Finite Group Databasebeta

← #27183176

Subgroups — #27183176

Order 12928.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265861C211yesmaximal, characteristic
2Q8×C22⋊D101C211yesmaximal, characteristic
3#20265544C211yesmaximal, characteristic, complemented
4#20265519C211yesmaximal, characteristic, complemented
5#20265506C211yesmaximal, characteristic, complemented
6#20265486C211yesmaximal, characteristic, complemented
7#20265433C211yesmaximal, characteristic, complemented
8#20265415C211yesmaximal, characteristic, complemented
9#20265385C211yesmaximal, characteristic, complemented
10#20265331C211yesmaximal, characteristic, complemented
11#20265284C211yesmaximal, characteristic, complemented
12#20265251C211yesmaximal, characteristic
13#20265186C211yesmaximal, characteristic, complemented
14#103×D101C211yesmaximal, characteristic, complemented
15#451×C101C211yesmaximal, characteristic, complemented
16C2×Q8×D101C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108058C2211yescharacteristic
19C2×C8×D101C2211yescharacteristic
20#15692253—21no—
21#15692253C2211yescharacteristic
22#15692248—21no—
23#15692246C2211yescharacteristic
24#15692245—21no—
25#15692237—21no—
26#15692234—21no—
27#15692234C2211yescharacteristic
28#15692231—21no—
29#15692231C2211yescharacteristic
30#15692231C2211yescharacteristic
31Q8×Dic101C2211yescharacteristic
32Q8×Dic101C2211yescharacteristic
33C4×C22⋊D101—21no—
34C4×C22⋊D101C2211yescharacteristic
35#15692214C2211yescharacteristic
36#15692213—21no—
37C4⋊C4×D101C2211yescharacteristic
38#15692189C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692186C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692179C2211yescharacteristic
46#15692179C2211yescharacteristic
47#15692179C2211yescharacteristic
48#15692179C2211yescharacteristic
49#15692178C2211yescharacteristic
50#15692167C2211yescharacteristic
51#15692165C2211yescharacteristic
52C22×Q8×C101C2211yescharacteristic
53C2×Q16×C101C2211yescharacteristic
54C2×Q16×C101C2211yescharacteristic
55C22⋊Q8×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61Q8×D101—22no—
62Q8×D101—41no—
63C2×C22⋊D101D411yescharacteristic
64C2×C4×D101—21no—
65C2×C4×D101—21no—
66C2×C4×D101C2311yescharacteristic
67#13108064—41no—
68#13108058—41no—
69C2×#11717579C2311yescharacteristic
70C2×#11717579C2311yescharacteristic
71C8×D101—41no—
72C4⋊Dic101—21no—
73C4⋊Dic101—21no—
74C4⋊Dic101—21no—
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77C4⋊Dic101C2311yescharacteristic
78#13108053—21no—
79#13108053—21no—
80#13108053—21no—
81#13108053—21no—
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85#13108053D411yescharacteristic
86C22⋊Dic101—21no—
87C22⋊Dic101—21no—
88C22⋊Dic101D411yescharacteristic
89#13108051—21no—
90#13108051—21no—
91#13108051D411yescharacteristic
92C4×Dic101—21no—
93C4×Dic101—21no—
94C4×Dic101—21no—
95C4×Dic101C2311yescharacteristic
96C2×Q8×C101—21no—
97C2×Q8×C101—22no—
98C2×Q8×C101—41no—
99C2×Q8×C101C2311yescharacteristic
100C2×Q8×C101C2311yescharacteristic
101C2×Q8×C101C2311yescharacteristic
102C22×C4×C101—21no—
103C22×C4×C101C2311yescharacteristic
104Q16×C101—22no—
105Q16×C101—41no—
106C2×C8×C101C2311yescharacteristic
107C2×C8×C101C2311yescharacteristic
108C4⋊C4×C101—21no—
109C4⋊C4×C101C2311yescharacteristic
110C22⋊C4×C101—21no—
111C23×C101C2×D411yescharacteristic
112C22×D101C2×D411yescharacteristic
113Q8×C101—21no—
114Q8×C101—21no—
115Q8×C101—21no—
116Q8×C101—21no—
117Q8×C101—21no—
118Q8×C101—42no—
119Q8×C101#4112yes—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—42no—
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2411yescharacteristic
129C22⋊D101—22no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136C2×Dic101C2×D411yescharacteristic
137C4×D101—21no—
138C4×D101—21no—
139C4×D101—42no—
140C4×D101—42no—
141Dic202—21no—
142Dic202—21no—
143Dic202—42no—
144C8×C101—21no—
145C8×C101—21no—
146#11717579—21no—
147#11717579—21no—
148C22×C101—21no—
149C22×C101—21no—
150C22×C101—21no—
151C22×C101C22×D411yescharacteristic
152C2×D101—21no—
153C2×D101—21no—
154C2×D101#13712yes—
155C4×C101—22no—
156C4×C101—21no—
157C4×C101—41no—
158C4×C101—41no—
159C4×C101—41no—
160C4×C101C2×#4111yescharacteristic
161C4×C101H(5,𝔽2)11yescharacteristic
