Finite Group Databasebeta

← Q8×#14556550 #33245413

Subgroups — Q8×#14556550 #33245413

Order 18688.

2434 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612888C216yesmaximal, complemented
2Q8×C4⋊Dic73C212yesmaximal, complemented
3#22612808C216yesmaximal
4Q8×C22⋊Dic73C212yesmaximal
5#22612409C216yesmaximal, complemented
6#22612338C213yesmaximal
7C4×Q8×Dic73C212yesmaximal, complemented
8C4×#14556550C213yesmaximal
9C22×C4×Q8×C73C211yesmaximal, characteristic
10C2×Q8×Dic73C2214yes—
11C2×#14556550C2213yes—
12C2×#14556550C2211yescharacteristic
13#17988494C22112yes—
14#17988486C22112yes—
15#17988473C22124yes—
16#17988457C2216yes—
17#17988453C2216yes—
18#17988447C2216yes—
19#17988447C2216yes—
20C4⋊C4×Dic73C22112yes—
21C4×C4⋊Dic73C2216yes—
22C4×C4⋊Dic73C2216yes—
23C4×C4⋊Dic73C2216yes—
24C4×C4⋊Dic73C2216yes—
25C4×C22⋊Dic73C2216yes—
26C4×C22⋊Dic73C2216yes—
27#17988433C2216yes—
28C42×Dic73C2216yes—
29C23×Q8×C73C2211yescharacteristic
30C2×C4×Q8×C73C4116yescomplemented
31C2×C4×Q8×C73C2211yescharacteristic
32C2×C4×Q8×C73C2211yescharacteristic
33C2×C4×Q8×C73C2213yes—
34C2×C4×Q8×C73C2213yes—
35C22×C4⋊C4×C73C2213yes—
36C22×C42×C73C2213yes—
37Q8×Dic73—216no—
38#14556550—212no—
39#14556550Q818yesdirect product factor
40C2×C4⋊Dic73C2316yes—
41C2×C4⋊Dic73C23112yes—
42C2×C4⋊Dic73C23112yes—
43C2×C4⋊Dic73C2312yes—
44C2×C22⋊Dic73C2316yes—
45C2×C22⋊Dic73C2312yes—
46C2×C4×Dic73C2316yes—
47C2×C4×Dic73C23112yes—
48C2×C4×Dic73C23112yes—
49C2×C4×Dic73C2312yes—
50#14556498C23112yes—
51#14556498C23112yes—
52#14556498C23112yes—
53#14556498C23112yes—
54C22×Q8×C73—28no—
55C22×Q8×C73C2×C418yes—
56C22×Q8×C73C2313yes—
57C22×Q8×C73C2311yescharacteristic
58C23×C4×C73C2313yes—
59C23×C4×C73C2313yes—
60C23×C4×C73C2311yescharacteristic
61C4×Q8×C73C2×C4124yes—
62C4×Q8×C73C2×C4124yes—
63C4×Q8×C73C2×C418yes—
64C4×Q8×C73C2×C418yes—
65C2×C4⋊C4×C73C2×C4124yes—
66C2×C4⋊C4×C73C2313yes—
67C2×C4⋊C4×C73C2316yes—
68C2×C4⋊C4×C73C2313yes—
69C2×C42×C73C2×C4124yes—
70C2×C42×C73C2313yes—
71C2×C42×C73C2313yes—
72C2×C42×C73C2313yes—
73C2×C42×C73C2313yes—
74C22×Dic73C24112yes—
75C22×Dic73C2414yes—
76C4⋊Dic73—212no—
77C4⋊Dic73—224no—
78C4⋊Dic73—412no—
79C4⋊Dic73C2×Q818yescomplemented
80C22⋊Dic73—212no—
81C22⋊Dic73C2×Q818yes—
82C4×Dic73—212no—
83C4×Dic73—224no—
84C4×Dic73—412no—
85C4×Dic73C2×Q818yescomplemented
86C24×C73C2411yescharacteristic
87C2×Q8×C73—232no—
88C2×Q8×C73—212no—
89C2×Q8×C73C22×C4112yes—
90C2×Q8×C73C22×C414yes—
91C2×Q8×C73#4118yes—
92C22×C4×C73—212no—
93C22×C4×C73—46no—
94C22×C4×C73C22×C4112yes—
95C22×C4×C73C22×C4112yes—
96C22×C4×C73C22×C414yes—
97C22×C4×C73C2×Q814yes—
98C22×C4×C73#4114yes—
99C22×C4×C73C2413yes—
100C22×C4×C73C2413yes—
101C22×C4×C73C2413yes—
102C22×C4×C73C2411yescharacteristic
103C22×C4×C73C2413yes—
104C22×C4×C73C2411yescharacteristic
105C4⋊C4×C73C22×C4124yes—
106C4⋊C4×C73C22×C4112yes—
107C4⋊C4×C73C22×C4112yes—
108C42×C73C22×C4112yes—
109C42×C73C22×C4112yes—
110C42×C73C22×C4112yes—
