Finite Group Databasebeta

← #27174322

Subgroups — #27174322

Order 12928.

456 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692250C211yesmaximal, characteristic
2#20265076C211yesmaximal, characteristic, complemented
3#20265075C211yesmaximal, characteristic, complemented
4#20265074C211yesmaximal, characteristic, complemented
5#20265069C211yesmaximal, characteristic
6#20265068C211yesmaximal, characteristic
7#20265066C211yesmaximal, characteristic
8#20265060C211yesmaximal, characteristic, complemented
9#20265059C211yesmaximal, characteristic, complemented
10#20265051C211yesmaximal, characteristic
11#20265051C211yesmaximal, characteristic
12#20265048C211yesmaximal, characteristic
13#20265046C211yesmaximal, characteristic, complemented
14#20265044C211yesmaximal, characteristic, complemented
15#387×C101C211yesmaximal, characteristic, complemented
16C22×C4×D101C2211yescharacteristic
17#15692250C418yescomplemented
18C2×C4⋊Dic101C2211yescharacteristic
19C2×C4⋊Dic101C2211yescharacteristic
20C2×C4⋊Dic101C2211yescharacteristic
21C2×C4⋊Dic101C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×C4×Dic101C2211yescharacteristic
35C2×C4×Dic101C2211yescharacteristic
36C2×C4×Dic101C2211yescharacteristic
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—41no—
61C2×C4×D101—41no—
62C2×C4×D101—41no—
63C4⋊Dic101—41no—
64C4⋊Dic101—41no—
65C4⋊Dic101—41no—
66C4⋊Dic101C2×C414yescomplemented
67#13108053—41no—
68#13108053—41no—
69#13108053—41no—
70#13108053—41no—
71#13108053—41no—
72#13108053C2×C414yescomplemented
73#13108051—41no—
74#13108051—41no—
75#13108051—41no—
76#13108051C2×C414yes—
77#13108051C2×C414yes—
78#13108051C2×C414yes—
79C4×Dic101—41no—
80C4×Dic101—41no—
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×D101—21no—
98C22×D101—21no—
99C22×D101C22×C412yes—
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×C101—21no—
118C2×C4×C101C22×C412yes—
119C2×C4×C101C22×C412yes—
120C2×C4×C101C22×C412yes—
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×Dic101—21no—
139C2×Dic101C22×C412yes—
140C2×Dic101C22×C412yes—
141C2×Dic101C22×C412yes—
142C4×D101—44no—
143C22×C101C23×C411yescharacteristic
144C22×C101H(5,𝔽2)11yescharacteristic
145C22×C101H(5,𝔽2)11yescharacteristic
146C22×C101H(5,𝔽2)11yescharacteristic
147C22×C1012−1+411yescharacteristic
148C22×C1012−1+411yescharacteristic
149C22×C1012−1+411yescharacteristic
150C2×D101—22no—
151C2×D101—22no—
152C2×D101—22no—
153C2×D101—41no—
154C2×D101—41no—
155C2×D101—41no—
156C2×D101—41no—
157C4×C101—22no—
158C4×C101—22no—
159C4×C101—22no—
160C4×C101—41no—
161C4×C101—41no—
162C4×C101—41no—
163C4×C101—41no—
164Dic101—22no—
165Dic101—22no—
166Dic101—22no—
167Dic101—41no—
168Dic101—41no—
169Dic101—41no—
170Dic101—41no—
171C2×C101#52011yescharacteristic
172C2×C101#52011yescharacteristic
173C2×C101#52011yescharacteristic
174C2×C101#56011yescharacteristic
175C2×C101#56311yescharacteristic
176C2×C101#56311yescharacteristic
177C2×C101#56311yescharacteristic
178D101—42no—
179#2383—1011nomaximal, Sylow
180C101#238311yescharacteristic, complemented, Sylow
181C2×#126—1011no—
182#389—1011no—
183#389—1011no—
184#389—1011no—
185#388—1011no—
186#388—1011no—
187#388—1011no—
188#387—1011no—
189#387—1011no—
190#387D10111yescharacteristic, complemented
191#385—1011no—
192#385—1011no—
193#385—1011no—
194#384—1011no—
195#383—1011no—
196C23×C4—1011no—
197#126—1018no—
198C2×C4⋊C4—1011no—
199C2×C4⋊C4—1011no—
200C2×C4⋊C4—1011no—
201C2×C4⋊C4—1011no—
202C2×C4⋊C4—1011no—
