Finite Group Databasebeta

← Q8×#88049 #15879504

Subgroups — Q8×#88049 #15879504

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×D52C211yesmaximal, characteristic
2C2×Q8×#1055C211yesmaximal, characteristic, complemented
3#13115796C213yesmaximal, complemented
4#13115788C213yesmaximal, complemented
5#13114833C213yesmaximal
6C4×#88049C213yesmaximal
7Q8×C13×C4⋊C4C211yesmaximal, characteristic, complemented
8C22×Q8×D13—21no—
9C2×C4×Q8×C13—21no—
10C2×C4×Q8×C13C2211yescharacteristic
11Q8×D52—24no—
12Q8×D52C418yescomplemented
13C2×#88151—23no—
14C2×C4×D52C2213yes—
15C2×#88123C2213yes—
16C13×#389—23no—
17C13×#384C2213yes—
18C4×C13×C4⋊C4C2213yes—
19Q8×#1055—24no—
20C2×#88049C2213yes—
21C2×#88049C2211yescharacteristic
22C2×C4⋊#1055C2213yes—
23C2×C4×#1055C2213yes—
24#11718121C2213yes—
25#11718121C2213yes—
26#11718121C2216yes—
27C22×Q8×C13D411yescharacteristic
28C2×Q8×D13—28no—
29C2×Q8×D13—44no—
30C2×Q8×D13—44no—
31C22×D52C2311yescharacteristic
32C22×C4×D13—23no—
33C22×Dic26—23no—
34C4×Q8×C13—44no—
35C4×Q8×C13C2×C414yes—
36C4×Q8×C13D414yes—
37C2×C13×C4⋊C4—23no—
38C2×C13×C4⋊C4—23no—
39C2×C13×C4⋊C4D413yes—
40C2×C13×C4⋊C4C2313yes—
41C2×C13×C4⋊C4C2311yescharacteristic
42C2×C42×C13—23no—
43C2×C42×C13C2313yes—
44C2×#6103—23no—
45C2×C4⋊Dic13—23no—
46C2×C4⋊Dic13C2313yes—
47C22×#1055C2313yes—
48C22×#1055C2311yescharacteristic
49#88151—224no—
50C4×D52—26no—
51C4×D52C2×C4112yes—
52#88123—26no—
53#88123C2×C4112yes—
54C13×#95—23no—
55#88049—26no—
56#88049Q814yesdirect product factor
57C4⋊#1055—26no—
58C4×#1055—26no—
59Q8×#102—131nomaximal, Sylow
60C23×D13—21no—
61C2×Q8×C13C22⋊C414yes—
62C2×Q8×C13D814yes—
63C2×Q8×C13SD1614yes—
64C22×C4×C13—21no—
65C22×C4×C13—23no—
66C22×C4×C13C2×D413yes—
67C22×C4×C13C2×D413yes—
68C22×C4×C13C2411yescharacteristic
69C22×Dic13—23no—
70Q8×D13—432no—
71C2×D52—21no—
72C2×D52—21no—
73C2×D52—21no—
74C2×D52C22×C412yes—
75C2×D52C2×Q812yes—
76C2×D52#4112yes—
77C2×C4×D13—212no—
78C2×C4×D13—46no—
79C2×C4×D13—46no—
80C2×Dic26—212no—
81C2×Dic26—46no—
82C2×Dic26—46no—
83C13×C4⋊C4—21no—
84C13×C4⋊C4—23no—
85C13×C4⋊C4—23no—
86C13×C4⋊C4—46no—
87C13×C4⋊C4—83no—
88C13×C4⋊C4C22⋊C4112yes—
89C13×C4⋊C4C2×Q812yescomplemented
90C42×C13—46no—
91C42×C13C22×C416yes—
92C42×C13C2×D416yes—
93#6103—212no—
94C4⋊Dic13—212no—
95C4⋊Dic13—23no—
96C4⋊Dic13C22×C416yes—
97C2×#1055—21no—
98C2×#1055—21no—
99C2×#1055—23no—
100C2×#1055—23no—
101C2×#1055—23no—
102C2×#1055C2×Q812yescomplemented
103C2×D4×Q8—131no—
104C2×C8×Q8—131no—
105Q8×C4⋊C4D1311yescharacteristic, complemented
106#1990—133no—
107#1988—133no—
108#1802—133no—
109C4×#102—133no—
110C23×C13C22×D411yescharacteristic
111C22×D13—22no—
112C22×D13—22no—
113C22×D13—22no—
114C22×D13—41no—
115C22×D13—41no—
116C22×D13—41no—
117C22×D13—41no—
