Finite Group Databasebeta

← #49123960

Subgroups — #49123960

Order 36224.

696 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583585C211yesmaximal, characteristic
2C2×C4×D1132C211yesmaximal, characteristic, complemented
3C2×#22583572C211yesmaximal, characteristic, complemented
4#31826356C211yesmaximal, characteristic, complemented
5#31826354C211yesmaximal, characteristic, complemented
6#31826351C211yesmaximal, characteristic, complemented
7#31826343C211yesmaximal, characteristic, complemented
8#31826343C211yesmaximal, characteristic, complemented
9#31826343C211yesmaximal, characteristic, complemented
10#31826343C211yesmaximal, characteristic, complemented
11#31826339C211yesmaximal, characteristic, complemented
12#31826336C211yesmaximal, characteristic, complemented
13#31826324C211yesmaximal, characteristic, complemented
14#95×D283C211yesmaximal, characteristic, complemented
15#387×C283C211yesmaximal, characteristic, complemented
16C22×D1132C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18C22×D1132C2211yescharacteristic
19C22×C4×D283C2211yescharacteristic
20C22×C4×D283C2211yescharacteristic
21C22×C4×D283C2211yescharacteristic
22C22×C4×D283C2211yescharacteristic
23C22×C4×D283C2211yescharacteristic
24#22583585C418yescomplemented
25C4×D1132—24no—
26#22583572—24no—
27C2×C4⋊Dic283C2211yescharacteristic
28C2×Dic283⋊C4C2211yescharacteristic
29C2×Dic283⋊C4C2211yescharacteristic
30C2×(C2×D283)⋊C4C2211yescharacteristic
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
39C2×(C2×D283)⋊C4C2211yescharacteristic
40C2×C4×Dic283C2211yescharacteristic
41C283⋊#95C2211yescharacteristic
42C283⋊#95C2211yescharacteristic
43C283⋊#95C2211yescharacteristic
44C283⋊#95C2211yescharacteristic
45C283⋊#95C2211yescharacteristic
46C283⋊#95C2211yescharacteristic
47C2×C4⋊C4×C283C2211yescharacteristic
48C2×C4⋊C4×C283C2211yescharacteristic
49C2×C42×C283C2211yescharacteristic
50#95×C283C2211yescharacteristic
51#95×C283C2211yescharacteristic
52#95×C283C2211yescharacteristic
53#95×C283C2211yescharacteristic
54C23×D283C2311yescharacteristic
55C23×D283C2311yescharacteristic
56C23×D283C2311yescharacteristic
57C2×D1132—22no—
58C2×D1132—22no—
59C2×D1132—22no—
60C2×D1132—22no—
61C2×D1132—41no—
62C2×D1132—41no—
63C2×D1132C2×C414yescomplemented
64C2×D1132C2×C414yescomplemented
65C2×D1132C2×C414yescomplemented
66C22×Dic283C2311yescharacteristic
67C22×Dic283C2311yescharacteristic
68C22×Dic283C2311yescharacteristic
69C22×Dic283C2311yescharacteristic
70C22×Dic283C2311yescharacteristic
71C2×C4×D283—22no—
72C2×C4×D283—22no—
73C2×C4×D283—22no—
74C2×C4×D283—41no—
75C2×C4×D283—41no—
76C2×C4×D283—41no—
77C2×C4×D283—41no—
78C2×C4×D283—41no—
79C2×C4×D283—41no—
80C2×C4×D283—41no—
81C2×C4×D283—41no—
82C2×C4×D283—41no—
83C2×C4×D283—41no—
84C2×C4×D283C2×C414yescomplemented
85C2×C4×D283D414yescomplemented
86C4⋊Dic283—22no—
87Dic283⋊C4—41no—
88Dic283⋊C4—41no—
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)⋊C4—41no—
96(C2×D283)⋊C4—41no—
97(C2×D283)⋊C4C2×C414yescomplemented
98(C2×D283)⋊C4C2×C414yescomplemented
99C4×Dic283—22no—
100C22×C4×C283C2311yescharacteristic
101C22×C4×C283C2311yescharacteristic
102C22×C4×C283C2311yescharacteristic
103C22×C4×C283C2311yescharacteristic
104C22×C4×C283C2311yescharacteristic
105C22×C4×C283C2311yescharacteristic
106C22×C4×C283C2311yescharacteristic
107C4⋊C4×C283—22no—
108C4⋊C4×C283C2×C414yescomplemented
109C42×C283—22no—
110C22×D283—21no—
111C22×D283—21no—
112C22×D283—21no—
113C22×D283—21no—
114C22×D283—21no—
115C22×D283—21no—
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×D283C22×C412yes—
127C22×D283C22×C412yes—
128C22×D283C22×C412yes—
