Finite Group Databasebeta

← C2×Q8×#11924037 #34031490

Subgroups — C2×Q8×#11924037 #34031490

Order 18688.

2154 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11924037C2116yesmaximal, direct product factor
2C2×#17989638C216yesmaximal, complemented
3C2×Q8×#10596833C212yesmaximal, complemented
4C2×#17989590C213yesmaximal
5C2×C4×#11924037C213yesmaximal
6C22×Q8×Dic73C211yesmaximal, characteristic
7C22×#11924037C2213yes—
8C22×#11924037C2211yescharacteristic
9#17989638C22148yescomplemented
10Q8×#10596833C22116yescomplemented
11#17989590C22124yes—
12C2×#14556658C2216yes—
13C2×C4⋊#10596833C2216yes—
14C4×#11924037C22124yes—
15C2×#14556646C2216yes—
16C2×C4×#10596833C2216yes—
17C2×Q8×Dic73C418yes—
18C2×Q8×Dic73C413yes—
19C2×Q8×Dic73C411yescharacteristic
20C2×Q8×Dic73C2211yescharacteristic
21C2×Q8×Dic73C2213yes—
22C2×Q8×Dic73C2218yes—
23C22×C4⋊Dic73C413yes—
24C22×C4×Dic73C2213yes—
25C23×Q8×C73C411yescharacteristic
26C2×#11924037—212no—
27C2×#11924037Q818yesdirect product factor
28C2×#11924037C23112yes—
29C2×#11924037C2314yes—
30C22×#10596833C2316yes—
31C22×#10596833C2312yes—
32#14556658C23124yes—
33C4⋊#10596833C23124yes—
34#14556646C23124yes—
35C4×#10596833C23124yes—
36C23×Dic73C2×C413yes—
37C23×Dic73C2311yescharacteristic
38Q8×Dic73C2×C4112yes—
39Q8×Dic73C2×C4124yes—
40Q8×Dic73C2×C414yes—
41Q8×Dic73C2×C418yes—
42Q8×Dic73C2314yes—
43Q8×Dic73C23112yes—
44C2×C4⋊Dic73C2×C416yes—
45C2×C4⋊Dic73C2×C4124yes—
46C2×C4⋊Dic73C2×C413yes—
47C2×C4⋊Dic73C2×C413yes—
48C2×C4×Dic73C2×C413yes—
49C2×C4×Dic73C2×C4112yes—
50C2×C4×Dic73C2×C413yes—
51C2×C4×Dic73C2313yes—
52C2×C4×Dic73C2313yes—
53C2×C4×Dic73C23112yes—
54C22×Q8×C73—28no—
55C22×Q8×C73C2×C418yes—
56C22×Q8×C73C2×C413yes—
57C22×Q8×C73C2×C411yescharacteristic
58C23×C4×C73C2×C413yes—
59#11924037—224no—
60#11924037C2×Q8116yesdirect product factor
61C2×#10596833—212no—
62C2×#10596833C2×Q818yescomplemented
63C2×#10596833C24112yes—
64C2×#10596833C2414yes—
65C22×Dic73—46no—
66C22×Dic73C22×C413yes—
67C22×Dic73C22×C413yes—
68C22×Dic73C22×C4112yes—
69C22×Dic73C22×C412yes—
70C22×Dic73C22×C411yescharacteristic
71C22×Dic73C2×Q814yes—
72C22×Dic73#4114yes—
73C22×Dic73C2411yescharacteristic
74C22×Dic73C2412yes—
75C4⋊Dic73C22×C4124yes—
76C4⋊Dic73C22×C4112yes—
77C4⋊Dic73C22×C4112yes—
78C4×Dic73C22×C416yes—
79C4×Dic73C22×C4112yes—
80C4×Dic73C22×C416yes—
81C4×Dic73C22×C4112yes—
82C4×Dic73C2416yes—
83C4×Dic73C2416yes—
84C24×C73C22×C411yescharacteristic
85C2×Q8×C73—232no—
86C2×Q8×C73—212no—
87C2×Q8×C73M1618yescomplemented
88C2×Q8×C73C22×C4112yes—
