Finite Group Databasebeta

← Q8×#88072 #15879508

Subgroups — Q8×#88072 #15879508

Order 3328.

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×#1055C211yesmaximal, characteristic, complemented
2Q8×C4⋊Dic13C211yesmaximal, characteristic, complemented
3#13115798C213yesmaximal, complemented
4#13115789C213yesmaximal, complemented
5#13114836C213yesmaximal
6C4×#88072C213yesmaximal
7C2×D4×Q8×C13C211yesmaximal, characteristic
8C23×Q8×C13—21no—
9D4×Q8×C13—24no—
10D4×Q8×C13C418yescomplemented
11C2×C13×C4⋊Q8C2213yes—
12C2×C13×C22⋊Q8—23no—
13C2×C4×Q8×C13C2211yescharacteristic
14C2×C4×D4×C13C2213yes—
15C2×Q8×Dic13—21no—
16C2×#88072C2213yes—
17C2×#88072C2211yescharacteristic
18Q8×#1055—24no—
19#11718646—23no—
20#11718596C2213yes—
21C4×C4⋊Dic13C2213yes—
22C2×C4⋊#1055C2213yes—
23C2×C4×#1055C2213yes—
24#11718121C2213yes—
25#11718121C2213yes—
26#11718121C2216yes—
27C22×Q8×C13—28no—
28C22×Q8×C13—23no—
29C22×Q8×C13—44no—
30C22×Q8×C13—44no—
31C22×Q8×C13D411yescharacteristic
32C22×D4×C13C2311yescharacteristic
33C23×C4×C13—23no—
34C13×C4⋊Q8—26no—
35C13×C4⋊Q8C2×C4112yes—
36C13×C22⋊Q8—224no—
37C4×Q8×C13C2×C414yes—
38C4×Q8×C13D414yes—
39C4×D4×C13—26no—
40C4×D4×C13C2×C4112yes—
41C2×C13×C4⋊C4—23no—
42C2×C13×C4⋊C4D413yes—
43C2×C13×C4⋊C4C2313yes—
44C2×C13×C22⋊C4—23no—
45C2×C42×C13C2313yes—
46Q8×Dic13—44no—
47C2×C4⋊Dic13—23no—
48C2×C4⋊Dic13—23no—
49C2×C4⋊Dic13C2313yes—
50C2×C4⋊Dic13C2311yescharacteristic
51C2×C4×Dic13—23no—
52C22×#1055C2313yes—
53C22×#1055C2311yescharacteristic
54#88072—26no—
55#88072Q814yesdirect product factor
56#88071—23no—
57C4⋊#1055—26no—
58C4×#1055—26no—
59Q8×#102—131nomaximal, Sylow
60C24×C13—21no—
61C2×Q8×C13—212no—
62C2×Q8×C13—432no—
63C2×Q8×C13—46no—
64C2×Q8×C13—46no—
65C2×Q8×C13C22⋊C414yes—
66C2×Q8×C13D814yes—
67C2×Q8×C13SD1614yes—
68C2×D4×C13—21no—
69C2×D4×C13—21no—
70C2×D4×C13—21no—
71C2×D4×C13C22×C412yes—
72C2×D4×C13C2×Q812yes—
73C2×D4×C13#4112yes—
74C22×C4×C13—23no—
75C22×C4×C13—212no—
76C22×C4×C13—46no—
77C22×C4×C13—46no—
78C22×C4×C13C2×D413yes—
79C22×C4×C13C2×D413yes—
80C22×C4×C13C2411yescharacteristic
81C22×Dic13—21no—
82C22×Dic13—23no—
83C13×C4⋊C4—212no—
84C13×C4⋊C4—23no—
85C13×C4⋊C4C22⋊C4112yes—
86C13×C4⋊C4C22×C416yes—
87C13×C22⋊C4—212no—
88C42×C13C22×C416yes—
89C42×C13C2×D416yes—
90C4⋊Dic13—21no—
91C4⋊Dic13—23no—
92C4⋊Dic13—23no—
93C4⋊Dic13—46no—
94C4⋊Dic13—83no—
95C4⋊Dic13C2×Q812yescomplemented
96C4×Dic13—46no—
97C2×#1055—21no—
98C2×#1055—21no—
99C2×#1055—23no—
100C2×#1055—23no—
101C2×#1055—23no—
102C2×#1055C2×Q812yescomplemented
103C2×D4×Q8D1311yescharacteristic
104C2×C8×Q8—131no—
105Q8×C4⋊C4—131no—
106#1990—133no—
107#1988—133no—
108#1802—133no—
109C4×#102—133no—
110C23×C13—22no—
111C23×C13—22no—
112C23×C13—22no—
113C23×C13—41no—
114C23×C13—41no—
115C23×C13—41no—
116C23×C13—41no—
117C23×C13C22×D411yescharacteristic
118Q8×C13—424no—
