Finite Group Databasebeta

← Q8×#117×C103 #39939399

Subgroups — Q8×#117×C103 #39939399

Order 26368.

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
1C22×C4×Q8×C103C211yesmaximal, characteristic
2C103×#2355C216yesmaximal, complemented
3Q8×C4⋊C4×C103C212yesmaximal, complemented
4C103×#2345C216yesmaximal
5Q8×C22⋊C4×C103C212yesmaximal
6C103×#2295C216yesmaximal, complemented
7C103×#2285C213yesmaximal
8C42×Q8×C103C212yesmaximal, complemented
9C4×#117×C103C213yesmaximal
10C23×Q8×C103C2211yescharacteristic
11C2×C4×Q8×C103C4116yescomplemented
12C2×C4×Q8×C103C2214yes—
13C2×C4×Q8×C103C2211yescharacteristic
14C2×C4×Q8×C103C2213yes—
15C2×C4×Q8×C103C2211yescharacteristic
16C2×C4×Q8×C103C2213yes—
17C2×#117×C103C2213yes—
18C2×#117×C103C2211yescharacteristic
19C22×C4⋊C4×C103C2213yes—
20C22×C42×C103C2213yes—
21#391×C103C22112yes—
22#389×C103C22112yes—
23#387×C103C22124yes—
24#384×C103C2216yes—
25#382×C103C2216yes—
26#380×C103C2216yes—
27#380×C103C2216yes—
28C4×C4⋊C4×C103C2216yes—
29C4×C4⋊C4×C103C2216yes—
30C4×C4⋊C4×C103C2216yes—
31C4×C4⋊C4×C103C22112yes—
32C4×C4⋊C4×C103C2216yes—
33C4×C22⋊C4×C103C2216yes—
34C4×C22⋊C4×C103C2216yes—
35#376×C103C2216yes—
36C43×C103C2216yes—
37C22×Q8×C103—28no—
38C22×Q8×C103C2×C418yes—
39C22×Q8×C103C2313yes—
40C22×Q8×C103C2311yescharacteristic
41C23×C4×C103C2313yes—
42C23×C4×C103C2313yes—
43C23×C4×C103C2311yescharacteristic
44C4×Q8×C103—216no—
45C4×Q8×C103C2×C4124yes—
46C4×Q8×C103C2×C4124yes—
47C4×Q8×C103C2×C418yes—
48C4×Q8×C103C2×C418yes—
49#117×C103—212no—
50#117×C103Q818yesdirect product factor
51C2×C4⋊C4×C103C2×C4124yes—
52C2×C4⋊C4×C103C23112yes—
53C2×C4⋊C4×C103C2316yes—
54C2×C4⋊C4×C103C23112yes—
55C2×C4⋊C4×C103C2313yes—
56C2×C4⋊C4×C103C2316yes—
57C2×C4⋊C4×C103C2313yes—
58C2×C4⋊C4×C103C2312yes—
59C2×C22⋊C4×C103C2316yes—
60C2×C22⋊C4×C103C2312yes—
61C2×C42×C103C2×C4124yes—
62C2×C42×C103C23112yes—
63C2×C42×C103C2316yes—
64C2×C42×C103C23112yes—
65C2×C42×C103C2313yes—
66C2×C42×C103C2313yes—
67C2×C42×C103C2313yes—
68C2×C42×C103C2313yes—
69C2×C42×C103C2312yes—
70#95×C103C23112yes—
71#95×C103C23112yes—
72#95×C103C23112yes—
73#95×C103C23112yes—
74C24×C103C2411yescharacteristic
75C2×Q8×C103—232no—
76C2×Q8×C103—212no—
77C2×Q8×C103C22×C4112yes—
78C2×Q8×C103C22×C414yes—
79C2×Q8×C103#4118yes—
80C22×C4×C103—212no—
81C22×C4×C103—46no—
82C22×C4×C103C22×C4112yes—
83C22×C4×C103C22×C4112yes—
84C22×C4×C103C22×C414yes—
85C22×C4×C103C2×Q814yes—
86C22×C4×C103#4114yes—
87C22×C4×C103C24112yes—
88C22×C4×C103C2413yes—
89C22×C4×C103C2413yes—
90C22×C4×C103C2413yes—
91C22×C4×C103C2414yes—
92C22×C4×C103C2411yescharacteristic
93C22×C4×C103C2413yes—
94C22×C4×C103C2411yescharacteristic
95C4⋊C4×C103—212no—
96C4⋊C4×C103—224no—
97C4⋊C4×C103—412no—
98C4⋊C4×C103C22×C4124yes—
99C4⋊C4×C103C22×C4112yes—
100C4⋊C4×C103C22×C4112yes—
101C4⋊C4×C103C2×Q818yescomplemented
102C22⋊C4×C103—212no—
103C22⋊C4×C103C2×Q818yes—
104C42×C103—212no—
105C42×C103—224no—
106C42×C103—412no—
107C42×C103C22×C4112yes—