89C2×Q8×C73C22×C414yes—
90C22×C4×C73—212no—
91C22×C4×C73C22×C4112yes—
92C22×C4×C73C22×C413yes—
93C22×C4×C73C22×C413yes—
94C23×C73—22no—
95C23×C73—22no—
96C23×C73C4×Q814yes—
97C23×C73C23×C412yes—
98C23×C73C23×C411yescharacteristic
99Q8×C73—224no—
100Q8×C73C2×M16116yescomplemented
101C2×C4×C73—26no—
102C2×C4×C73—224no—
103C2×C4×C73C2×M16112yes—
104C2×C4×C73C23×C416yes—
105C2×C4×C73C23×C416yes—
106C2×Dic73—26no—
107C2×Dic73—26no—
108C2×Dic73—412no—
109C2×Dic73C4×Q818yes—
110C2×Dic73C4×Q814yes—
111C2×Dic73C23×C416yes—
112C2×Dic73C23×C416yes—
113C2×Dic73C23×C411yescharacteristic
114C2×Dic73C23×C412yes—
115C2×Dic73C22×Q812yes—
116C2×Dic73C22×Q814yes—
117C2×Dic73C2×#4112yes—
118C2×Dic73C2×#4114yes—
119C2×Dic73C2511yescharacteristic
120#10596833—212no—
121#10596833C22×Q818yescomplemented
122C22×C73—28no—
123C22×C73—22no—
124C22×C73—22no—
125C22×C73C2×C4×Q814yes—
126C22×C73C2×C4×Q812yes—
127C22×C73C22×M1612yes—
128C22×C73#56812yes—
129C22×C73C24×C411yescharacteristic
130C4×C73—26no—
131C4×C73C22×M16112yes—
132Dic73—26no—
133Dic73—26no—
134Dic73C2×C4×Q812yes—
135Dic73C2×C4×Q814yes—
136Dic73C23×Q811yescharacteristic
137Dic73C22×#4111yescharacteristic
138C2×Q8×M16—731nomaximal, Sylow
139C2×C73—24no—
140C2×C73Q8×M1614yescomplemented
141C2×C73C22×C4×Q811yescharacteristic
142C2×C73C23×M1611yescharacteristic
143C2×C73C2×#56811yescharacteristic
144C22×C4×Q8—731no—
145Q8×M16—7316no—
146C2×#446—736no—
147C2×C8×Q8—732no—
148C2×#423—733no—
149C2×C4×M16—733no—
150C73C2×Q8×M1611yescharacteristic, complemented, Sylow
151C23×Q8#6440811yescharacteristic
152C22×M16—731no—
153C22×M16—733no—
154C2×C4×Q8—731no—
155C2×C4×Q8—733no—
156C2×C4×Q8—738no—
157C2×C4×Q8—738no—
158C2×C4×Q8—733no—
159C2×C4×Q8—731no—
160C22×C4⋊C4—733no—
161C22×C42—733no—
162#446—7348no—
163C8×Q8—7316no—
164#423—7324no—
165C2×C4⋊C8—736no—
166C2×C4⋊C8—736no—
167C4×M16—7324no—
168C2×#97—736no—
169C2×C4×C8—736no—
170C22×Q8—28no—
171C22×Q8C2×#6440818yes—
172C22×Q8C2×#6440813yes—
173C22×Q8C2×#6440811yescharacteristic
174C23×C4—733no—
175C23×C4—731no—
176C23×C4C2×#6440813yes—
177C2×M16—738no—
178C2×M16—734no—
179C2×M16—7312no—
180C2×M16—14612no—
181C22×C8—732no—
182C22×C8—736no—
183C4×Q8—7312no—
184C4×Q8—7324no—
185C4×Q8—734no—
186C4×Q8—738no—
187C4×Q8—734no—
188C4×Q8—7312no—
189C2×C4⋊C4—7324no—
190C2×C4⋊C4—736no—
191C2×C4⋊C4—733no—
192C2×C4⋊C4—733no—
193C2×C42—7312no—
194C2×C42—733no—
195C2×C42—733no—
196C2×C42—733no—
197C2×C42—733no—
198C2×C42—7312no—
199C4⋊C8—7324no—
200C4⋊C8—7324no—
201#97—7324no—
202C4×C8—7324no—
203C24C22×#6440811yescharacteristic