119Q8×C13#102116yescomplemented
120D4×C13—22no—
121D4×C13—22no—
122D4×C13—22no—
123D4×C13C4×Q814yescomplemented
124C2×C4×C13—23no—
125C2×C4×C13—23no—
126C2×C4×C13—26no—
127C2×C4×C13—43no—
128C2×C4×C13—43no—
129C2×C4×C13—424no—
130C2×C4×C13C2×C22⋊C416yes—
131C2×C4×C13C2×C22⋊C416yes—
132C2×C4×C13C2×D816yes—
133C2×C4×C13C2×SD1616yes—
134C2×C4×C13C23×C411yescharacteristic
135C2×C4×C13C22×D411yescharacteristic
136C2×C4×C13C22×Q811yescharacteristic
137C2×C4×C13C2×#4111yescharacteristic
138C2×C4×C132−1+411yescharacteristic
139C2×C4×C132−1+411yescharacteristic
140C2×Dic13—22no—
141C2×Dic13—41no—
142C2×Dic13—41no—
143C2×Dic13—43no—
144C2×Dic13—43no—
145C2×Dic13—43no—
146#1055—22no—
147#1055—43no—
148C23×Q8—21no—
149D4×Q8—24no—
150D4×Q8Dic1318yescomplemented
151C2×C4⋊Q8C2×D1313yes—
152C2×C22⋊Q8—23no—
153C2×C4×Q8—261no—
154C2×C4×Q8C2×D1311yescharacteristic
155C2×C4×D4C2×D1313yes—
156C8×Q8—264no—
157C2×C4⋊C8—133no—
158C2×#102—131no—
159C2×#102—133no—
160C2×C4×C8—133no—
161#389—263no—
162#384—133no—
163C4×C4⋊C4—133no—
164#340—136no—
165#340—133no—
166#340—133no—
167C22×C13—24no—
168C22×C13—42no—
169C22×C13—42no—
170C22×C13—42no—
171C22×C13—42no—
172C22×C13—42no—
173C22×C13—42no—
174C22×C13C22×C22⋊C411yescharacteristic
175C22×C13#54811yescharacteristic
176C22×C13D4×Q811yescharacteristic
177C22×C13C22×D811yescharacteristic
178C22×C13C22×SD1611yescharacteristic
179C22×C13#57711yescharacteristic
180C22×C13#57811yescharacteristic
181C4×C13—26no—
182C4×C13—46no—
183C4×C13C2×#102112yes—
184C4×C13C2×C4×Q811yescharacteristic
185C4×C13#51911yescharacteristic
186C4×C13#54811yescharacteristic
187C4×C13D4×Q811yescharacteristic
188Dic13—42no—
189Dic13—83no—
190C22×Q8—28no—
191C22×Q8—23no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8C22⋊D1311yescharacteristic
195C22×D4C22×D1311yescharacteristic
196C23×C4—23no—
197C22×C8—131no—
198C22×C8—133no—
199C4⋊Q8—26no—
200C4⋊Q8C2×Dic13112yes—
201C22⋊Q8—224no—
202C4×Q8—524no—
203C4×Q8C2×Dic1314yes—
204C4×Q8C22⋊D1314yes—
205C4×D4—26no—
206C4×D4C2×Dic13112yes—
207C2×C4⋊C4—23no—
208C2×C4⋊C4—131no—
209C2×C4⋊C4—133no—
210C2×C4⋊C4—263no—
211C2×C4⋊C4—263no—
212C2×C4⋊C4C22⋊D1313yes—
213C2×C4⋊C4C22×D1313yes—
214C2×C22⋊C4—23no—
215C2×C42—263no—
216C2×C42C22×D1313yes—
217C4⋊C8—266no—
218#102—134no—
219#102—266no—
220C4×C8—266no—
221#95—263no—
222C2×C13—44no—
223C2×C13Q8×C22⋊C411yescharacteristic
224C2×C13C22×#10211yescharacteristic
225C2×C13#289811yescharacteristic
226C2×C13#335811yescharacteristic
227C2×C13#335911yescharacteristic
228C2×C13Q8×D811yescharacteristic
229C2×C13Q8×SD1611yescharacteristic
230C24—21no—
231C2×Q8—212no—
232C2×Q8—432no—
233C2×Q8—46no—
234C2×Q8—46no—
235C2×Q8#610414yes—
236C2×Q8#610514yes—
237C2×Q8C22⋊Dic1314yes—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4Q8×D1312yes—
242C2×D4#613112yes—
243C2×D4C22×Dic1312yes—
