Finite Group Databasebeta

← #49124406

Subgroups — #49124406

Order 36224.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583595C211yesmaximal, characteristic
2#31826366C211yesmaximal, characteristic
3#31826360C211yesmaximal, characteristic, complemented
4#31826357C211yesmaximal, characteristic, complemented
5#31826356C211yesmaximal, characteristic, complemented
6#31826356C211yesmaximal, characteristic, complemented
7#31826355C211yesmaximal, characteristic, complemented
8#31826351C211yesmaximal, characteristic, complemented
9#31826347C211yesmaximal, characteristic, complemented
10#31826343C211yesmaximal, characteristic
11#31826336C211yesmaximal, characteristic, complemented
12#31826336C211yesmaximal, characteristic, complemented
13#31826324C211yesmaximal, characteristic, complemented
14#31826322C211yesmaximal, characteristic, complemented
15#389×C283C211yesmaximal, characteristic, complemented
16C22×D1132C2211yescharacteristic
17C22×C4×D283C2211yescharacteristic
18C22×C4×D283C2211yescharacteristic
19#22583595C418yescomplemented
20C2×C4⋊Dic283C2211yescharacteristic
21C2×Dic283⋊C4C2211yescharacteristic
22C2×Dic283⋊C4C2211yescharacteristic
23C2×Dic283⋊C4C2211yescharacteristic
24C2×Dic283⋊C4C2211yescharacteristic
25C2×(C2×D283)⋊C4C2211yescharacteristic
26C2×(C2×D283)⋊C4C2211yescharacteristic
27C2×(C2×D283)⋊C4C2211yescharacteristic
28C2×(C2×D283)⋊C4C2211yescharacteristic
29C2×(C2×D283)⋊C4C2211yescharacteristic
30C2×(C2×D283)⋊C4C2211yescharacteristic
31C2×(C2×D283)⋊C4C2211yescharacteristic
32C2×(C2×D283)⋊C4C2211yescharacteristic
33C2×(C2×D283)⋊C4C2211yescharacteristic
34C2×(C2×D283)⋊C4C2211yescharacteristic
35C2×C4×Dic283C2211yescharacteristic
36C2×C4×Dic283C2211yescharacteristic
37C283⋊#95C2211yescharacteristic
38C283⋊#95C2211yescharacteristic
39C283⋊#95C2211yescharacteristic
40C283⋊#95C2211yescharacteristic
41C283⋊#95C2211yescharacteristic
42C283⋊#95C2211yescharacteristic
43C283⋊#95C2211yescharacteristic
44C283⋊#95C2211yescharacteristic
45C2×C4⋊C4×C283C2211yescharacteristic
46C2×C4⋊C4×C283C2211yescharacteristic
47C2×C4⋊C4×C283C2211yescharacteristic
48C2×C4⋊C4×C283C2211yescharacteristic
49C2×C42×C283C2211yescharacteristic
50#95×C283C2211yescharacteristic
51#95×C283C2211yescharacteristic
52C23×D283C2311yescharacteristic
53C23×D283C2311yescharacteristic
54C2×D1132—41no—
55C2×D1132—41no—
56C2×D1132C2×C414yes—
57C22×Dic283C2311yescharacteristic
58C22×Dic283C2311yescharacteristic
59C22×Dic283C2311yescharacteristic
60C22×Dic283C2311yescharacteristic
61C22×Dic283C2311yescharacteristic
62C22×Dic283C2311yescharacteristic
63C2×C4×D283—41no—
64C2×C4×D283—41no—
65C2×C4×D283—41no—
66C2×C4×D283—41no—
67C2×C4×D283—41no—
68C2×C4×D283C2×C414yescomplemented
69C4⋊Dic283—41no—
70Dic283⋊C4—41no—
71Dic283⋊C4—41no—
72Dic283⋊C4—41no—
73Dic283⋊C4C2×C414yescomplemented
74(C2×D283)⋊C4—41no—
75(C2×D283)⋊C4—41no—
76(C2×D283)⋊C4—41no—
77(C2×D283)⋊C4—41no—
78(C2×D283)⋊C4—41no—
79(C2×D283)⋊C4—41no—
80(C2×D283)⋊C4—41no—
81(C2×D283)⋊C4C2×C414yes—
82(C2×D283)⋊C4C2×C414yes—
83(C2×D283)⋊C4C2×C414yescomplemented
84C4×Dic283—41no—
85C4×Dic283—41no—
86C22×C4×C283C2311yescharacteristic
87C22×C4×C283C2311yescharacteristic
88C22×C4×C283C2311yescharacteristic
89C22×C4×C283C2311yescharacteristic
90C22×C4×C283C2311yescharacteristic
91C22×C4×C283C2311yescharacteristic
92C22×C4×C283C2311yescharacteristic
