Finite Group Databasebeta

← #27173991

Subgroups — #27173991

Order 12928.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692213C211yesmaximal, characteristic
2#20265063C211yesmaximal, characteristic, complemented
3#20265062C211yesmaximal, characteristic, complemented
4#20265060C211yesmaximal, characteristic, complemented
5#20265051C211yesmaximal, characteristic
6#20265048C211yesmaximal, characteristic
7#20265044C211yesmaximal, characteristic, complemented
8#20265042C211yesmaximal, characteristic, complemented
9#20265039C211yesmaximal, characteristic
10C4×#13108053C211yesmaximal, characteristic, complemented
11C4×#13108051C211yesmaximal, characteristic
12#20265024C211yesmaximal, characteristic
13#20265022C211yesmaximal, characteristic, complemented
14#95×D101C211yesmaximal, characteristic
15#387×C101C211yesmaximal, characteristic, complemented
16C22×C4×D101C2211yescharacteristic
17C22×C4×D101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19#15692213C418yescomplemented
20C2×C4⋊Dic101C2211yescharacteristic
21C2×C4⋊Dic101C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108051C2211yescharacteristic
25C2×#13108051C2211yescharacteristic
26C2×#13108051C2211yescharacteristic
27C2×#13108051C2211yescharacteristic
28C2×C4×Dic101C2211yescharacteristic
29C2×C4×Dic101C2211yescharacteristic
30C2×C4×Dic101C2211yescharacteristic
31C2×C4×Dic101C2211yescharacteristic
32C2×C4×Dic101C2211yescharacteristic
33#15692162C2211yescharacteristic
34#15692162C2211yescharacteristic
35#15692162C2211yescharacteristic
36#15692162C2211yescharacteristic
37#15692162C2211yescharacteristic
38#15692162C2211yescharacteristic
39#15692162C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45C2×C4⋊C4×C101C2211yescharacteristic
46C2×C4⋊C4×C101C2211yescharacteristic
47C2×C42×C101C2211yescharacteristic
48#95×C101C2211yescharacteristic
49#95×C101C2211yescharacteristic
50#95×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52C23×D101C2311yescharacteristic
53C22×Dic101C2311yescharacteristic
54C22×Dic101C2311yescharacteristic
55C22×Dic101C2311yescharacteristic
56C22×Dic101C2311yescharacteristic
57C22×Dic101C2311yescharacteristic
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C2×C4×D101—22no—
61C2×C4×D101—22no—
62C2×C4×D101—22no—
63C2×C4×D101—22no—
64C2×C4×D101—22no—
65C2×C4×D101—22no—
66C2×C4×D101—41no—
67C2×C4×D101—41no—
68C2×C4×D101C2×C414yes—
69C4⋊Dic101—41no—
70C4⋊Dic101C2×C414yescomplemented
71#13108053—22no—
72#13108053—22no—
73#13108051—22no—
74#13108051—22no—
75#13108051C2×C414yes—
76#13108051C2×C414yes—
77C4×Dic101—22no—
78C4×Dic101—22no—
79C4×Dic101—41no—
80C4×Dic101C2×C414yescomplemented
81C4×Dic101C2×C414yescomplemented
82C22×C4×C101C2311yescharacteristic
83C22×C4×C101C2311yescharacteristic
84C22×C4×C101C2311yescharacteristic
85C22×C4×C101C2311yescharacteristic
86C22×C4×C101C2311yescharacteristic
87C22×C4×C101C2311yescharacteristic
88C22×C4×C101C2311yescharacteristic
89C4⋊C4×C101—41no—
90C4⋊C4×C101C2×C414yescomplemented
91C42×C101—41no—
92C23×C101C2411yescharacteristic
93C22×D101—21no—
94C22×D101—21no—
95C22×D101—21no—
96C22×D101—21no—
97C22×D101C22×C412yes—
98C22×D101#4112yes—
99C22×D101#4112yes—
100C2×C4×C101—21no—
101C2×C4×C101—21no—
102C2×C4×C101—21no—
103C2×C4×C101—21no—
104C2×C4×C101—21no—
105C2×C4×C101—21no—
106C2×C4×C101—21no—
107C2×C4×C101—21no—
108C2×C4×C101—21no—
109C2×C4×C101—21no—
110C2×C4×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101C22×C412yes—
118C2×C4×C101C22×C412yes—
119C2×C4×C101C22×C412yes—
120C2×C4×C101#4112yes—
121C2×Dic101—21no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101C22×C412yes—
139C2×Dic101C22×C412yes—
140C2×Dic101C22×C412yes—
141C2×Dic101#4112yes—
142C4×D101—28no—
143C4×D101—44no—
144C4×D101—44no—
145C22×C101C23×C411yescharacteristic
146C22×C101C2×#4111yescharacteristic
147C22×C101C2×#4111yescharacteristic
148C22×C101C2×#4111yescharacteristic
149C22×C101C2×#4111yescharacteristic
150C22×C101H(5,𝔽2)11yescharacteristic
151C22×C1012−1+411yescharacteristic
152C2×D101—22no—
153C2×D101—22no—
154C2×D101—22no—
155C2×D101—22no—
156C2×D101—22no—
157C2×D101—22no—
158C2×D101#11714yes—
159C4×C101—22no—
160C4×C101—22no—
161C4×C101—22no—
162C4×C101—22no—
163C4×C101—41no—
164C4×C101—41no—
165C4×C101—41no—
166Dic101—22no—
167Dic101—22no—
168Dic101—22no—
169Dic101—22no—
170Dic101—41no—
171Dic101—41no—
172Dic101—41no—
173C2×C101C2×#11711yescharacteristic