129C22×D283C2×D412yes—
130C22×D283#4112yes—
131D1132—28no—
132D1132—28no—
133D1132—44no—
134C2×Dic283—21no—
135C2×Dic283—21no—
136C2×Dic283—21no—
137C2×Dic283—21no—
138C2×Dic283—21no—
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×Dic283C22×C412yes—
148C2×Dic283C2×D412yes—
149C4×D283—44no—
150C4×D283—44no—
151C4×D283—44no—
152C4×D283—44no—
153C4×D283—44no—
154C23×C283C2411yescharacteristic
155C2×C4×C283—21no—
156C2×C4×C283—21no—
157C2×C4×C283—21no—
158C2×C4×C283—21no—
159C2×C4×C283—21no—
160C2×C4×C283—21no—
161C2×C4×C283—21no—
162C2×C4×C283—21no—
163C2×C4×C283—21no—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—21no—
167C2×C4×C283—21no—
168C2×C4×C283—21no—
169C2×C4×C283—21no—
170C2×C4×C283—21no—
171C2×C4×C283—21no—
172C2×C4×C283C22×C412yes—
173C2×C4×C283C22×C412yes—
174C2×C4×C283C22×C412yes—
175C2×C4×C283C2×D412yes—
176C22×C283C23×C411yescharacteristic
177C22×C283C22×D411yescharacteristic
178C22×C283C2×#4111yescharacteristic
179C22×C283H(5,𝔽2)11yescharacteristic
180C22×C283H(5,𝔽2)11yescharacteristic
181C22×C283H(5,𝔽2)11yescharacteristic
182C22×C2832−1+411yescharacteristic
183C2×D283—22no—
184C2×D283—22no—
185C2×D283—22no—
186C2×D283—22no—
187C2×D283—22no—
188C2×D283—22no—
189C2×D283—22no—
190C2×D283—22no—
191C2×D283—22no—
192C2×D283—22no—
193C2×D283—22no—
194C2×D283—22no—
195C2×D283—41no—
196C2×D283—41no—
197C2×D283—41no—
198C2×D283—41no—
199C2×D283—41no—
200C2×D283—41no—
201C2×D283—41no—
202C2×D283—41no—
203C2×D283C4×D414yes—
204C4×C283—22no—
205C4×C283—22no—
206C4×C283—22no—
207C4×C283—22no—
208C4×C283—41no—
209C4×C283—41no—
210C4×C283—41no—
211Dic283—22no—
212Dic283—22no—
213Dic283—41no—
214Dic283—41no—
215Dic283—41no—
216C2×C283C2×C4×D411yescharacteristic
217C2×C283#51811yescharacteristic
218C2×C283#52011yescharacteristic
219C2×C283#53411yescharacteristic
220C2×C283#53711yescharacteristic
221C2×C283#53811yescharacteristic
222C2×C283#53911yescharacteristic
223D283—24no—
224D283—42no—
225D283—42no—
226C283#232511yescharacteristic, complemented, Sylow
227#2325—2831nomaximal, Sylow
228C2×#121—2831no—
229C2×C4×D4—2831no—
230C2×C4×D4—2831no—
231#388—2831no—
232#388—2831no—
233#387—2831no—
234#387D28311yescharacteristic, complemented
235#386—2831no—
236#386—2831no—
237#386—2831no—
238#386—2831no—
239#385—2831no—
240#385—2831no—
241#380—2831no—
242C2×#95—2831no—
243C22×D4—2831no—
244C22×D4—2831no—
245C22×D4—2831no—
246C23×C4—2831no—
247C23×C4—2831no—
248C23×C4—2831no—
249C23×C4—2831no—
250C23×C4—2831no—
251#121—2838no—
252C4×D4—5664no—
253C4×D4—5664no—
254C2×C4⋊C4—2831no—
255C2×C4⋊C4—2831no—
256C2×C4⋊C4—2831no—
257C2×C4⋊C4C2×D28311yescharacteristic
258C2×C4⋊C4C2×D28311yescharacteristic
259C2×C22⋊C4—2831no—
260C2×C22⋊C4—2831no—
261C2×C22⋊C4—2831no—
262C2×C22⋊C4—2831no—
263C2×C22⋊C4—2831no—
264C2×C22⋊C4—2831no—
265C2×C22⋊C4—2831no—
266C2×C22⋊C4—2831no—
267C2×C22⋊C4—2831no—
268C2×C22⋊C4—2831no—
269C2×C42—2831no—
270C2×C42C2×D28311yescharacteristic
271#95—2831no—
272#95—2831no—
273#95—2831no—
274#95—2831no—
275#95—2831no—
276#95—2831no—
277#95C2×D28311yescharacteristic
278#95C2×D28311yescharacteristic
279#95C2×D28311yescharacteristic
280#95C2×D28311yescharacteristic
281C24—2831no—
282C24—2831no—
283C24—2831no—
284C2×D4—2834no—
285C2×D4—2834no—
286C2×D4—2834no—
287C2×D4—5662no—
288C2×D4—5662no—
289C2×D4—5662no—
290C2×D4—5662no—
291C2×D4—11321no—
292C2×D4—11321no—
293C22×C4—2831no—
294C22×C4—2831no—
295C22×C4—2834no—
296C22×C4—2834no—
297C22×C4—2831no—
298C22×C4—2831no—
299C22×C4—2831no—
300C22×C4—5662no—
301C22×C4—5662no—
302C22×C4—5662no—
303C22×C4—11321no—
304C22×C4—11321no—
305C22×C4—11321no—
306C22×C4—11321no—