111C42×C73C22×C4112yes—
112C23×C73—22no—
113C23×C73—22no—
114C23×C73C23×C412yes—
115C23×C73C22×Q812yes—
116C23×C73C2×#4112yes—
117C23×C73C2511yescharacteristic
118Q8×C73—224no—
119Q8×C73#117116yescomplemented
120C2×C4×C73—26no—
121C2×C4×C73—26no—
122C2×C4×C73—26no—
123C2×C4×C73—224no—
124C2×C4×C73—412no—
125C2×C4×C73C4×Q8116yescomplemented
126C2×C4×C73C23×C416yes—
127C2×C4×C73C23×C416yes—
128C2×C4×C73C23×C416yes—
129C2×C4×C73C23×C416yes—
130C2×C4×C73C23×C412yes—
131C2×C4×C73C23×C412yes—
132C2×C4×C73C22×Q812yes—
133C2×C4×C73C22×Q812yes—
134C2×C4×C73C2×#41112yes—
135C2×C4×C73C2×#4112yes—
136C2×C4×C73C2×#4112yes—
137C2×Dic73—24no—
138C2×Dic73—24no—
139C2×Dic73—212no—
140C2×Dic73—212no—
141C2×Dic73—212no—
142C2×Dic73C22×Q818yes—
143C22×C73—22no—
144C22×C73—22no—
145C22×C73—24no—
146C22×C73—24no—
147C22×C73C2×C4×Q814yes—
148C22×C73C24×C411yescharacteristic
149C22×C73C23×Q811yescharacteristic
150C22×C73C22×#4112yes—
151C22×C73C22×#4111yescharacteristic
152C22×C73#58512yes—
153C4×C73—26no—
154C4×C73—26no—
155C4×C73—26no—
156C4×C73C2×#117112yes—
157C4×C73C2×C4×Q814yes—
158C4×C73C2×C4×Q814yes—
159Dic73—28no—
160Dic73—412no—
161Q8×#117—731nomaximal, Sylow
162C2×C73—24no—
163C2×C73C22×#11711yescharacteristic
164C2×C73C22×C4×Q811yescharacteristic
165C2×C73#342811yescharacteristic
166C2×C73Q8×#4112yes—
167C2×C73#348012yes—
168C22×C4×Q8D7311yescharacteristic
169#2355—736no—
170Q8×C4⋊C4—732no—
171#2345—736no—
172Q8×C22⋊C4—732no—
173#2295—736no—
174#2285—733no—
175C42×Q8—732no—
176C4×#117—733no—
177C73Q8×#11711yescharacteristic, complemented, Sylow
178C23×Q8C2×D7311yescharacteristic
179C2×C4×Q8—734no—
180C2×C4×Q8Dic73116yescomplemented
181C2×C4×Q8C2×D7311yescharacteristic
182C2×C4×Q8C2×D7313yes—
183C2×C4×Q8C2×D7311yescharacteristic
184C2×C4×Q8C2×D7313yes—
185C2×#117—731no—
186C2×#117—733no—
187C22×C4⋊C4C2×D7313yes—
188C22×C42C2×D7313yes—
189#391—7312no—
190#389—7312no—
191#387—7324no—
192#384—736no—
193#382—736no—
194#380—736no—
195#380—736no—
196C4×C4⋊C4—7312no—
197C4×C4⋊C4—736no—
198C4×C4⋊C4—736no—
199C4×C4⋊C4—736no—
200C4×C4⋊C4—736no—
201C4×C22⋊C4—736no—
202C4×C22⋊C4—736no—
203#376—736no—
204C43—736no—
205C22×Q8—28no—
206C22×Q8C2×Dic7318yes—
207C22×Q8C22×D7313yes—
208C22×Q8C22×D7311yescharacteristic
209C23×C4C22×D7313yes—
210C23×C4C22×D7313yes—
211C23×C4C22×D7311yescharacteristic
212C4×Q8—14616no—
213C4×Q8C2×Dic73124yes—
214C4×Q8C2×Dic73124yes—
215C4×Q8C2×Dic7318yes—
216C4×Q8C2×Dic7318yes—
217#117—738no—
218#117—14612no—
219C2×C4⋊C4—7312no—
220C2×C4⋊C4—732no—
221C2×C4⋊C4—736no—
222C2×C4⋊C4—7312no—
223C2×C4⋊C4C2×Dic73124yes—
224C2×C4⋊C4C22×D7313yes—
225C2×C4⋊C4C22×D7316yes—
226C2×C4⋊C4C22×D7313yes—
227C2×C22⋊C4—732no—
228C2×C22⋊C4—736no—
229C2×C42—7312no—
230C2×C42—732no—
231C2×C42—7312no—
232C2×C42—736no—
233C2×C42C2×Dic73124yes—
234C2×C42C22×D7313yes—
235C2×C42C22×D7313yes—
236C2×C42C22×D7313yes—
237C2×C42C22×D7313yes—
238#95—7312no—
239#95—7312no—
240#95—7312no—