174C2×C101C4×#4111yescharacteristic
175C2×C101#52011yescharacteristic
176C2×C101#55111yescharacteristic
177C2×C101#55111yescharacteristic
178C2×C101#55211yescharacteristic
179C2×C101#55211yescharacteristic
180D101—24no—
181#2343—1011nomaximal, Sylow
182C101#234311yescharacteristic, complemented, Sylow
183C2×#117—1011no—
184#387—1011no—
185#387—1011no—
186#387—1011no—
187#387D10111yescharacteristic, complemented
188#385—1011no—
189#385—1011no—
190#383—1011no—
191#382—1011no—
192#381—1011no—
193C4×C4⋊C4—1011no—
194C4×C22⋊C4—1011no—
195C4×C22⋊C4—1011no—
196#376—1011no—
197C2×#95—1011no—
198C23×C4—1011no—
199C23×C4—1011no—
200C23×C4—1011no—
201#117—1018no—
202C2×C4⋊C4—1011no—
203C2×C4⋊C4—1011no—
204C2×C4⋊C4—1011no—
205C2×C4⋊C4—1011no—
206C2×C4⋊C4C2×D10111yescharacteristic
207C2×C4⋊C4C2×D10111yescharacteristic
208C2×C22⋊C4—1011no—
209C2×C22⋊C4—1011no—
210C2×C22⋊C4—1011no—
211C2×C22⋊C4—1011no—
212C2×C42—1011no—
213C2×C42—1011no—
214C2×C42—1011no—
215C2×C42—1011no—
216C2×C42—1011no—
217C2×C42C2×D10111yescharacteristic
218#95—1011no—
219#95—1011no—
220#95—1011no—
221#95—1011no—
222#95—1011no—
223#95—1011no—
224#95—1011no—
225#95—1011no—
226#95—1011no—
227#95—1011no—
228#95—1011no—
229#95—1011no—
230#95C2×D10111yescharacteristic
231#95C2×D10111yescharacteristic
232#95C2×D10111yescharacteristic
233#95C2×D10111yescharacteristic
234C24—1011no—
235C22×C4—1011no—
236C22×C4—1011no—
237C22×C4—1011no—
238C22×C4—1011no—
239C22×C4—1014no—
240C22×C4—1011no—
241C22×C4—1011no—
242C22×C4—1011no—
243C22×C4—4041no—
244C22×C4—4041no—
245C22×C4—2022no—
246C22×C4—2022no—
247C22×C4—2022no—
248C22×C4—2022no—
249C22×C4—2022no—
250C22×C4—2022no—
251C22×C4C22×D10111yescharacteristic
252C22×C4C22×D10111yescharacteristic
253C22×C4C22×D10111yescharacteristic
254C22×C4C22×D10111yescharacteristic
255C22×C4C22×D10111yescharacteristic
256C22×C4C22×D10111yescharacteristic
257C22×C4C22×D10111yescharacteristic
258C4⋊C4—41no—
259C4⋊C4—1014no—
260C4⋊C4—4041no—
261C4⋊C4—2022no—
262C4⋊C4—2022no—
263C4⋊C4C4×D10114yescomplemented
264C22⋊C4—1014no—
265C22⋊C4—1014no—
266C22⋊C4—2022no—
267C22⋊C4—2022no—
268C42—41no—
269C42—1014no—
270C42—1014no—
271C42—4041no—
272C42—2022no—
273C42—2022no—
274C23—1012no—
275C23—1012no—
276C23—1012no—
277C23—2021no—
278C23—2021no—
279C23—2021no—
280C23—2021no—
281C23C23×D10111yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—1012no—
300C2×C4—1012no—
301C2×C4—1012no—
302C2×C4—1012no—
303C2×C4—4044no—
304C2×C4—4044no—
305C2×C4—2028no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4—2021no—
310C2×C4—2021no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4—2021no—
316C2×C4—2021no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4—2021no—
323C2×C4C2×C4×D10112yes—
324C2×C4C2×C4×D10112yes—
325C2×C4C2×C4×D10112yes—
326C2×C4#1310807512yes—
327C22—1014no—
328C22—2022no—
329C22—2022no—
330C22—2022no—
331C22—2022no—
332C22—2022no—
333C22—2022no—
334C22C22×C4×D10111yescharacteristic
335C22#41×D10111yescharacteristic
336C22#41×D10111yescharacteristic
337C22#41×D10111yescharacteristic
338C22C2×#1310807511yescharacteristic
339C22#1569230011yescharacteristic
340C22#1569230211yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—4041no—
349C4—4041no—
350C4—4041no—
351C4—2022no—
352C4—2022no—
353C4—2022no—
354C4—2022no—
355C2—2024no—
356C2#117×D10111yescharacteristic
357C2C4×#1310807511yescharacteristic
358C2#2026565611yescharacteristic
359C2#2026585411yescharacteristic
360C2#2026585511yescharacteristic
361C2#2026586311yescharacteristic
362C2#2026586511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#387×C101C2
socle, cosocleC23×C101C24
prime radicalC23×D101C23
hypercenter#387D101
hypocenterC101#2343
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #20265063C2semidirect
3. #20265062C2semidirect
4. #20265060C2semidirect
7. #20265044C2semidirect
8. #20265042C2semidirect
10. C4×#13108053C2semidirect
13. #20265022C2semidirect
15. #387×C101C2semidirect
19. #15692213C4semidirect
70. C4⋊Dic101C2×C4semidirect
80. C4×Dic101C2×C4semidirect
81. C4×Dic101C2×C4semidirect
90. C4⋊C4×C101C2×C4semidirect
182. C101#2343semidirect
187. #387D101semidirect
263. C4⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922132
#202650632
#202650622
#202650602
#202650512
#202650482
#202650442
#202650422
#202650392
C4×#131080532
C4×#131080512
#202650242
#202650222
#95×D1012
#387×C1012
#2343101