Finite Group Databasebeta

← #27173875

Subgroups — #27173875

Order 12928.

460 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692253C211yesmaximal, characteristic
2#20265084C211yesmaximal, characteristic
3#20265081C211yesmaximal, characteristic
4#20265081C211yesmaximal, characteristic
5#20265074C211yesmaximal, characteristic
6#20265072C211yesmaximal, characteristic
7#20265064C211yesmaximal, characteristic
8#20265061C211yesmaximal, characteristic
9#20265059C211yesmaximal, characteristic
10#20265058C211yesmaximal, characteristic
11#20265041C211yesmaximal, characteristic
12C4×#13108053C211yesmaximal, characteristic
13#20265028C211yesmaximal, characteristic
14#20265022C211yesmaximal, characteristic
15#391×C101C211yesmaximal, characteristic
16C22×Dic202C2211yescharacteristic
17#15692253C418yescomplemented
18C2×C4⋊Dic101C2211yescharacteristic
19C2×#13108053C2211yescharacteristic
20C2×#13108053C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×C4×Dic101C2211yescharacteristic
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
45C22×Q8×C101C2211yescharacteristic
46C2×C4⋊C4×C101C2211yescharacteristic
47C2×C42×C101C2211yescharacteristic
48#95×C101C2211yescharacteristic
49#95×C101C2211yescharacteristic
50#95×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52C2×Dic202—41no—
53C2×Dic202—41no—
54C2×Dic202C2×C414yes—
55C22×Dic101C2311yescharacteristic
56C22×Dic101C2311yescharacteristic
57C22×Dic101C2311yescharacteristic
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C4⋊Dic101—41no—
64#13108053—22no—
65#13108053—22no—
66#13108053—41no—
67#13108053—41no—
68#13108053C2×C414yes—
69#13108053C2×C414yes—
70#13108053C2×C414yes—
71#13108053C2×C414yes—
72C4×Dic101—22no—
73C4×Dic101—41no—
74C4×Dic101—41no—
75C4×Dic101—41no—
76C4×Dic101C2×C414yes—
77C4×Dic101Q814yescomplemented
78C2×Q8×C101—41no—
79C2×Q8×C101—41no—
80C2×Q8×C101C2×C414yes—
81C22×C4×C101C2311yescharacteristic
82C22×C4×C101C2311yescharacteristic
83C22×C4×C101C2311yescharacteristic
84C22×C4×C101C2311yescharacteristic
85C22×C4×C101C2311yescharacteristic
86C22×C4×C101C2311yescharacteristic
87C22×C4×C101C2311yescharacteristic
88C4⋊C4×C101—41no—
89C42×C101—41no—
90C23×C101C2411yescharacteristic
91Q8×C101—44no—
92C2×C4×C101—21no—
93C2×C4×C101—21no—
94C2×C4×C101—21no—
95C2×C4×C101—21no—
96C2×C4×C101—21no—
97C2×C4×C101—21no—
98C2×C4×C101—21no—
99C2×C4×C101—21no—
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×C101C22×C412yes—
110C2×C4×C101C22×C412yes—
111C2×C4×C101C22×C412yes—
112C2×C4×C101C2×Q812yes—
113C2×Dic101—21no—
114C2×Dic101—21no—
115C2×Dic101—21no—
116C2×Dic101—21no—
117C2×Dic101—21no—
118C2×Dic101—21no—
119C2×Dic101—21no—
120C2×Dic101—21no—
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×Dic101C22×C412yes—
131C2×Dic101C22×C412yes—
132C2×Dic101C22×C412yes—
133C2×Dic101C22×C412yes—
134C2×Dic101C2×Q812yes—
135C2×Dic101C2×Q812yes—
136C2×Dic101#4112yes—
137Dic202—44no—
138C22×C101C23×C411yescharacteristic
139C22×C101C22×Q811yescharacteristic
140C22×C101C2×#4111yescharacteristic
141C22×C101H(5,𝔽2)11yescharacteristic
142C22×C101H(5,𝔽2)11yescharacteristic
143C22×C1012−1+411yescharacteristic
144C22×C1012−1+411yescharacteristic
145C4×C101—22no—
146C4×C101—22no—
147C4×C101—22no—
148C4×C101—22no—
149C4×C101—41no—
150C4×C101—41no—
151C4×C101—41no—
152Dic101—22no—
153Dic101—22no—
154Dic101—22no—
155Dic101—22no—
156Dic101—41no—
157Dic101—41no—
158Dic101—41no—
159Dic101C4×Q814yes—
160C2×C101C2×C4×Q811yescharacteristic
161C2×C101#51811yescharacteristic
162C2×C101#51911yescharacteristic
163C2×C101#54211yescharacteristic
164C2×C101#54211yescharacteristic
165C2×C101#54411yescharacteristic
166C2×C101#54411yescharacteristic
