Finite Group Databasebeta

← Q8×#14556679 #34031507

Subgroups — Q8×#14556679 #34031507

Order 18688.

1882 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11924038C213yesmaximal, complemented
2#22626727C219yesmaximal, complemented
3#22626720C219yesmaximal, complemented
4Q8×#11924036C213yesmaximal, complemented
5#22626577C213yesmaximal
6C4×#14556679C213yesmaximal
7Q8×#11924029C211yesmaximal, characteristic
8C2×#14556679C2213yes—
9C2×#14556679C2211yescharacteristic
10#17989638C22118yes—
11#17989637C22118yes—
12#17989636C2219yes—
13#17989636C2219yes—
14Q8×#10596833C2211yescharacteristic
15Q8×#10596833C2213yes—
16Q8×#10596833C2216yes—
17#17989594C22118yes—
18#17989591C2219yes—
19#17989588C2219yes—
20#17989552C2219yes—
21#17989552C2219yes—
22C4×#11924038C2219yes—
23C4×#11924036C2219yes—
24#17989316C413yes—
25#17989298C419yes—
26Q8×D292C413yes—
27#17989142C2219yes—
28C4×#11924029C2213yes—
29C4×Q8×D73C2213yes—
30Q82×C73C411yescharacteristic
31#14556679—212no—
32#14556679Q818yesdirect product factor
33C2×#11924038C2319yes—
34C2×#11924038C2313yes—
35C2×#11924036C2319yes—
36C2×#11924036C2313yes—
37#14556658C23118yes—
38C4⋊#10596833C2313yes—
39C4⋊#10596833C2319yes—
40C4⋊#10596833C23118yes—
41#14556646C23118yes—
42C4×#10596833C2319yes—
43C4×#10596833C2313yes—
44C4×#10596833C2319yes—
45C4×#10596833C2319yes—
46C2×#11924029C2311yescharacteristic
47C2×Q8×D73C2×C413yes—
48#14556591C2×C419yes—
49#14556582C2×C4118yes—
50#14556571C2×C4118yes—
51#14556567C2×C4118yes—
52Q8×Dic73C2×C413yes—
53Q8×Dic73C2311yescharacteristic
54Q8×Dic73C2316yes—
55C4×Dic146C2×C419yes—
56#14556556C2319yes—
57C4×D292C2×C419yes—
58#14556553C2×C419yes—
59#14556549C2319yes—
60C4⋊C4×D73C2×C419yes—
61C42×D73C2319yes—
62C4⋊Q8×C73C2×C419yes—
63C4×Q8×C73C2×C413yes—
64C4×Q8×C73C2×C413yes—
65#11924038—218no—
66#11924038C2×Q8112yescomplemented
67#11924036—218no—
68#11924036C2×Q8112yescomplemented
69C2×#10596833C2413yes—
70C2×#10596833C2419yes—
71C2×#10596833C2411yescharacteristic
72C2×#10596833C2413yes—
73#11924029C2×Q814yes—
74#11924029#4114yes—
75C2×Dic146C22×C419yes—
76Q8×D73—212no—
77C2×D292C22×C413yes—
78C2×C4×D73C22×C419yes—
79C2×C4×D73C22×C419yes—
80C2×C4×D73C2413yes—
81C4⋊Dic73C22×C413yes—
82C4⋊Dic73C22×C419yes—
83C4⋊Dic73C22×C4118yes—
84#11924007C22×C4118yes—
85#11924005C22×C4118yes—
86C4×Dic73C22×C419yes—
87C4×Dic73C2413yes—
88C4×Dic73C2419yes—
89C2×Q8×C73C22×C416yes—
90C2×Q8×C73C22×C411yescharacteristic
91C2×Q8×C73C22×C411yescharacteristic
92C4⋊C4×C73C22×C419yes—
93C4⋊C4×C73C22×C419yes—
94C42×C73C22×C419yes—
95C22×D73C23×C413yes—
96Q8×C73—46no—
97Q8×C73C4×Q814yes—
98Q8×C73#13114yescomplemented
99C2×C4×C73C23×C419yes—
100C2×C4×C73C23×C413yes—
101C2×C4×C73C23×C413yes—
102C2×Dic73C23×C413yes—
103C2×Dic73C23×C419yes—
104C2×Dic73C2511yescharacteristic
105D292C4×Q8112yes—
106C4×D73—218no—
107C4×D73—49no—
108C4×D73C22×Q816yes—
109C4×D73C2×#4116yes—
110Dic146—218no—
111#10596833—23no—
112#10596833—29no—
113#10596833C22×Q812yes—
114#10596833C22×Q816yes—
115C22×C73C24×C411yescharacteristic
116C2×D73—23no—
117C2×D73—23no—
118C2×D73C2×C4×Q816yes—
119C4×C73—29no—
120C4×C73C2×C4×Q816yes—
121C4×C73C2×#13116yes—
122Dic73—29no—
123Dic73—23no—
124Dic73C23×Q811yescharacteristic
125Dic73C22×#4111yescharacteristic
126Q8×#131—731nomaximal, Sylow