204C2×Q8—232no—
205C2×Q8—212no—
206C2×Q8#1192403718yesdirect product factor
207C2×Q8C22×#64408112yes—
208C2×Q8C22×#6440814yes—
209C22×C4—212no—
210C22×C4—2926no—
211C22×C4—7312no—
212C22×C4—734no—
213C22×C4—734no—
214C22×C4—732no—
215C22×C4—732no—
216C22×C4—731no—
217C22×C4—731no—
218C22×C4—733no—
219C22×C4—733no—
220C22×C4C22×#64408112yes—
221C22×C4C22×#6440813yes—
222C22×C4C22×#6440813yes—
223M16—7316no—
224M16—14624no—
225C2×C8—738no—
226C2×C8—734no—
227C2×C8—7312no—
228C2×C8—14612no—
229C4⋊C4—7324no—
230C4⋊C4—7312no—
231C4⋊C4—7312no—
232C42—736no—
233C42—7312no—
234C42—736no—
235C42—7312no—
236C42—736no—
237C42—736no—
238C23—22no—
239C23—22no—
240C23Q8×#6440814yes—
241C23C23×#6440812yes—
242C23C23×#6440811yescharacteristic
243Q8—224no—
244Q8C2×#11924037116yesdirect product factor
245C2×C4—26no—
246C2×C4—224no—
247C2×C4—29212no—
248C2×C4—732no—
249C2×C4—732no—
250C2×C4—734no—
251C2×C4—734no—
252C2×C4—738no—
253C2×C4—734no—
254C2×C4—731no—
255C2×C4—731no—
256C2×C4—732no—
257C2×C4—736no—
258C2×C4—736no—
259C2×C4—1466no—
260C2×C4—1466no—
261C2×C4C2×#11924037112yes—
262C2×C4C23×#6440816yes—
263C2×C4C23×#6440816yes—
264C8—738no—
265C8—14612no—
266C22—22no—
267C22—22no—
268C22—28no—
269C22C2×Q8×#6440814yes—
270C22C2×Q8×#6440812yes—
271C22C22×#1192403712yes—
272C22#1798966712yes—
273C22C24×#6440811yescharacteristic
274C4—26no—
275C4—731no—
276C4—731no—
277C4—732no—
278C4—734no—
279C4—1466no—
280C4—1466no—
281C4C22×#11924037112yes—
282C2—24no—
283C2Q8×#1192403714yesdirect product factor
284C2C22×Q8×#6440811yescharacteristic
285C2C23×#1192403711yescharacteristic
286C2C2×#1798966711yescharacteristic
subgroupG/H
centerC23C23×#64408
commutatorC22×C73C24×C4
FrattiniC22C24×#64408
FittingC23×Q8×C73C4
socleC23×C73C23×C4
prime radicalC24×C73C22×C4
cosocleC2×Dic73C25
hypercenterC23×Q8#64408
hypocenterC73C2×Q8×M16
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#11924037C2direct (N × H)
2. C2×#17989638C2semidirect
3. C2×Q8×#10596833C2semidirect
9. #17989638C22semidirect
10. Q8×#10596833C22semidirect
27. C2×#11924037Q8direct (N × H)
60. #11924037C2×Q8direct (N × H)
62. C2×#10596833C2×Q8semidirect
87. C2×Q8×C73M16semidirect
100. Q8×C73C2×M16semidirect
121. #10596833C22×Q8semidirect
140. C2×C73Q8×M16semidirect
150. C73C2×Q8×M16semidirect
206. C2×Q8#11924037direct (N × H)
244. Q8C2×#11924037direct (N × H)
283. C2Q8×#11924037direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#119240372
C2×#179896382
C2×Q8×#105968332
C2×#179895902
C2×C4×#119240372
C22×Q8×Dic732
C2×Q8×M1673