Finite Group Databasebeta

← Q8×#66873 #14770358

Subgroups — Q8×#66873 #14770358

Order 2816.

1314 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D44C211yesmaximal, characteristic
2C2×Q8×C11⋊C8C211yesmaximal, characteristic, complemented
3#13069576C213yesmaximal, complemented
4#13069568C213yesmaximal, complemented
5#13068613C213yesmaximal
6C4×#66873C213yesmaximal
7Q8×C11×C4⋊C4C211yesmaximal, characteristic, complemented
8C22×Q8×D11—21no—
9C2×C4×Q8×C11—21no—
10C2×C4×Q8×C11C2211yescharacteristic
11Q8×D44—24no—
12Q8×D44C418yescomplemented
13C2×#66950—23no—
14C2×C4×D44C2213yes—
15C2×#66922C2213yes—
16C11×#389—23no—
17C11×#384C2213yes—
18C4×C11×C4⋊C4C2213yes—
19Q8×C11⋊C8—24no—
20C2×#66873C2213yes—
21C2×#66873C2211yescharacteristic
22C2×C4⋊(C11⋊C8)C2213yes—
23C2×C4×C11⋊C8C2213yes—
24#10622167C2213yes—
25#10622167C2216yes—
26#10622167C2213yes—
27C22×Q8×C11D411yescharacteristic
28C2×Q8×D11—28no—
29C2×Q8×D11—44no—
30C2×Q8×D11—44no—
31C22×D44C2311yescharacteristic
32C22×C4×D11—23no—
33C22×Dic22—23no—
34C4×Q8×C11—44no—
35C4×Q8×C11C2×C414yes—
36C4×Q8×C11D414yes—
37C2×C11×C4⋊C4—23no—
38C2×C11×C4⋊C4—23no—
39C2×C11×C4⋊C4D413yes—
40C2×C11×C4⋊C4C2313yes—
41C2×C11×C4⋊C4C2311yescharacteristic
42C2×C42×C11—23no—
43C2×C42×C11C2313yes—
44C2×(C2×D11)⋊C4—23no—
45C2×C4⋊Dic11—23no—
46C2×C4⋊Dic11C2313yes—
47C22×C11⋊C8C2313yes—
48C22×C11⋊C8C2311yescharacteristic
49#66950—224no—
50C4×D44—26no—
51C4×D44C2×C4112yes—
52#66922—26no—
53#66922C2×C4112yes—
54C11×#95—23no—
55#66873—26no—
56#66873Q814yesdirect product factor
57C4⋊(C11⋊C8)—26no—
58C4×C11⋊C8—26no—
59Q8×#102—111nomaximal, Sylow
60C23×D11—21no—
61C2×Q8×C11C22⋊C414yes—
62C2×Q8×C11D814yes—
63C2×Q8×C11SD1614yes—
64C22×C4×C11—21no—
65C22×C4×C11—23no—
66C22×C4×C11C2×D413yes—
67C22×C4×C11C2×D413yes—
68C22×C4×C11C2411yescharacteristic
69C22×Dic11—23no—
70Q8×D11—432no—
71C2×D44—21no—
72C2×D44—21no—
73C2×D44—21no—
74C2×D44C22×C412yes—
75C2×D44C2×Q812yes—
76C2×D44#4112yes—
77C2×C4×D11—212no—
78C2×C4×D11—46no—
79C2×C4×D11—46no—
80C2×Dic22—212no—
81C2×Dic22—46no—
82C2×Dic22—46no—
83C11×C4⋊C4—21no—
84C11×C4⋊C4—23no—
85C11×C4⋊C4—23no—
86C11×C4⋊C4—46no—
87C11×C4⋊C4—83no—
88C11×C4⋊C4C22⋊C4112yes—
89C11×C4⋊C4C2×Q812yescomplemented
90C42×C11—46no—
91C42×C11C22×C416yes—
92C42×C11C2×D416yes—
93(C2×D11)⋊C4—212no—
94C4⋊Dic11—212no—
95C4⋊Dic11—23no—
96C4⋊Dic11C22×C416yes—
97C2×C11⋊C8—21no—
98C2×C11⋊C8—23no—
99C2×C11⋊C8—23no—
100C2×C11⋊C8—21no—
101C2×C11⋊C8—23no—
102C2×C11⋊C8C2×Q812yescomplemented
103C2×D4×Q8—111no—
104C2×C8×Q8—111no—
105Q8×C4⋊C4D1111yescharacteristic, complemented
106#1990—113no—
107#1988—113no—
108#1802—113no—
109C4×#102—113no—
110C23×C11C22×D411yescharacteristic
111C22×D11—22no—
112C22×D11—22no—
113C22×D11—22no—
114C22×D11—41no—
115C22×D11—41no—
116C22×D11—41no—
117C22×D11—41no—
118Q8×C11#102116yescomplemented
119C2×C4×C11—23no—