93C4⋊C4×C283—41no—
94C4⋊C4×C283—41no—
95C4⋊C4×C283—41no—
96C4⋊C4×C283C2×C414yescomplemented
97C42×C283—41no—
98C22×D283—21no—
99C22×D283—21no—
100C22×D283—21no—
101C22×D283—21no—
102C22×D283—21no—
103C22×D283—21no—
104C22×D283—21no—
105C22×D283—21no—
106C22×D283—21no—
107C22×D283—21no—
108C22×D283—21no—
109C22×D283—21no—
110C22×D283C22×C412yes—
111C22×D283C22×C412yes—
112D1132—44no—
113C2×Dic283—21no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283—21no—
117C2×Dic283—21no—
118C2×Dic283—21no—
119C2×Dic283—21no—
120C2×Dic283—21no—
121C2×Dic283—21no—
122C2×Dic283—21no—
123C2×Dic283—21no—
124C2×Dic283—21no—
125C2×Dic283—21no—
126C2×Dic283—21no—
127C2×Dic283—21no—
128C2×Dic283—21no—
129C2×Dic283C22×C412yes—
130C2×Dic283C22×C412yes—
131C4×D283—44no—
132C4×D283—44no—
133C23×C283C2411yescharacteristic
134C2×C4×C283—21no—
135C2×C4×C283—21no—
136C2×C4×C283—21no—
137C2×C4×C283—21no—
138C2×C4×C283—21no—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283—21no—
149C2×C4×C283—21no—
150C2×C4×C283—21no—
151C2×C4×C283—21no—
152C2×C4×C283C22×C412yes—
153C2×C4×C283C22×C412yes—
154C2×C4×C283C22×C412yes—
155C22×C283C23×C411yescharacteristic
156C22×C283H(5,𝔽2)11yescharacteristic
157C22×C283H(5,𝔽2)11yescharacteristic
158C22×C283H(5,𝔽2)11yescharacteristic
159C22×C283H(5,𝔽2)11yescharacteristic
160C22×C2832−1+411yescharacteristic
161C22×C2832−1+411yescharacteristic
162C2×D283—22no—
163C2×D283—22no—
164C2×D283—22no—
165C2×D283—22no—
166C2×D283—22no—
167C2×D283—22no—
168C2×D283—41no—
169C2×D283—41no—
170C2×D283—41no—
171C2×D283—41no—
172C2×D283—41no—
173C2×D283—41no—
174C2×D283—41no—
175C2×D283—41no—
176C4×C283—22no—
177C4×C283—22no—
178C4×C283—22no—
179C4×C283—41no—
180C4×C283—41no—
181C4×C283—41no—
182C4×C283—41no—
183Dic283—22no—
184Dic283—22no—
185Dic283—41no—
186Dic283—41no—
187Dic283—41no—
188Dic283—41no—
189C2×C283#51811yescharacteristic
190C2×C283#52011yescharacteristic
191C2×C283#52011yescharacteristic
192C2×C283#56011yescharacteristic
193C2×C283#56211yescharacteristic
194C2×C283#56211yescharacteristic
195C2×C283#56311yescharacteristic
196D283—42no—
197D283—42no—
198C283#237611yescharacteristic, complemented, Sylow
199#2376—2831nomaximal, Sylow
200C2×#123—2831no—
201#390—2831no—
202#389—2831no—
203#389D28311yescharacteristic, complemented
204#388—2831no—
205#388—2831no—
206#388—2831no—
207#388—2831no—
208#387—2831no—
209#387—2831no—
210#386—2831no—
211#385—2831no—
212#385—2831no—
213#380—2831no—
214#380—2831no—
215C22×D4—2831no—
216C23×C4—2831no—
217C23×C4—2831no—
218#123—2838no—
219C2×C4⋊C4—2831no—
220C2×C4⋊C4—2831no—
221C2×C4⋊C4—2831no—
222C2×C4⋊C4—2831no—
223C2×C4⋊C4—2831no—
224C2×C4⋊C4C2×D28311yescharacteristic
225C2×C4⋊C4C2×D28311yescharacteristic
226C2×C4⋊C4C2×D28311yescharacteristic
227C2×C4⋊C4C2×D28311yescharacteristic
228C2×C22⋊C4—2831no—
229C2×C22⋊C4—2831no—
230C2×C22⋊C4—2831no—
231C2×C22⋊C4—2831no—
232C2×C22⋊C4—2831no—
233C2×C22⋊C4—2831no—
234C2×C22⋊C4—2831no—
235C2×C22⋊C4—2831no—
236C2×C22⋊C4—2831no—
237C2×C22⋊C4—2831no—
238C2×C42—2831no—
239C2×C42—2831no—
240C2×C42C2×D28311yescharacteristic
241#95—2831no—
242#95—2831no—
243#95—2831no—
244#95—2831no—
245#95—2831no—
246#95—2831no—
247#95—2831no—
248#95—2831no—
249#95C2×D28311yescharacteristic
250#95C2×D28311yescharacteristic
251C24—2831no—
252C24—2831no—
253C2×D4—2834no—
254C2×D4—11321no—
255C2×D4—11321no—
256C22×C4—2831no—