127C2×C73C22×C4×Q811yescharacteristic
128C2×C73C22×#13111yescharacteristic
129C2×C73#357311yescharacteristic
130D73—26no—
131Q8×#41—731no—
132Q8×M16—733no—
133#2962—739no—
134#2960—739no—
135C2×C8×Q8—733no—
136#2906—733no—
137C4×#131—733no—
138C73Q8×#13111yescharacteristic, complemented, Sylow
139C2×#131—731no—
140C2×#131—733no—
141Q82#6440811yescharacteristic
142#557—733no—
143#554—739no—
144D4×Q8—733no—
145#533—739no—
146C4×#41—733no—
147C2×C4×Q8—733no—
148#446—7318no—
149#446—7318no—
150C8×Q8—731no—
151C8×Q8—733no—
152C8×Q8—736no—
153C8×Q8—739no—
154C8×Q8—739no—
155#424—7318no—
156#423—739no—
157C2×C4⋊C8—739no—
158#405—739no—
159#405—739no—
160C4×M16—739no—
161C2×C4×C8—739no—
162C2×#41—731no—
163C22×Q8—733no—
164#131—738no—
165#131—14612no—
166C2×M16—733no—
167C2×M16—739no—
168C22×C8—733no—
169C22×C8—739no—
170C4⋊Q8—739no—
171C4⋊Q8C2×#6440819yes—
172#125—7318no—
173C22⋊Q8—7318no—
174C22⋊Q8—7318no—
175C4×Q8—731no—
176C4×Q8—733no—
177C4×Q8—736no—
178C4×Q8—739no—
179C4×Q8—739no—
180C4×Q8C2×#6440813yes—
181C4×Q8C2×#6440813yes—
182C4×D4—739no—
183C4×D4—739no—
184#117—739no—
185C2×C4⋊C4—739no—
186C2×C42—739no—
187C4⋊C8—733no—
188C4⋊C8—739no—
189C4⋊C8—7318no—
190C4⋊C8—7318no—
191#97—7318no—
192C4×C8—733no—
193C4×C8—739no—
194C4×C8—739no—
195C4×C8—739no—
196#41—734no—
197#41—734no—
198C2×Q8—739no—
199C2×Q8—14612no—
200C2×Q8C22×#6440816yes—
201C2×Q8C22×#6440811yescharacteristic
202C2×Q8C22×#6440811yescharacteristic
203C2×D4—733no—
204C22×C4—739no—
205C22×C4—733no—
206C22×C4—739no—
207M16—7312no—
208M16—14618no—
209C2×C8—731no—
210C2×C8—733no—
211C2×C8—7312no—
212C2×C8—733no—
213C2×C8—739no—
214C2×C8—14618no—
215C4⋊C4—7318no—
216C4⋊C4—733no—
217C4⋊C4—739no—
218C4⋊C4—7318no—
219C4⋊C4C22×#6440819yes—
220C4⋊C4C22×#6440819yes—
221C22⋊C4—7318no—
222C42—739no—
223C42—733no—
224C42—739no—
225C42C22×#6440819yes—
226C23—733no—
227Q8—46no—
228Q8—14618no—
229Q8Q8×#6440814yes—
230Q8#1455667914yesdirect product factor
231D4—7312no—
232C2×C4—2929no—
233C2×C4—731no—
234C2×C4—736no—
235C2×C4—736no—
236C2×C4—733no—
237C2×C4—739no—
238C2×C4—14618no—
239C2×C4C23×#6440819yes—
240C2×C4C23×#6440813yes—
241C2×C4C23×#6440813yes—
242C8—732no—
243C8—736no—
244C8—1463no—
245C8—1469no—
246C22—736no—
247C22—1463no—
248C22—1463no—
249C22C24×#6440811yescharacteristic
250C4—29no—
251C4—731no—
252C4—731no—
253C4—1469no—
254C4—1463no—
255C4C2×Q8×#6440816yes—
256C4C2×#1455667916yes—
257C2—1466no—
258C2C22×Q8×#6440811yescharacteristic
259C2C22×#1455667911yescharacteristic
260C2#2262692611yescharacteristic
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
FittingQ82×C73C4
prime radicalC2×#11924029C23
cosocleC2×Dic73C25
hypercenterQ82#64408
hypocenterC73Q8×#131
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ Q82

Complements

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

NH ≅ G/Ntype
1. Q8×#11924038C2semidirect
2. #22626727C2semidirect
3. #22626720C2semidirect
4. Q8×#11924036C2semidirect
32. #14556679Q8direct (N × H)
66. #11924038C2×Q8semidirect
68. #11924036C2×Q8semidirect
98. Q8×C73#131semidirect
138. C73Q8×#131semidirect
230. Q8#14556679direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#119240382
#226267272
#226267202
Q8×#119240362
#226265772
C4×#145566792
Q8×#119240292
Q8×#13173