. #31826320C2semidirect
14. #31826312C2semidirect
15. #386×C283C2semidirect
21. #22583578C4semidirect
23. C2×C4⋊Dic283C22semidirect
24. C2×Dic283⋊C4C22semidirect
25. C2×Dic283⋊C4C22semidirect
29. C2×C22⋊Dic283C22semidirect
30. C2×C22⋊Dic283C22semidirect
40. C2×C4×Dic283C22semidirect
41. C2×C4×Dic283C22semidirect
42. C2×C4×Dic283C22semidirect
43. C283⋊#95C22semidirect
44. C283⋊#95C22semidirect
45. C283⋊#95C22semidirect
46. C283⋊#95C22semidirect
48. C23×C4×C283C22semidirect
51. C2×C22⋊C4×C283C22semidirect
52. #95×C283C22semidirect
53. #95×C283C22semidirect
60. C23×D283C2×C4semidirect
68. C2×C22⋊D283C2×C4semidirect
80. C2×C4×D283D4semidirect
88. C22⋊Dic283C2×C4semidirect
95. (C2×D283)⋊C4C2×C4semidirect
96. (C2×D283)⋊C4C2×C4semidirect
104. C2×D4×C283C2×C4semidirect
249. C283#2321semidirect
262. #386D283semidirect
272. C23×C4C2×D283semidirect
294. C2×C22⋊C4C2×D283semidirect
302. #95C2×D283semidirect
303. #95C2×D283semidirect
323. C2×D4C4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835782
C2×C4×C22⋊D2832
C2×C22⋊C4×D2832
#318263682
#318263652
#318263572
#318263542
#318263532
#318263442
#318263412
#318263392
#318263202
#318263122
#386×C2832
#2321283