203C2×C4⋊C4—1011no—
204C2×C4⋊C4—1011no—
205C2×C4⋊C4—1011no—
206C2×C4⋊C4—1011no—
207C2×C4⋊C4—1011no—
208C2×C4⋊C4C2×D10111yescharacteristic
209C2×C4⋊C4C2×D10111yescharacteristic
210C2×C22⋊C4—1011no—
211C2×C22⋊C4—1011no—
212C2×C22⋊C4—1011no—
213C2×C22⋊C4—1011no—
214C2×C22⋊C4—1011no—
215C2×C22⋊C4—1011no—
216C2×C42—1011no—
217C2×C42—1011no—
218C2×C42—1011no—
219C2×C42C2×D10111yescharacteristic
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#95C2×D10111yescharacteristic
229#95C2×D10111yescharacteristic
230#95C2×D10111yescharacteristic
231#95C2×D10111yescharacteristic
232C24—1011no—
233C22×C4—1011no—
234C22×C4—1011no—
235C22×C4—1011no—
236C22×C4—1011no—
237C22×C4—1011no—
238C22×C4—1011no—
239C22×C4—1011no—
240C22×C4—4041no—
241C22×C4—4041no—
242C22×C4—4041no—
243C22×C4C22×D10111yescharacteristic
244C22×C4C22×D10111yescharacteristic
245C22×C4C22×D10111yescharacteristic
246C22×C4C22×D10111yescharacteristic
247C22×C4C22×D10111yescharacteristic
248C22×C4C22×D10111yescharacteristic
249C22×C4C22×D10111yescharacteristic
250C4⋊C4—41no—
251C4⋊C4—1014no—
252C4⋊C4—1014no—
253C4⋊C4—4041no—
254C4⋊C4—4041no—
255C4⋊C4—4041no—
256C4⋊C4—4041no—
257C4⋊C4—4041no—
258C4⋊C4—4041no—
259C4⋊C4—4041no—
260C4⋊C4—4041no—
261C4⋊C4C4×D10114yescomplemented
262C22⋊C4—1014no—
263C22⋊C4—1014no—
264C22⋊C4—1014no—
265C22⋊C4—4041no—
266C22⋊C4—4041no—
267C22⋊C4—4041no—
268C42—41no—
269C42—1014no—
270C42—4041no—
271C42—4041no—
272C23—1012no—
273C23—2021no—
274C23—2021no—
275C23—2021no—
276C23—2021no—
277C23—2021no—
278C23—2021no—
279C23C23×D10111yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
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—1012no—
299C2×C4—1012no—
300C2×C4—1012no—
301C2×C4—4044no—
302C2×C4—2021no—
303C2×C4—2021no—
304C2×C4—2021no—
305C2×C4—2021no—
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×C4C2×C4×D10112yes—
321C2×C4C2×C4×D10112yes—
322C2×C4C2×C4×D10112yes—
323C22—4041no—
324C22—4041no—
325C22—4041no—
326C22—4041no—
327C22—2022no—
328C22—2022no—
329C22—2022no—
330C22C22×C4×D10111yescharacteristic
331C22#1569230011yescharacteristic
332C22#1569230111yescharacteristic
333C22#1569230111yescharacteristic
334C22#1569230211yescharacteristic
335C22#1569230211yescharacteristic
336C22#1569230311yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—4041no—
345C4—4041no—
346C4—4041no—
347C4—4041no—
348C4—2022no—
349C4—2022no—
350C4—2022no—
351C2—4042no—
352C2#2026565611yescharacteristic
353C2#2026565711yescharacteristic
354C2#2026565911yescharacteristic
355C2#2026591711yescharacteristic
356C2#2026592811yescharacteristic
357C2#2026592911yescharacteristic
358C2#2026593111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#387×C101C2
socle, cosocleC23×C101C24
prime radicalC23×D101C23
hypercenter#387D101
hypocenterC101#2383
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #20265076C2semidirect
3. #20265075C2semidirect
4. #20265074C2semidirect
8. #20265060C2semidirect
9. #20265059C2semidirect
13. #20265046C2semidirect
14. #20265044C2semidirect
15. #387×C101C2semidirect
17. #15692250C4semidirect
66. C4⋊Dic101C2×C4semidirect
72. #13108053C2×C4semidirect
81. C4×Dic101C2×C4semidirect
90. C4⋊C4×C101C2×C4semidirect
180. C101#2383semidirect
190. #387D101semidirect
261. C4⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922502
#202650762
#202650752
#202650742
#202650692
#202650682
#202650662
#202650602
#202650592
#202650512
#202650482
#202650462
#202650442
#387×C1012
#2383101