162Dic101—21no—
163Dic101—21no—
164Dic101—21no—
165Dic101—41no—
166Dic101—41no—
167Dic101—41no—
168Dic101—41no—
169C2×C101—41no—
170C2×C101#54611yescharacteristic
171C2×C101C2×#13711yescharacteristic
172C2×C101#57811yescharacteristic
173D101—22no—
174#3312—1011nomaximal, Sylow
175C101#331211yescharacteristic, complemented, Sylow
176#552—1011no—
177D4×Q8—1011no—
178#491—1011no—
179#485—1011no—
180#483—1011no—
181#479—1011no—
182#470—1011no—
183#467—1011no—
184#462—1011no—
185#451—1011no—
186#451D10111yescharacteristic, complemented
187#441—1011no—
188#435—1011no—
189#419—1011no—
190C2×#103—1011no—
191C22×Q8—1011no—
192C22×Q8C2×D10111yescharacteristic
193C2×Q16—1011no—
194C2×Q16C2×D10111yescharacteristic
195C2×Q16C2×D10111yescharacteristic
196C2×M16—1011no—
197C22×C8—1011no—
198C4⋊Q8—1011no—
199C4⋊Q8—2021no—
200#126—2021no—
201#125—1011no—
202#125—2021no—
203#123—2021no—
204C22⋊Q8—1011no—
205C22⋊Q8—1011no—
206C22⋊Q8—1011no—
207C22⋊Q8—2021no—
208C22⋊Q8—2021no—
209C22⋊Q8C2×D10111yescharacteristic
210C4×Q8—1011no—
211C4×Q8—1011no—
212C4×D4—1011no—
213C4×D4—2021no—
214#117—1011no—
215#117—2021no—
216C2×C4⋊C4—1011no—
217#107—1011no—
218#106—1011no—
219#106—1011no—
220C4⋊C8—1011no—
221#103—1011no—
222#103—1011no—
223#103—1011no—
224#103—1011no—
225#103—1011no—
226#103—1011no—
227#103—1011no—
228#103—1011no—
229#103C2×D10111yescharacteristic
230#103C2×D10111yescharacteristic
231C22⋊C8—1011no—
232C22⋊C8C2×D10111yescharacteristic
233#97—1011no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—1011no—
238C2×Q8—4041no—
239C2×Q8—2021no—
240C2×Q8—2022no—
241C2×Q8C22×D10111yescharacteristic
242C2×Q8C22×D10111yescharacteristic
243C2×Q8C22×D10111yescharacteristic
244C2×D4—1011no—
245C22×C4—21no—
246C22×C4—1011no—
247C22×C4—2021no—
248C22×C4—2021no—
249C22×C4C22×D10111yescharacteristic
250Q16—22no—
251Q16—41no—
252Q16—4041no—
253M16—4041no—
254C2×C8—1011no—
255C2×C8—1011no—
256C2×C8—4041no—
257C2×C8C22×D10111yescharacteristic
258C2×C8C22×D10111yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—1011no—
261C4⋊C4—1011no—
262C4⋊C4—1011no—
263C4⋊C4—1011no—
264C4⋊C4—2021no—
265C4⋊C4—2021no—
266C4⋊C4—2021no—
267C4⋊C4—2021no—
268C4⋊C4—2021no—
269C4⋊C4—2021no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4C22×D10111yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—1011no—
277C22⋊C4—1011no—
278C22⋊C4—2021no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C42—1011no—
283C42—2021no—
284C42—2021no—
285C42—2021no—
286C23—1011no—
287C23D4×D10111yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—4042no—
295Q8—2021no—
296Q8—2021no—
297Q8#1310807312yes—
298D4—2022no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—1011no—
307C2×C4—1011no—
308C2×C4—1011no—
309C2×C4—4042no—
310C2×C4—4042no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4—2021no—
316C2×C4—2021no—
317C2×C4D4×D10111yescharacteristic
318C2×C4C23×D10111yescharacteristic
319C8—21no—
320C8—21no—
321C8—2021no—
322C8—2021no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—1012no—
327C22—2021no—
328C22—2021no—
329C22C2×D4×D10111yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—4041no—
336C4—4041no—
337C4—4041no—
338C4—4041no—
339C4—2021no—
340C4—2021no—
341C4—2021no—
342C4C2×#1310807311yescharacteristic
343C4#1569230011yescharacteristic
344C2—41no—
345C2—2022no—
346C2#2026581411yescharacteristic
347C2#137×D10111yescharacteristic
348C2#2026599011yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#451×C101C2
socleC22×C101C22×D4
prime radicalC2×C22⋊D101D4
hypercenter#451D101
hypocenterC101#3312
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #20265544C2semidirect
4. #20265519C2semidirect
5. #20265506C2semidirect
6. #20265486C2semidirect
7. #20265433C2semidirect
8. #20265415C2semidirect
9. #20265385C2semidirect
10. #20265331C2semidirect
11. #20265284C2semidirect
13. #20265186C2semidirect
14. #103×D101C2semidirect
15. #451×C101C2semidirect
175. C101#3312semidirect
186. #451D101semidirect

Maximal subgroups

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

subgroupindex
#202658612
Q8×C22⋊D1012
#202655442
#202655192
#202655062
#202654862
#202654332
#202654152
#202653852
#202653312
#202652842
#202652512
#202651862
#103×D1012
#451×C1012
#3312101