241#95—7312no—
242C24C23×D7311yescharacteristic
243C2×Q8—232no—
244C2×Q8—212no—
245C2×Q8C22×Dic73112yes—
246C2×Q8C22×Dic7314yes—
247C2×Q8#1192402818yes—
248C22×C4—212no—
249C22×C4—46no—
250C22×C4—734no—
251C22×C4—7312no—
252C22×C4C22×Dic73112yes—
253C22×C4C22×Dic73112yes—
254C22×C4C22×Dic7314yes—
255C22×C4Q8×D7314yes—
256C22×C4#1192402914yes—
257C22×C4C23×D7313yes—
258C22×C4C23×D7313yes—
259C22×C4C23×D7313yes—
260C22×C4C23×D7311yescharacteristic
261C22×C4C23×D7313yes—
262C22×C4C23×D7311yescharacteristic
263C4⋊C4—29212no—
264C4⋊C4—738no—
265C4⋊C4—14624no—
266C4⋊C4—14612no—
267C4⋊C4C22×Dic73124yes—
268C4⋊C4C22×Dic73112yes—
269C4⋊C4C22×Dic73112yes—
270C22⋊C4—738no—
271C22⋊C4—14612no—
272C42—29212no—
273C42—738no—
274C42—14624no—
275C42—14612no—
276C42C22×Dic73112yes—
277C42C22×Dic73112yes—
278C42C22×Dic73112yes—
279C42C22×Dic73112yes—
280C23—22no—
281C23—22no—
282C23C23×Dic7312yes—
283C23C2×Q8×D7312yes—
284C23C2×#1192402912yes—
285C23C24×D7311yescharacteristic
286Q8—224no—
287Q8#14556550116yesdirect product factor
288C2×C4—26no—
289C2×C4—26no—
290C2×C4—26no—
291C2×C4—224no—
292C2×C4—412no—
293C2×C4—738no—
294C2×C4—14612no—
295C2×C4—14612no—
296C2×C4—14612no—
297C2×C4—1464no—
298C2×C4—1464no—
299C2×C4Q8×Dic73116yescomplemented
300C2×C4C23×Dic7316yes—
301C2×C4C23×Dic7316yes—
302C2×C4C23×Dic7316yes—
303C2×C4C23×Dic7316yes—
304C2×C4C23×Dic7312yes—
305C2×C4C23×Dic7312yes—
306C2×C4C2×Q8×D7312yes—
307C2×C4C2×Q8×D7312yes—
308C2×C4C2×#11924028112yes—
309C2×C4C2×#1192402912yes—
310C2×C4C2×#1192402912yes—
311C22—24no—
312C22—24no—
313C22—22no—
314C22—22no—
315C22C2×Q8×Dic7314yes—
316C22C24×Dic7311yescharacteristic
317C22C22×Q8×D7311yescharacteristic
318C22C22×#1192402812yes—
319C22C22×#1192402911yescharacteristic
320C22#1798942312yes—
321C4—26no—
322C4—26no—
323C4—26no—
324C4—29212no—
325C4—1468no—
326C4C2×#14556550112yes—
327C4C2×Q8×Dic7314yes—
328C4C2×Q8×Dic7314yes—
329C2—24no—
330C2C22×#1455655011yescharacteristic
331C2C22×Q8×Dic7311yescharacteristic
332C2#2262313411yescharacteristic
333C2Q8×#1192402812yes—
334C2#2262352612yes—
subgroupG/H
centerC22×C4C23×D73
commutatorC22×C73C24×C4
FrattiniC23C24×D73
FittingC22×C4×Q8×C73C2
socle, cosocleC23×C73C25
prime radicalC24×C73C24
hypercenterC22×C4×Q8D73
hypocenterC73Q8×#117
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #22612888C2semidirect
2. Q8×C4⋊Dic73C2semidirect
5. #22612409C2semidirect
7. C4×Q8×Dic73C2semidirect
30. C2×C4×Q8×C73C4semidirect
39. #14556550Q8direct (N × H)
79. C4⋊Dic73C2×Q8semidirect
85. C4×Dic73C2×Q8semidirect
119. Q8×C73#117semidirect
125. C2×C4×C73C4×Q8semidirect
177. C73Q8×#117semidirect
180. C2×C4×Q8Dic73semidirect
287. Q8#14556550direct (N × H)
299. C2×C4Q8×Dic73semidirect

Maximal subgroups

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

subgroupindex
#226128882
Q8×C4⋊Dic732
#226128082
Q8×C22⋊Dic732
#226124092
#226123382
C4×Q8×Dic732
C4×#145565502
C22×C4×Q8×C732
Q8×#11773