120C2×C4×C11—23no—
121C2×C4×C11—22no—
122C2×C4×C11—41no—
123C2×C4×C11—41no—
124C2×C4×C11—43no—
125C2×C4×C11—43no—
126C2×C4×C11—43no—
127C2×C4×C11C2×C22⋊C416yes—
128C2×C4×C11C2×C22⋊C416yes—
129C2×C4×C11C2×D816yes—
130C2×C4×C11C2×SD1616yes—
131C2×C4×C11C23×C411yescharacteristic
132C2×C4×C11C22×D411yescharacteristic
133C2×C4×C11C22×Q811yescharacteristic
134C2×C4×C11C2×#4111yescharacteristic
135C2×C4×C112−1+411yescharacteristic
136C2×C4×C112−1+411yescharacteristic
137C2×Dic11—26no—
138C2×Dic11—43no—
139C2×Dic11—43no—
140D44—22no—
141D44—22no—
142D44—22no—
143D44C4×Q814yescomplemented
144C4×D11—424no—
145Dic22—424no—
146C11⋊C8—22no—
147C11⋊C8—43no—
148C23×Q8—221no—
149D4×Q8—118no—
150D4×Q8—224no—
151C2×C4⋊Q8—113no—
152C2×C22⋊Q8—223no—
153C2×C4×Q8—21no—
154C2×C4×Q8C2×D1111yescharacteristic
155C2×C4×D4—113no—
156C8×Q8—224no—
157C2×C4⋊C8—113no—
158C2×#102—111no—
159C2×#102—113no—
160C2×C4×C8—113no—
161#389—23no—
162#384C2×D1113yes—
163C4×C4⋊C4C2×D1113yes—
164#340—116no—
165#340—113no—
166#340—113no—
167C22×C11C22×C22⋊C411yescharacteristic
168C22×C11#54811yescharacteristic
169C22×C11D4×Q811yescharacteristic
170C22×C11C22×D811yescharacteristic
171C22×C11C22×SD1611yescharacteristic
172C22×C11#57711yescharacteristic
173C22×C11#57811yescharacteristic
174C2×D11—24no—
175C2×D11—42no—
176C2×D11—42no—
177C2×D11—42no—
178C2×D11—42no—
179C2×D11—42no—
180C2×D11—42no—
181C4×C11—26no—
182C4×C11—42no—
183C4×C11—83no—
184C4×C11C2×#102112yes—
185C4×C11C2×C4×Q811yescharacteristic
186C4×C11#51911yescharacteristic
187C4×C11#54811yescharacteristic
188C4×C11D4×Q811yescharacteristic
189Dic11—46no—
190C22×Q8—228no—
191C22×Q8—223no—
192C22×Q8—444no—
193C22×Q8—444no—
194C22×Q8C22⋊D1111yescharacteristic
195C22×D4—111no—
196C23×C4—223no—
197C22×C8—111no—
198C22×C8—113no—
199C4⋊Q8—1112no—
200C4⋊Q8—226no—
201C22⋊Q8—2224no—
202C4×Q8—44no—
203C4×Q8C4×D1114yes—
204C4×Q8D4414yes—
205C4×D4—1112no—
206C4×D4—226no—
207C2×C4⋊C4—23no—
208C2×C4⋊C4—23no—
209C2×C4⋊C4—113no—
210C2×C4⋊C4—223no—
211C2×C4⋊C4C22⋊D1113yes—
212C2×C4⋊C4C22×D1113yes—
213C2×C4⋊C4C22×D1111yescharacteristic
214C2×C22⋊C4—223no—
215C2×C42—23no—
216C2×C42C22×D1113yes—
217C4⋊C8—226no—
218#102—114no—
219#102—226no—
220C4×C8—226no—
221#95—23no—
222C2×C11Q8×C22⋊C411yescharacteristic
223C2×C11C22×#10211yescharacteristic
224C2×C11#289811yescharacteristic
225C2×C11#335811yescharacteristic
226C2×C11#335911yescharacteristic
227C2×C11Q8×D811yescharacteristic
228C2×C11Q8×SD1611yescharacteristic
229D11—44no—
230C24—221no—
231C2×Q8—2212no—
232C2×Q8—4432no—
233C2×Q8—446no—
234C2×Q8—446no—
235C2×Q8(C2×D11)⋊C414yes—
236C2×Q8#432314yes—
237C2×Q8#432514yes—
238C2×D4—112no—
239C2×D4—112no—
240C2×D4—112no—
241C2×D4—221no—
242C2×D4—221no—
243C2×D4—221no—
244C22×C4—21no—
245C22×C4—23no—
246C22×C4—223no—
247C22×C4—2212no—
248C22×C4—446no—
249C22×C4—446no—
250C22×C4C2×C22⋊D1113yes—