307C22×C4—11321no—
308C22×C4—11321no—
309C22×C4—11321no—
310C22×C4—11321no—
311C22×C4—11321no—
312C22×C4—11321no—
313C22×C4C22×D28311yescharacteristic
314C22×C4C22×D28311yescharacteristic
315C22×C4C22×D28311yescharacteristic
316C22×C4C22×D28311yescharacteristic
317C22×C4C22×D28311yescharacteristic
318C22×C4C22×D28311yescharacteristic
319C22×C4C22×D28311yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—5662no—
322C4⋊C4—11321no—
323C4⋊C4—11321no—
324C4⋊C4C4×D28314yescomplemented
325C22⋊C4—2834no—
326C22⋊C4—2834no—
327C22⋊C4—5662no—
328C22⋊C4—5662no—
329C22⋊C4—5662no—
330C22⋊C4—5662no—
331C22⋊C4—11321no—
332C22⋊C4—11321no—
333C22⋊C4—11321no—
334C22⋊C4—11321no—
335C42—22no—
336C42—5662no—
337C23—2832no—
338C23—2832no—
339C23—2832no—
340C23—2832no—
341C23—2832no—
342C23—5661no—
343C23—5661no—
344C23—5661no—
345C23—5661no—
346C23—5661no—
347C23—5661no—
348C23—5661no—
349C23—5661no—
350C23—5661no—
351C23—5661no—
352C23—5661no—
353C23—5661no—
354C23—5661no—
355C23—5661no—
356C23—5661no—
357C23—5661no—
358C23C23×D28311yescharacteristic
359D4—5668no—
360D4—5668no—
361D4—11324no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—2832no—
380C2×C4—2832no—
381C2×C4—5661no—
382C2×C4—5661no—
383C2×C4—5661no—
384C2×C4—5661no—
385C2×C4—5661no—
386C2×C4—5661no—
387C2×C4—5661no—
388C2×C4—5661no—
389C2×C4—5661no—
390C2×C4—5661no—
391C2×C4—5661no—
392C2×C4—5661no—
393C2×C4—5661no—
394C2×C4—11324no—
395C2×C4—11324no—
396C2×C4—11324no—
397C2×C4—11324no—
398C2×C4—11324no—
399C2×C4C2×C4×D28312yes—
400C2×C4C2×C4×D28312yes—
401C2×C4C2×C4×D28312yes—
402C2×C4D4×D28312yes—
403C22—2834no—
404C22—5662no—
405C22—5662no—
406C22—5662no—
407C22—5662no—
408C22—5662no—
409C22—5662no—
410C22—5662no—
411C22—5662no—
412C22—5662no—
413C22—5662no—
414C22—5662no—
415C22—5662no—
416C22—11321no—
417C22—11321no—
418C22—11321no—
419C22—11321no—
420C22—11321no—
421C22—11321no—
422C22—11321no—
423C22—11321no—
424C22C22×C4×D28311yescharacteristic
425C22C2×D4×D28311yescharacteristic
426C22#41×D28311yescharacteristic
427C22#2258365611yescharacteristic
428C22#2258365611yescharacteristic
429C22#2258365611yescharacteristic
430C22#2258365811yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—5662no—
439C4—5662no—
440C4—11321no—
441C4—11321no—
442C4—11321no—
443C2—5664no—
444C2—11322no—
445C2—11322no—
446C2C4×D4×D28311yescharacteristic
447C2#3182693911yescharacteristic
448C2#3182694711yescharacteristic
449C2#3182702611yescharacteristic
450C2#3182704511yescharacteristic
451C2#3182705411yescharacteristic
452C2#3182706311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#387×C283C2
socle, cosocleC23×C283C24
prime radicalC2×#22583585C2
hypercenter#387D283
hypocenterC283#2325
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×D1132C2semidirect
3. C2×#22583572C2semidirect
4. #31826356C2semidirect
5. #31826354C2semidirect
6. #31826351C2semidirect
7. #31826343C2semidirect
8. #31826343C2semidirect
9. #31826343C2semidirect
10. #31826343C2semidirect
11. #31826339C2semidirect
12. #31826336C2semidirect
13. #31826324C2semidirect
14. #95×D283C2semidirect
15. #387×C283C2semidirect
24. #22583585C4semidirect
63. C2×D1132C2×C4semidirect
64. C2×D1132C2×C4semidirect
65. C2×D1132C2×C4semidirect
84. C2×C4×D283C2×C4semidirect
85. C2×C4×D283D4semidirect
97. (C2×D283)⋊C4C2×C4semidirect
98. (C2×D283)⋊C4C2×C4semidirect
108. C4⋊C4×C283C2×C4semidirect
226. C283#2325semidirect
234. #387D283semidirect
324. C4⋊C4C4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835852
C2×C4×D11322
C2×#225835722
#318263562
#318263542
#318263512
#318263432
#318263392
#318263362
#318263242
#95×D2832
#387×C2832
#2325283