Finite Group Databasebeta

← #25256837

Subgroups — #25256837

Order 11008.

2350 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19137552C216yesmaximal, complemented
2Q8×C4⋊Dic43C212yesmaximal, complemented
3#19137464C216yesmaximal
4Q8×C22⋊Dic43C212yesmaximal
5#19137246C216yesmaximal, complemented
6#19137002C213yesmaximal
7#19136993C213yesmaximal
8#19136960C212yesmaximal, complemented
9C22×C4×Q8×C43C211yesmaximal, characteristic
10C2×Q8×Dic43C2214yes—
11C2×#13063284C2216yes—
12#14582806C22112yes—
13#14582798C22112yes—
14#14582785C22124yes—
15#14582769C2216yes—
16#14582767C2216yes—
17#14582765C2216yes—
18#14582765C2216yes—
19#14582762C2216yes—
20#14582759C2216yes—
21#14582759C2211yescharacteristic
22#14582759C2213yes—
23C4⋊C4×Dic43C22112yes—
24C4×C4⋊Dic43C2216yes—
25C4×C4⋊Dic43C2216yes—
26C4×C4⋊Dic43C2216yes—
27C4×C22⋊Dic43C2216yes—
28#14582745C2216yes—
29C23×Q8×C43C2211yescharacteristic
30C2×C4×Q8×C43C4116yescomplemented
31C2×C4×Q8×C43C2213yes—
32C2×C4×Q8×C43C2211yescharacteristic
33C2×C4×Q8×C43C2213yes—
34C2×C4×Q8×C43C2211yescharacteristic
35C22×C4⋊C4×C43C2213yes—
36C22×C42×C43C2213yes—
37Q8×Dic43—216no—
38#13063284—412no—
39C2×C4⋊Dic43C23112yes—
40C2×C4⋊Dic43C2312yes—
41C2×C4⋊Dic43C2316yes—
42C2×C4⋊Dic43C23112yes—
43C2×C22⋊Dic43C2316yes—
44C2×C22⋊Dic43C2312yes—
45C2×C4×Dic43C23112yes—
46C2×C4×Dic43C23112yes—
47C43⋊#95C23112yes—
48C43⋊#95C23112yes—
49C43⋊#95C23112yes—
50C43⋊#95C23112yes—
51C43⋊#95C2312yes—
52C43⋊#95C2316yes—
53C22×Q8×C43—28no—
54C22×Q8×C43C2×C418yes—
55C22×Q8×C43C2313yes—
56C22×Q8×C43C2311yescharacteristic
57C23×C4×C43C2311yescharacteristic
58C23×C4×C43C2313yes—
59C23×C4×C43C2313yes—
60C4×Q8×C43C2×C4124yes—
61C4×Q8×C43C2×C4124yes—
62C4×Q8×C43C2×C418yes—
63C4×Q8×C43C2×C418yes—
64C2×C4⋊C4×C43C2×C4124yes—
65C2×C4⋊C4×C43C2316yes—
66C2×C4⋊C4×C43C2313yes—
67C2×C4⋊C4×C43C2313yes—
68C2×C42×C43C2×C4124yes—
69C2×C42×C43C2313yes—
70C2×C42×C43C2313yes—
71C2×C42×C43C2313yes—
72C2×C42×C43C2313yes—
73C24×C43C2411yescharacteristic
74C2×Q8×C43—232no—
75C2×Q8×C43—212no—
76C2×Q8×C43C22×C4112yes—
77C2×Q8×C43C22×C414yes—
78C2×Q8×C43#4118yes—
79C22×C4×C43—212no—
80C22×C4×C43—24no—
81C22×C4×C43—26no—
82C22×C4×C43—26no—
83C22×C4×C43C22×C414yes—
84C22×C4×C43C22×C4112yes—
85C22×C4×C43C22×C4112yes—
86C22×C4×C43C2411yescharacteristic
87C22×C4×C43C2413yes—
88C22×C4×C43C2413yes—
89C22×C4×C43C2413yes—
90C22×C4×C43C2413yes—
91C22×C4×C43C2411yescharacteristic
92C22×Dic43C2414yes—
93C22×Dic43C24112yes—
94C4⋊C4×C43C22×C4124yes—
95C4⋊C4×C43C22×C4112yes—
96C4⋊C4×C43C22×C4112yes—
97C42×C43C22×C4112yes—
98C42×C43C22×C4112yes—
99C42×C43C22×C4112yes—
100C42×C43C22×C4112yes—
101C22⋊Dic43—212no—
102C22⋊Dic43—24no—
103C4⋊Dic43—224no—
104C4⋊Dic43—24no—
105C4⋊Dic43—212no—
106C4⋊Dic43—412no—