251C22×C4C2×C22⋊D1113yes—
252C22×C4C23×D1111yescharacteristic
253C2×C8—112no—
254C2×C8—221no—
255C2×C8—221no—
256C2×C8—223no—
257C2×C8—223no—
258C2×C8—223no—
259C4⋊C4—21no—
260C4⋊C4—23no—
261C4⋊C4—23no—
262C4⋊C4—46no—
263C4⋊C4—83no—
264C4⋊C4—116no—
265C4⋊C4—2212no—
266C4⋊C4—223no—
267C4⋊C4(C2×D11)⋊C4112yes—
268C4⋊C4Q8×D1112yescomplemented
269C22⋊C4—2212no—
270C42—46no—
271C42C2×C4×D1116yes—
272C42C2×D4416yes—
273C11Q8×#10211yescharacteristic, complemented, Sylow
274C23—222no—
275C23—222no—
276C23—222no—
277C23—441no—
278C23—441no—
279C23—441no—
280C23—441no—
281C23C22×C22⋊D1111yescharacteristic
282Q8—4424no—
283Q8#66873116yesdirect product factor
284D4—114no—
285D4—222no—
286D4—222no—
287D4—222no—
288C2×C4—23no—
289C2×C4—23no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—43no—
294C2×C4—43no—
295C2×C4—43no—
296C2×C4—226no—
297C2×C4—443no—
298C2×C4—443no—
299C2×C4—4424no—
300C2×C4C2×(C2×D11)⋊C416yes—
301C2×C4C2×(C2×D11)⋊C416yes—
302C2×C4C2×#432316yes—
303C2×C4C2×#432516yes—
304C2×C4C22×C4×D1111yescharacteristic
305C2×C4C22×D4411yescharacteristic
306C2×C4C2×Q8×D1111yescharacteristic
307C2×C4#6704011yescharacteristic
308C2×C4#41×D1111yescharacteristic
309C2×C4#6704311yescharacteristic
310C8—222no—
311C8—443no—
312C22—224no—
313C22—442no—
314C22—442no—
315C22—442no—
316C22—442no—
317C22—442no—
318C22—442no—
319C22C22×(C2×D11)⋊C411yescharacteristic
320C22C22×#432311yescharacteristic
321C22C22×#432511yescharacteristic
322C22Q8×C22⋊D1111yescharacteristic
323C22#1062336211yescharacteristic
324C22#1062336711yescharacteristic
325C22#1062336911yescharacteristic
326C4—26no—
327C4—42no—
328C4—83no—
329C4—446no—
330C4C2×#66873112yes—
331C4C4×Q8×D1111yescharacteristic
332C4#1062310311yescharacteristic
333C4Q8×D4411yescharacteristic
334C4#1062310711yescharacteristic
335C2—444no—
336C2Q8×(C2×D11)⋊C411yescharacteristic
337C2C22×#6687311yescharacteristic
338C2#1307714611yescharacteristic
339C2#1308124711yescharacteristic
340C2#1308125411yescharacteristic
341C2Q8×#432311yescharacteristic
342C2Q8×#432511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D11
commutatorC2×C4×C11C23×C4
FrattiniC22×C4C23×D11
FittingQ8×C11×C4⋊C4C2
socleC23×C11C22×D4
prime radicalC22×D44C23
cosocleC22×C4×C11C24
hypercenterQ8×C4⋊C4D11
hypocenterC11Q8×#102
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C11⋊C8C2semidirect
3. #13069576C2semidirect
4. #13069568C2semidirect
7. Q8×C11×C4⋊C4C2semidirect
12. Q8×D44C4semidirect
56. #66873Q8direct (N × H)
89. C11×C4⋊C4C2×Q8semidirect
102. C2×C11⋊C8C2×Q8semidirect
105. Q8×C4⋊C4D11semidirect
118. Q8×C11#102semidirect
143. D44C4×Q8semidirect
268. C4⋊C4Q8×D11semidirect
273. C11Q8×#102semidirect
283. Q8#66873direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×D442
C2×Q8×C11⋊C82
#130695762
#130695682
#130686132
C4×#668732
Q8×C11×C4⋊C42
Q8×#10211