118Q8×C13#102116yescomplemented
119C2×C4×C13—23no—
120C2×C4×C13—23no—
121C2×C4×C13—22no—
122C2×C4×C13—41no—
123C2×C4×C13—41no—
124C2×C4×C13—43no—
125C2×C4×C13—43no—
126C2×C4×C13—43no—
127C2×C4×C13C2×C22⋊C416yes—
128C2×C4×C13C2×C22⋊C416yes—
129C2×C4×C13C2×D816yes—
130C2×C4×C13C2×SD1616yes—
131C2×C4×C13C23×C411yescharacteristic
132C2×C4×C13C22×D411yescharacteristic
133C2×C4×C13C22×Q811yescharacteristic
134C2×C4×C13C2×#4111yescharacteristic
135C2×C4×C132−1+411yescharacteristic
136C2×C4×C132−1+411yescharacteristic
137C2×Dic13—26no—
138C2×Dic13—43no—
139C2×Dic13—43no—
140D52—22no—
141D52—22no—
142D52—22no—
143D52C4×Q814yescomplemented
144C4×D13—424no—
145Dic26—424no—
146#1055—22no—
147#1055—43no—
148C23×Q8—261no—
149D4×Q8—138no—
150D4×Q8—264no—
151C2×C4⋊Q8—133no—
152C2×C22⋊Q8—263no—
153C2×C4×Q8—21no—
154C2×C4×Q8C2×D1311yescharacteristic
155C2×C4×D4—133no—
156C8×Q8—264no—
157C2×C4⋊C8—133no—
158C2×#102—131no—
159C2×#102—133no—
160C2×C4×C8—133no—
161#389—23no—
162#384C2×D1313yes—
163C4×C4⋊C4C2×D1313yes—
164#340—136no—
165#340—133no—
166#340—133no—
167C22×C13C22×C22⋊C411yescharacteristic
168C22×C13#54811yescharacteristic
169C22×C13D4×Q811yescharacteristic
170C22×C13C22×D811yescharacteristic
171C22×C13C22×SD1611yescharacteristic
172C22×C13#57711yescharacteristic
173C22×C13#57811yescharacteristic
174C2×D13—24no—
175C2×D13—42no—
176C2×D13—42no—
177C2×D13—42no—
178C2×D13—42no—
179C2×D13—42no—
180C2×D13—42no—
181C4×C13—26no—
182C4×C13—42no—
183C4×C13—83no—
184C4×C13C2×#102112yes—
185C4×C13C2×C4×Q811yescharacteristic
186C4×C13#51911yescharacteristic
187C4×C13#54811yescharacteristic
188C4×C13D4×Q811yescharacteristic
189Dic13—46no—
190C22×Q8—268no—
191C22×Q8—263no—
192C22×Q8—524no—
193C22×Q8—524no—
194C22×Q8C22⋊D1311yescharacteristic
195C22×D4—131no—
196C23×C4—263no—
197C22×C8—131no—
198C22×C8—133no—
199C4⋊Q8—1312no—
200C4⋊Q8—266no—
201C22⋊Q8—2624no—
202C4×Q8—44no—
203C4×Q8C4×D1314yes—
204C4×Q8D5214yes—
205C4×D4—1312no—
206C4×D4—266no—
207C2×C4⋊C4—23no—
208C2×C4⋊C4—23no—
209C2×C4⋊C4—133no—
210C2×C4⋊C4—263no—
211C2×C4⋊C4C22⋊D1313yes—
212C2×C4⋊C4C22×D1313yes—
213C2×C4⋊C4C22×D1311yescharacteristic
214C2×C22⋊C4—263no—
215C2×C42—23no—
216C2×C42C22×D1313yes—
217C4⋊C8—266no—
218#102—134no—
219#102—266no—
220C4×C8—266no—
221#95—23no—
222C2×C13Q8×C22⋊C411yescharacteristic
223C2×C13C22×#10211yescharacteristic
224C2×C13#289811yescharacteristic
225C2×C13#335811yescharacteristic
226C2×C13#335911yescharacteristic
227C2×C13Q8×D811yescharacteristic
228C2×C13Q8×SD1611yescharacteristic
229D13—44no—
230C24—261no—
231C2×Q8—2612no—
232C2×Q8—5232no—
233C2×Q8—526no—
234C2×Q8—526no—
235C2×Q8#610314yes—
236C2×Q8#610414yes—
237C2×Q8#610614yes—
238C2×D4—132no—
239C2×D4—132no—
240C2×D4—132no—
241C2×D4—261no—