107C4×Dic43—224no—
108C4×Dic43—412no—
109C23×C43—22no—
110C23×C43—22no—
111C23×C43C23×C412yes—
112C23×C43H(5,𝔽2)12yes—
113C23×C432−1+412yes—
114C23×C43C2511yescharacteristic
115Q8×C43—224no—
116Q8×C43#117116yescomplemented
117C2×C4×C43—224no—
118C2×C4×C43—26no—
119C2×C4×C43—23no—
120C2×C4×C43—23no—
121C2×C4×C43—23no—
122C2×C4×C43—23no—
123C2×C4×C43—22no—
124C2×C4×C43—224no—
125C2×C4×C43—28no—
126C2×C4×C43C23×C412yes—
127C2×C4×C43C23×C416yes—
128C2×C4×C43C23×C416yes—
129C2×C4×C43C23×C416yes—
130C2×C4×C43C23×C416yes—
131C2×C4×C43C23×C412yes—
132C2×C4×C43C2×#41112yes—
133C2×C4×C432−1+414yes—
134C2×Dic43—24no—
135C2×Dic43—212no—
136C2×Dic43—212no—
137C2×Dic43—212no—
138C2×Dic43—24no—
139C2×Dic43—24no—
140#34196—431nomaximal, Sylow
141C22×C43—22no—
142C22×C43—22no—
143C22×C43—24no—
144C22×C43—24no—
145C22×C43#52014yes—
146C22×C43C24×C411yescharacteristic
147C22×C43C22×#4112yes—
148C22×C43C2×H(5,𝔽2)11yescharacteristic
149C22×C43C2×2−1+411yescharacteristic
150C22×C43C2×2−1+412yes—
151C4×C43—26no—
152C4×C43—22no—
153C4×C43—26no—
154C4×C43—26no—
155C4×C43C2×#117112yes—
156C4×C43#51914yes—
157Dic43—412no—
158Dic43—44no—
159C22×C4×Q8D4311yescharacteristic
160#2355—436no—
161Q8×C4⋊C4—432no—
162#2344—436no—
163Q8×C22⋊C4—432no—
164#2320—436no—
165#2285—433no—
166#2283—433no—
167#2276—432no—
168C2×C43—24no—
169C2×C43C22×#11711yescharacteristic
170C2×C43C2×#51911yescharacteristic
171C2×C43C2×#52011yescharacteristic
172C2×C43#352212yes—
173C2×C43#352312yes—
174C23×Q8C2×D4311yescharacteristic
175C2×C4×Q8—434no—
176C2×C4×Q8Dic43116yescomplemented
177C2×C4×Q8C2×D4313yes—
178C2×C4×Q8C2×D4311yescharacteristic
179C2×C4×Q8C2×D4313yes—
180C2×C4×Q8C2×D4311yescharacteristic
181C2×#117—436no—
182C22×C4⋊C4C2×D4313yes—
183C22×C42C2×D4313yes—
184#391—4312no—
185#389—4312no—
186#387—4324no—
187#384—436no—
188#383—436no—
189#382—436no—
190#382—436no—
191#381—436no—
192#380—436no—
193#380—431no—
194#380—433no—
195C4×C4⋊C4—4312no—
196C4×C4⋊C4—436no—
197C4×C4⋊C4—436no—
198C4×C4⋊C4—436no—
199C4×C22⋊C4—436no—
200#376—436no—
201C43#3419611yescharacteristic, complemented, Sylow
202C22×Q8—28no—
203C22×Q8C2×Dic4318yes—
204C22×Q8C22×D4313yes—
205C22×Q8C22×D4311yescharacteristic
206C23×C4C22×D4313yes—
207C23×C4C22×D4313yes—
208C23×C4C22×D4311yescharacteristic
209C4×Q8—8616no—
210C4×Q8C2×Dic43124yes—
211C4×Q8C2×Dic43124yes—
212C4×Q8C2×Dic4318yes—
213C4×Q8C2×Dic4318yes—
214#117—17212no—
215C2×C4⋊C4—4312no—
216C2×C4⋊C4—432no—
217C2×C4⋊C4—436no—
218C2×C4⋊C4—4312no—
219C2×C4⋊C4C2×Dic43124yes—
220C2×C4⋊C4C22×D4313yes—
221C2×C4⋊C4C22×D4316yes—
222C2×C4⋊C4C22×D4313yes—
223C2×C22⋊C4—436no—
224C2×C22⋊C4—432no—
225C2×C42—4312no—
226C2×C42—4312no—
227C2×C42C2×Dic43124yes—
228C2×C42C22×D4313yes—
229C2×C42C22×D4313yes—
230C2×C42C22×D4313yes—