108C42×C103C22×C4112yes—
109C42×C103C22×C4112yes—
110C42×C103C22×C4112yes—
111C42×C103C2×Q818yescomplemented
112C23×C103—22no—
113C23×C103—22no—
114C23×C103C23×C412yes—
115C23×C103C22×Q812yes—
116C23×C103C2×#4112yes—
117C23×C103C2511yescharacteristic
118Q8×C103—224no—
119Q8×C103#117116yesdirect product factor
120C2×C4×C103—24no—
121C2×C4×C103—26no—
122C2×C4×C103—24no—
123C2×C4×C103—26no—
124C2×C4×C103—26no—
125C2×C4×C103—224no—
126C2×C4×C103—212no—
127C2×C4×C103—212no—
128C2×C4×C103—212no—
129C2×C4×C103—412no—
130C2×C4×C103C4×Q8116yescomplemented
131C2×C4×C103C23×C416yes—
132C2×C4×C103C23×C416yes—
133C2×C4×C103C23×C416yes—
134C2×C4×C103C23×C416yes—
135C2×C4×C103C23×C412yes—
136C2×C4×C103C23×C412yes—
137C2×C4×C103C22×Q818yes—
138C2×C4×C103C22×Q812yes—
139C2×C4×C103C22×Q812yes—
140C2×C4×C103C2×#41112yes—
141C2×C4×C103C2×#4112yes—
142C2×C4×C103C2×#4112yes—
143C22×C103—22no—
144C22×C103—22no—
145C22×C103—24no—
146C22×C103—24no—
147C22×C103C2×C4×Q814yes—
148C22×C103C24×C411yescharacteristic
149C22×C103C23×Q811yescharacteristic
150C22×C103C22×#4112yes—
151C22×C103C22×#4111yescharacteristic
152C22×C103#58512yes—
153C4×C103—26no—
154C4×C103—26no—
155C4×C103—26no—
156C4×C103—28no—
157C4×C103—412no—
158C4×C103C2×#117112yes—
159C4×C103C2×C4×Q814yes—
160C4×C103C2×C4×Q814yes—
161Q8×#117C10311yesmaximal, characteristic, direct product factor, Sylow
162C2×C103—24no—
163C2×C103C22×#11711yescharacteristic
164C2×C103C22×C4×Q811yescharacteristic
165C2×C103#342811yescharacteristic
166C2×C103Q8×#4112yes—
167C2×C103#348012yes—
168C22×C4×Q8C2×C10311yescharacteristic
169#2355C2×C10316yescomplemented
170Q8×C4⋊C4C2×C10312yescomplemented
171#2345C2×C10316yes—
172Q8×C22⋊C4C2×C10312yes—
173#2295C2×C10316yescomplemented
174#2285C2×C10313yes—
175C42×Q8C2×C10312yescomplemented
176C4×#117C2×C10313yes—
177C103Q8×#11711yescharacteristic, direct product factor, Sylow
178C23×Q8C22×C10311yescharacteristic
179C2×C4×Q8C4×C103116yescomplemented
180C2×C4×Q8C22×C10314yes—
181C2×C4×Q8C22×C10311yescharacteristic
182C2×C4×Q8C22×C10313yes—
183C2×C4×Q8C22×C10311yescharacteristic
184C2×C4×Q8C22×C10313yes—
185C2×#117C22×C10313yes—
186C2×#117C22×C10311yescharacteristic
187C22×C4⋊C4C22×C10313yes—
188C22×C42C22×C10313yes—
189#391C22×C103112yes—
190#389C22×C103112yes—
191#387C22×C103124yes—
192#384C22×C10316yes—
193#382C22×C10316yes—
194#380C22×C10316yes—
195#380C22×C10316yes—
196C4×C4⋊C4C22×C103112yes—
197C4×C4⋊C4C22×C10316yes—
198C4×C4⋊C4C22×C10316yes—
199C4×C4⋊C4C22×C10316yes—
200C4×C4⋊C4C22×C10316yes—
201C4×C22⋊C4C22×C10316yes—
202C4×C22⋊C4C22×C10316yes—
203#376C22×C10316yes—
204C43C22×C10316yes—
205C22×Q8—28no—
206C22×Q8C2×C4×C10318yes—
207C22×Q8C23×C10313yes—
208C22×Q8C23×C10311yescharacteristic
209C23×C4C23×C10313yes—
210C23×C4C23×C10313yes—
211C23×C4C23×C10311yescharacteristic
212C4×Q8—216no—
213C4×Q8C2×C4×C103124yes—
214C4×Q8C2×C4×C103124yes—
215C4×Q8C2×C4×C10318yes—
216C4×Q8C2×C4×C10318yes—
217#117—212no—
218#117Q8×C10318yesdirect product factor