244C22×C4—23no—
245C22×C4—212no—
246C22×C4—46no—
247C22×C4—46no—
248C22×C4—261no—
249C22×C4—263no—
250C22×C4C2×C22⋊D1313yes—
251C22×C4C2×C22⋊D1313yes—
252C22×C4C23×D1311yescharacteristic
253C2×C8—132no—
254C2×C8—261no—
255C2×C8—261no—
256C2×C8—263no—
257C2×C8—263no—
258C2×C8—263no—
259C4⋊C4—212no—
260C4⋊C4—23no—
261C4⋊C4—132no—
262C4⋊C4—261no—
263C4⋊C4—263no—
264C4⋊C4—263no—
265C4⋊C4—526no—
266C4⋊C4—1043no—
267C4⋊C4C22⋊Dic13112yes—
268C4⋊C4C22×Dic1316yes—
269C22⋊C4—212no—
270C42—526no—
271C42C22×Dic1316yes—
272C42C2×C22⋊D1316yes—
273C13Q8×#10211yescharacteristic, complemented, Sylow
274C23—22no—
275C23—22no—
276C23—22no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C22×C22⋊D1311yescharacteristic
282Q8—424no—
283Q8#88072116yesdirect product factor
284D4—22no—
285D4—22no—
286D4—22no—
287D4Q8×Dic1314yescomplemented
288C2×C4—23no—
289C2×C4—23no—
290C2×C4—26no—
291C2×C4—43no—
292C2×C4—43no—
293C2×C4—424no—
294C2×C4—262no—
295C2×C4—521no—
296C2×C4—521no—
297C2×C4—523no—
298C2×C4—523no—
299C2×C4—523no—
300C2×C4C2×#610416yes—
301C2×C4C2×#610516yes—
302C2×C4C2×C22⋊Dic1316yes—
303C2×C4C2×C22⋊Dic1316yes—
304C2×C4C2×Q8×D1311yescharacteristic
305C2×C4C2×#613111yescharacteristic
306C2×C4#8825411yescharacteristic
307C2×C4#8825411yescharacteristic
308C2×C4C23×Dic1311yescharacteristic
309C2×C4C22×C22⋊D1311yescharacteristic
310C8—262no—
311C8—523no—
312C22—24no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22C22×#610411yescharacteristic
320C22C22×#610511yescharacteristic
321C22Q8×C22⋊D1311yescharacteristic
322C22#1171952211yescharacteristic
323C22#1171954211yescharacteristic
324C22#1171954311yescharacteristic
325C22C22×C22⋊Dic1311yescharacteristic
326C4—26no—
327C4—46no—
328C4—522no—
329C4—1043no—
330C4C2×#88072112yes—
331C4C2×Q8×Dic1311yescharacteristic
332C4#1171951711yescharacteristic
333C4Q8×C22⋊D1311yescharacteristic
334C4#1171952211yescharacteristic
335C2—44no—
336C2Q8×C22⋊Dic1311yescharacteristic
337C2C22×#8807211yescharacteristic
338C2#1312336911yescharacteristic
339C2#1312747111yescharacteristic
340C2#1312747811yescharacteristic
341C2Q8×#610411yescharacteristic
342C2Q8×#610511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×D4×Q8×C13C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×D4×Q8D13
hypocenterC13Q8×#102
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#1055C2semidirect
2. Q8×C4⋊Dic13C2semidirect
3. #13115798C2semidirect
4. #13115789C2semidirect
10. D4×Q8×C13C4semidirect
55. #88072Q8direct (N × H)
95. C4⋊Dic13C2×Q8semidirect
102. C2×#1055C2×Q8semidirect
119. Q8×C13#102semidirect
123. D4×C13C4×Q8semidirect
150. D4×Q8Dic13semidirect
273. C13Q8×#102semidirect
283. Q8#88072direct (N × H)
287. D4Q8×Dic13semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#10552
Q8×C4⋊Dic132
#131157982
#131157892
#131148362
C4×#880722
C2×D4×Q8×C132
Q8×#10213