231C2×C42C22×D4313yes—
232#95—432no—
233#95—436no—
234#95—4312no—
235#95—4312no—
236#95—4312no—
237#95—4312no—
238C24C23×D4311yescharacteristic
239C2×Q8—232no—
240C2×Q8—212no—
241C2×Q8#1062189818yes—
242C2×Q8C22×Dic43112yes—
243C2×Q8C22×Dic4314yes—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—212no—
247C22×C4—24no—
248C22×C4—434no—
249C22×C4—4312no—
250C22×C4C22×Dic4314yes—
251C22×C4C22×Dic43112yes—
252C22×C4C22×Dic43112yes—
253C22×C4C23×D4311yescharacteristic
254C22×C4C23×D4313yes—
255C22×C4C23×D4313yes—
256C22×C4C23×D4313yes—
257C22×C4C23×D4313yes—
258C22×C4C23×D4311yescharacteristic
259C4⋊C4—8624no—
260C4⋊C4—864no—
261C4⋊C4—8612no—
262C4⋊C4—17212no—
263C4⋊C4C22×Dic43124yes—
264C4⋊C4C22×Dic43112yes—
265C4⋊C4C22×Dic43112yes—
266C22⋊C4—8612no—
267C22⋊C4—864no—
268C42—8624no—
269C42—17212no—
270C42C22×Dic43112yes—
271C42C22×Dic43112yes—
272C42C22×Dic43112yes—
273C42C22×Dic43112yes—
274C23—22no—
275C23—22no—
276C23C23×Dic4312yes—
277C23#1306337112yes—
278C23#1306337212yes—
279C23C24×D4311yescharacteristic
280Q8—224no—
281Q8#13063284116yescomplemented
282C2×C4—224no—
283C2×C4—26no—
284C2×C4—23no—
285C2×C4—23no—
286C2×C4—23no—
287C2×C4—23no—
288C2×C4—22no—
289C2×C4—224no—
290C2×C4—28no—
291C2×C4—864no—
292C2×C4—8612no—
293C2×C4—8612no—
294C2×C4—864no—
295C2×C4—8612no—
296C2×C4—864no—
297C2×C4C23×Dic4312yes—
298C2×C4C23×Dic4316yes—
299C2×C4C23×Dic4316yes—
300C2×C4C23×Dic4316yes—
301C2×C4C23×Dic4316yes—
302C2×C4C23×Dic4312yes—
303C2×C4C2×#10621898112yes—
304C2×C4#1306337314yes—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—22no—
309C22#1458338214yes—
310C22C24×Dic4311yescharacteristic
311C22C22×#1062189812yes—
312C22C2×#1306337111yescharacteristic
313C22C2×#1306337211yescharacteristic
314C22C2×#1306337312yes—
315C4—22no—
316C4—26no—
317C4—26no—
318C4—26no—
319C4—17212no—
320C4—1724no—
321C4C2×#13063284112yes—
322C4#1458337714yes—
323C2—24no—
324C2C22×#1306328411yescharacteristic
325C2C2×#1458337711yescharacteristic
326C2C2×#1458338211yescharacteristic
327C2#1914868012yes—
328C2#1914869112yes—
subgroupG/H
center, FrattiniC23C24×D43
commutatorC22×C43C24×C4
FittingC22×C4×Q8×C43C2
socle, cosocleC23×C43C25
prime radicalC24×C43C24
hypercenterC22×C4×Q8D43
hypocenterC43#34196
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #19137552C2semidirect
2. Q8×C4⋊Dic43C2semidirect
5. #19137246C2semidirect
8. #19136960C2semidirect
30. C2×C4×Q8×C43C4semidirect
116. Q8×C43#117semidirect
176. C2×C4×Q8Dic43semidirect
201. C43#34196semidirect
281. Q8#13063284semidirect

Maximal subgroups

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

subgroupindex
#191375522
Q8×C4⋊Dic432
#191374642
Q8×C22⋊Dic432
#191372462
#191370022
#191369932
#191369602
C22×C4×Q8×C432
#3419643