219C2×C4⋊C4C2×C4×C103124yes—
220C2×C4⋊C4C23×C10316yes—
221C2×C4⋊C4C23×C103112yes—
222C2×C4⋊C4C23×C103112yes—
223C2×C4⋊C4C23×C10313yes—
224C2×C4⋊C4C23×C10316yes—
225C2×C4⋊C4C23×C10313yes—
226C2×C4⋊C4C23×C10312yes—
227C2×C22⋊C4C23×C10316yes—
228C2×C22⋊C4C23×C10312yes—
229C2×C42C2×C4×C103124yes—
230C2×C42C23×C10316yes—
231C2×C42C23×C103112yes—
232C2×C42C23×C103112yes—
233C2×C42C23×C10313yes—
234C2×C42C23×C10313yes—
235C2×C42C23×C10313yes—
236C2×C42C23×C10313yes—
237C2×C42C23×C10312yes—
238#95C23×C103112yes—
239#95C23×C103112yes—
240#95C23×C103112yes—
241#95C23×C103112yes—
242C24C24×C10311yescharacteristic
243C2×Q8—232no—
244C2×Q8—212no—
245C2×Q8C22×C4×C103112yes—
246C2×Q8C22×C4×C10314yes—
247C2×Q8#41×C10318yes—
248C22×C4—212no—
249C22×C4—46no—
250C22×C4C22×C4×C103112yes—
251C22×C4C22×C4×C103112yes—
252C22×C4C22×C4×C10314yes—
253C22×C4C2×Q8×C10314yes—
254C22×C4#41×C10314yes—
255C22×C4C24×C103112yes—
256C22×C4C24×C10313yes—
257C22×C4C24×C10313yes—
258C22×C4C24×C10313yes—
259C22×C4C24×C10314yes—
260C22×C4C24×C10311yescharacteristic
261C22×C4C24×C10311yescharacteristic
262C22×C4C24×C10313yes—
263C4⋊C4—212no—
264C4⋊C4—224no—
265C4⋊C4—412no—
266C4⋊C4C22×C4×C103124yes—
267C4⋊C4C22×C4×C103112yes—
268C4⋊C4C22×C4×C103112yes—
269C4⋊C4C2×Q8×C10318yescomplemented
270C22⋊C4—212no—
271C22⋊C4C2×Q8×C10318yes—
272C42—212no—
273C42—224no—
274C42—412no—
275C42C22×C4×C103112yes—
276C42C22×C4×C103112yes—
277C42C22×C4×C103112yes—
278C42C22×C4×C103112yes—
279C42C2×Q8×C10318yescomplemented
280C23—22no—
281C23—22no—
282C23C23×C4×C10312yes—
283C23C22×Q8×C10312yes—
284C23C2×#41×C10312yes—
285C23C25×C10311yescharacteristic
286Q8—224no—
287Q8#117×C103116yesdirect product factor
288C2×C4—24no—
289C2×C4—26no—
290C2×C4—24no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—224no—
294C2×C4—212no—
295C2×C4—212no—
296C2×C4—212no—
297C2×C4—412no—
298C2×C4C4×Q8×C103116yescomplemented
299C2×C4C23×C4×C10316yes—
300C2×C4C23×C4×C10316yes—
301C2×C4C23×C4×C10316yes—
302C2×C4C23×C4×C10316yes—
303C2×C4C23×C4×C10312yes—
304C2×C4C23×C4×C10312yes—
305C2×C4C22×Q8×C10318yes—
306C2×C4C22×Q8×C10312yes—
307C2×C4C22×Q8×C10312yes—
308C2×C4C2×#41×C103112yes—
309C2×C4C2×#41×C10312yes—
310C2×C4C2×#41×C10312yes—
311C22—24no—
312C22—24no—
313C22—22no—
314C22—22no—
315C22C2×C4×Q8×C10314yes—
316C22C24×C4×C10311yescharacteristic
317C22C23×Q8×C10311yescharacteristic
318C22C22×#41×C10312yes—
319C22C22×#41×C10311yescharacteristic
320C22#585×C10312yes—
321C4—28no—
322C4—26no—
323C4—26no—
324C4—26no—
325C4—412no—
326C4C2×#117×C103112yes—
327C4C2×C4×Q8×C10314yes—
328C4C2×C4×Q8×C10314yes—
329C2—24no—
330C2C22×#117×C10311yescharacteristic
331C2C22×C4×Q8×C10311yescharacteristic
332C2C103×#342811yescharacteristic
333C2Q8×#41×C10312yes—
334C2C103×#348012yes—
subgroupG/H
centerC22×C4×C103C24
commutatorC22C24×C4×C103
Frattini, cosocleC23C25×C103
socleC23×C103C25
prime radicalC24×C103C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C103 ◁ G