257C22×C4—2831no—
258C22×C4—2834no—
259C22×C4—2831no—
260C22×C4—2831no—
261C22×C4—2831no—
262C22×C4—2831no—
263C22×C4—11321no—
264C22×C4—11321no—
265C22×C4—11321no—
266C22×C4—11321no—
267C22×C4—11321no—
268C22×C4C22×D28311yescharacteristic
269C22×C4C22×D28311yescharacteristic
270C22×C4C22×D28311yescharacteristic
271C22×C4C22×D28311yescharacteristic
272C22×C4C22×D28311yescharacteristic
273C22×C4C22×D28311yescharacteristic
274C22×C4C22×D28311yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—2834no—
279C4⋊C4—11321no—
280C4⋊C4—11321no—
281C4⋊C4—11321no—
282C4⋊C4—11321no—
283C4⋊C4C4×D28314yescomplemented
284C22⋊C4—2834no—
285C22⋊C4—2834no—
286C22⋊C4—2834no—
287C22⋊C4—11321no—
288C22⋊C4—11321no—
289C22⋊C4—11321no—
290C22⋊C4—11321no—
291C22⋊C4—11321no—
292C22⋊C4—11321no—
293C22⋊C4—11321no—
294C42—41no—
295C42—11321no—
296C42—11321no—
297C23—2832no—
298C23—2832no—
299C23—5661no—
300C23—5661no—
301C23—5661no—
302C23—5661no—
303C23—5661no—
304C23—5661no—
305C23—5661no—
306C23—5661no—
307C23—5661no—
308C23—5661no—
309C23—5661no—
310C23—5661no—
311C23C23×D28311yescharacteristic
312D4—11324no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—2832no—
332C2×C4—2832no—
333C2×C4—5661no—
334C2×C4—5661no—
335C2×C4—5661no—
336C2×C4—5661no—
337C2×C4—5661no—
338C2×C4—5661no—
339C2×C4—5661no—
340C2×C4—5661no—
341C2×C4—5661no—
342C2×C4—5661no—
343C2×C4—5661no—
344C2×C4—5661no—
345C2×C4—5661no—
346C2×C4—5661no—
347C2×C4—5661no—
348C2×C4—5661no—
349C2×C4—11324no—
350C2×C4—11324no—
351C2×C4C2×C4×D28312yes—
352C2×C4C2×C4×D28312yes—
353C2×C4C2×C4×D28312yes—
354C22—5662no—
355C22—5662no—
356C22—5662no—
357C22—5662no—
358C22—5662no—
359C22—5662no—
360C22—11321no—
361C22—11321no—
362C22—11321no—
363C22—11321no—
364C22—11321no—
365C22—11321no—
366C22—11321no—
367C22—11321no—
368C22C22×C4×D28311yescharacteristic
369C22#2258365511yescharacteristic
370C22#2258365511yescharacteristic
371C22#2258365611yescharacteristic
372C22#2258365611yescharacteristic
373C22#2258365811yescharacteristic
374C22#2258365811yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—5662no—
383C4—5662no—
384C4—11321no—
385C4—11321no—
386C4—11321no—
387C4—11321no—
388C2—11322no—
389C2—11322no—
390C2#3182693711yescharacteristic
391C2#3182694611yescharacteristic
392C2#3182694711yescharacteristic
393C2#3182720311yescharacteristic
394C2#3182721111yescharacteristic
395C2#3182721311yescharacteristic
396C2#3182721411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#389×C283C2
socle, cosocleC23×C283C24
prime radicalC22×D1132C22
hypercenter#389D283
hypocenterC283#2376
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #31826360C2semidirect
4. #31826357C2semidirect
5. #31826356C2semidirect
6. #31826356C2semidirect
7. #31826355C2semidirect
8. #31826351C2semidirect
9. #31826347C2semidirect
11. #31826336C2semidirect
12. #31826336C2semidirect
13. #31826324C2semidirect
14. #31826322C2semidirect
15. #389×C283C2semidirect
19. #22583595C4semidirect
68. C2×C4×D283C2×C4semidirect
73. Dic283⋊C4C2×C4semidirect
83. (C2×D283)⋊C4C2×C4semidirect
96. C4⋊C4×C283C2×C4semidirect
198. C283#2376semidirect
203. #389D283semidirect
283. C4⋊C4C4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835952
#318263662
#318263602
#318263572
#318263562
#318263552
#318263512
#318263472
#318263432
#318263362
#318263242
#318263222
#389×C2832
#2376283