167#2331—1011nomaximal, Sylow
168C101#233111yescharacteristic, complemented, Sylow
169C2×C4⋊Q8—1011no—
170#391—1011no—
171#391—1011no—
172#391—1011no—
173#391D10111yescharacteristic
174#389—1011no—
175#389—1011no—
176#387—1011no—
177#387—1011no—
178#387—1011no—
179#387—1011no—
180#382—1011no—
181C4×C4⋊C4—1011no—
182C4×C4⋊C4—1011no—
183#376—1011no—
184C22×Q8—1011no—
185C22×Q8C2×D10111yescharacteristic
186C4⋊Q8—1018no—
187C2×C4⋊C4—1011no—
188C2×C4⋊C4—1011no—
189C2×C4⋊C4—1011no—
190C2×C4⋊C4—1011no—
191C2×C4⋊C4—1011no—
192C2×C4⋊C4—1011no—
193C2×C4⋊C4—1011no—
194C2×C4⋊C4—1011no—
195C2×C4⋊C4—1011no—
196C2×C4⋊C4C2×D10111yescharacteristic
197C2×C42—1011no—
198C2×C42—1011no—
199C2×C42—1011no—
200C2×C42—1011no—
201C2×C42—1011no—
202C2×C42—1011no—
203C2×C42C2×D10111yescharacteristic
204#95—1011no—
205#95—1011no—
206#95—1011no—
207#95—1011no—
208#95—1011no—
209#95—1011no—
210#95—1011no—
211#95—1011no—
212#95—1011no—
213#95—1011no—
214#95—1011no—
215#95—1011no—
216#95C2×D10111yescharacteristic
217#95C2×D10111yescharacteristic
218#95C2×D10111yescharacteristic
219#95C2×D10111yescharacteristic
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—1014no—
223C2×Q8—4041no—
224C2×Q8—4041no—
225C2×Q8C4×D10114yes—
226C22×C4—1011no—
227C22×C4—1011no—
228C22×C4—1011no—
229C22×C4—1011no—
230C22×C4—1011no—
231C22×C4—1011no—
232C22×C4—1011no—
233C22×C4—1011no—
234C22×C4C22×D10111yescharacteristic
235C22×C4C22×D10111yescharacteristic
236C22×C4C22×D10111yescharacteristic
237C22×C4C22×D10111yescharacteristic
238C22×C4C22×D10111yescharacteristic
239C22×C4C22×D10111yescharacteristic
240C22×C4C22×D10111yescharacteristic
241C4⋊C4—41no—
242C4⋊C4—1014no—
243C4⋊C4—1014no—
244C4⋊C4—1014no—
245C4⋊C4—1014no—
246C4⋊C4—4041no—
247C4⋊C4—4041no—
248C4⋊C4—4041no—
249C4⋊C4—2022no—
250C4⋊C4—2022no—
251C42—41no—
252C42—1014no—
253C42—1014no—
254C42—4041no—
255C42—4041no—
256C42—4041no—
257C42—2022no—
258C23C23×D10111yescharacteristic
259Q8—44no—
260Q8—4044no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—1012no—
279C2×C4—1012no—
280C2×C4—1012no—
281C2×C4—1012no—
282C2×C4—1012no—
283C2×C4—1012no—
284C2×C4—1012no—
285C2×C4—2021no—
286C2×C4—2021no—
287C2×C4—2021no—
288C2×C4—2021no—
289C2×C4—2021no—
290C2×C4—2021no—
291C2×C4—2021no—
292C2×C4—2021no—
293C2×C4—2021no—
294C2×C4—2021no—
295C2×C4—2021no—
296C2×C4—2021no—
297C2×C4—2021no—
298C2×C4—2021no—
299C2×C4—2021no—
300C2×C4—2021no—
301C2×C4—2021no—
302C2×C4C2×C4×D10112yes—
303C2×C4C2×C4×D10112yes—
304C2×C4C2×C4×D10112yes—
305C2×C4Q8×D10112yes—
306C22C22×C4×D10111yescharacteristic
307C22C2×Q8×D10111yescharacteristic
308C22#41×D10111yescharacteristic
309C22#1569230011yescharacteristic
310C22#1569230011yescharacteristic
311C22#1569230211yescharacteristic
312C22#1569230311yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—1014no—
321C4—4041no—
322C4—4041no—
323C4—4041no—
324C4—2022no—
325C4—2022no—
326C4—2022no—
327C4—2022no—
328C2C4×Q8×D10111yescharacteristic
329C2#2026565011yescharacteristic
330C2#2026565311yescharacteristic
331C2#2026579111yescharacteristic
332C2#2026579711yescharacteristic
333C2#2026580411yescharacteristic
334C2#2026580611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#391×C101C2
socle, prime radical, cosocleC23×C101C24
hypercenter#391D101
hypocenterC101#2331
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
17. #15692253C4semidirect
77. C4×Dic101Q8semidirect
168. C101#2331semidirect

Maximal subgroups

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

subgroupindex
C2×#156922532
#202650842
#202650812
#202650742
#202650722
#202650642
#202650612
#202650592
#202650582
#202650412
C4×#131080532
#202650282
#202650222
#391×C1012
#2331101