Complements

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

NH ≅ G/Ntype
2. C103×#2355C2semidirect
3. Q8×C4⋊C4×C103C2semidirect
6. C103×#2295C2semidirect
8. C42×Q8×C103C2semidirect
11. C2×C4×Q8×C103C4semidirect
50. #117×C103Q8direct (N × H)
101. C4⋊C4×C103C2×Q8semidirect
111. C42×C103C2×Q8semidirect
119. Q8×C103#117direct (N × H)
130. C2×C4×C103C4×Q8semidirect
161. Q8×#117C103direct (N × H)
169. #2355C2×C103semidirect
170. Q8×C4⋊C4C2×C103semidirect
173. #2295C2×C103semidirect
175. C42×Q8C2×C103semidirect
177. C103Q8×#117direct (N × H)
179. C2×C4×Q8C4×C103semidirect
218. #117Q8×C103direct (N × H)
269. C4⋊C4C2×Q8×C103semidirect
279. C42C2×Q8×C103semidirect
287. Q8#117×C103direct (N × H)
298. C2×C4C4×Q8×C103semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q8×C1032
C103×#23552
Q8×C4⋊C4×C1032
C103×#23452
Q8×C22⋊C4×C1032
C103×#22952
C103×#22852
C42×Q8×C1032
C4×#117×C1032
Q8×#117103