242C2×D4—261no—
243C2×D4—261no—
244C22×C4—21no—
245C22×C4—23no—
246C22×C4—263no—
247C22×C4—2612no—
248C22×C4—526no—
249C22×C4—526no—
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—21no—
260C4⋊C4—23no—
261C4⋊C4—23no—
262C4⋊C4—46no—
263C4⋊C4—83no—
264C4⋊C4—136no—
265C4⋊C4—2612no—
266C4⋊C4—263no—
267C4⋊C4#6103112yes—
268C4⋊C4Q8×D1312yescomplemented
269C22⋊C4—2612no—
270C42—46no—
271C42C2×C4×D1316yes—
272C42C2×D5216yes—
273C13Q8×#10211yescharacteristic, complemented, Sylow
274C23—262no—
275C23—262no—
276C23—262no—
277C23—521no—
278C23—521no—
279C23—521no—
280C23—521no—
281C23C22×C22⋊D1311yescharacteristic
282Q8—5224no—
283Q8#88049116yesdirect product factor
284D4—134no—
285D4—262no—
286D4—262no—
287D4—262no—
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—266no—
297C2×C4—523no—
298C2×C4—523no—
299C2×C4—5224no—
300C2×C4C2×#610316yes—
301C2×C4C2×#610316yes—
302C2×C4C2×#610416yes—
303C2×C4C2×#610616yes—
304C2×C4C22×C4×D1311yescharacteristic
305C2×C4C22×D5211yescharacteristic
306C2×C4C2×Q8×D1311yescharacteristic
307C2×C4#8825411yescharacteristic
308C2×C4#41×D1311yescharacteristic
309C2×C4#8825711yescharacteristic
310C8—262no—
311C8—523no—
312C22—264no—
313C22—522no—
314C22—522no—
315C22—522no—
316C22—522no—
317C22—522no—
318C22—522no—
319C22C22×#610311yescharacteristic
320C22C22×#610411yescharacteristic
321C22C22×#610611yescharacteristic
322C22Q8×C22⋊D1311yescharacteristic
323C22#1171953611yescharacteristic
324C22#1171954111yescharacteristic
325C22#1171954311yescharacteristic
326C4—26no—
327C4—42no—
328C4—83no—
329C4—526no—
330C4C2×#88049112yes—
331C4C4×Q8×D1311yescharacteristic
332C4#1171927711yescharacteristic
333C4Q8×D5211yescharacteristic
334C4#1171928111yescharacteristic
335C2—524no—
336C2Q8×#610311yescharacteristic
337C2C22×#8804911yescharacteristic
338C2#1312336611yescharacteristic
339C2#1312746711yescharacteristic
340C2#1312747411yescharacteristic
341C2Q8×#610411yescharacteristic
342C2Q8×#610611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingQ8×C13×C4⋊C4C2
socleC23×C13C22×D4
prime radicalC22×D52C23
cosocleC22×C4×C13C24
hypercenterQ8×C4⋊C4D13
hypocenterC13Q8×#102
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
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×#1055C2semidirect
3. #13115796C2semidirect
4. #13115788C2semidirect
7. Q8×C13×C4⋊C4C2semidirect
12. Q8×D52C4semidirect
56. #88049Q8direct (N × H)
89. C13×C4⋊C4C2×Q8semidirect
102. C2×#1055C2×Q8semidirect
105. Q8×C4⋊C4D13semidirect
118. Q8×C13#102semidirect
143. D52C4×Q8semidirect
268. C4⋊C4Q8×D13semidirect
273. C13Q8×#102semidirect
283. Q8#88049direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×D522
C2×Q8×#10552
#131157962
#131157882
#131148332
C4×#880492
Q8×C13×C4⋊C42
Q8×#10213