Finite Group Databasebeta

← C2×Q8×#11758189 #30544397

Subgroups — C2×Q8×#11758189 #30544397

Order 15616.

2154 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D61C211yesmaximal, characteristic
2Q8×#11758189C2116yesmaximal, direct product factor
3C2×#16839892C213yesmaximal, complemented
4C2×#16839889C213yesmaximal, complemented
5C2×Q8×#92418C211yesmaximal, characteristic, complemented
6C2×#16839784C213yesmaximal
7C2×C4×#11758189C213yesmaximal
8C2×C8×Q8×C61C211yesmaximal, characteristic, complemented
9C22×Q8×D61C411yescharacteristic
10C22×#11758189C2213yes—
11C22×#11758189C2211yescharacteristic
12C2×Q8×Dic61C411yescharacteristic
13C2×C4×Dic122C2213yes—
14C2×#14519742C413yes—
15C4×Q8×D61C418yes—
16C4×Q8×D61C2218yes—
17C2×C4⋊C4×D61C413yes—
18C2×C42×D61C2213yes—
19#16839892C22124yescomplemented
20#16839889C22124yescomplemented
21Q8×#92418C2218yescomplemented
22#16839784C22124yes—
23C2×C4⋊#92418C2213yes—
24C2×#14519706C2216yes—
25C4×#11758189C22124yes—
26C2×#14519687C2213yes—
27C2×#14519687C2213yes—
28C2×C4×#92418C2213yes—
29C2×C4×Q8×C61C2211yescharacteristic
30C8×Q8×C61C2218yescomplemented
31C2×C4⋊C8×C61C2213yes—
32C2×C4×C8×C61C2213yes—
33C22×Dic122C2×C413yes—
34C2×Q8×D61—28no—
35C2×Q8×D61C2×C418yes—
36C22×C4×D61C2×C413yes—
37C22×C4×D61C2×C413yes—
38C22×C4×D61C2311yescharacteristic
39C2×#11758189—212no—
40C2×#11758189Q818yesdirect product factor
41C2×#11758189C23112yes—
42C2×#11758189C2314yes—
43C22×#92418C2313yes—
44C22×#92418C2311yescharacteristic
45Q8×Dic61C2×C418yes—
46C4×Dic122C2×C4112yes—
47C4×Dic122C23112yes—
48#14519742C2×C4124yes—
49C2×C4⋊Dic61C2×C413yes—
50C2×#11758196C2×C416yes—
51C4⋊C4×D61C2×C4124yes—
52C2×C4×Dic61C2×C413yes—
53C2×C4×Dic61C2×C413yes—
54C2×C4×Dic61C2313yes—
55C42×D61C2×C4112yes—
56C42×D61C23112yes—
57C4⋊#92418C23112yes—
58#14519706C23124yes—
59#14519687C23112yes—
60#14519687C23112yes—
61C4×#92418C23112yes—
62C22×Q8×C61C2×C411yescharacteristic
63C22×C8×C61C2313yes—
64C22×C8×C61C2311yescharacteristic
65C4×Q8×C61C2×C414yes—
66C4×Q8×C61C2314yes—
67C2×C4⋊C4×C61C2×C413yes—
68C2×C42×C61C2313yes—
69C4⋊C8×C61C23112yes—
70C4×C8×C61C23112yes—
71C23×D61C22×C411yescharacteristic
72C2×Q8×C61M1618yescomplemented
73C2×Q8×C61C22×C414yes—
74C22×C4×C61C22×C413yes—
75C22×C4×C61C22×C413yes—
76C22×C4×C61C2411yescharacteristic
77C22×Dic61C22×C413yes—
78C22×Dic61C22×C413yes—
79C22×Dic61C22×C411yescharacteristic
80Q8×D61—232no—
81C2×C4×D61—212no—
82C2×C4×D61—46no—
83C2×C4×D61C22×C4112yes—
84C2×C4×D61C22×C4112yes—
85C2×C4×D61C22×C412yes—
86C2×C4×D61C2×Q814yes—
87C2×C4×D61#4114yes—
88C2×C4×D61C2412yes—
89C2×Dic122—212no—
90C2×Dic122C22×C4112yes—
91C2×C8×C61—26no—
92C2×C8×C61C2×Q814yescomplemented
93C2×C8×C61C2416yes—
94C2×C8×C61C2412yes—
95C4⋊C4×C61C22×C4112yes—
96C42×C61C22×C416yes—
97C42×C61C2416yes—
98C4⋊Dic61C22×C4112yes—
99#11758196C22×C4124yes—
100C4×Dic61C22×C416yes—
101C4×Dic61C22×C4112yes—
102C4×Dic61C22×C4112yes—
103C4×Dic61C2416yes—
104C2×#92418—26no—
105C2×#92418C2×Q814yescomplemented
106C2×#92418C2416yes—
107C2×#92418C2412yes—
108#11758189—224no—
109#11758189C2×Q8116yesdirect product factor
110C23×C61C23×C411yescharacteristic
111C22×D61—22no—
112C22×D61—22no—
113C22×D61C4×Q814yes—
114C22×D61C23×C412yes—
115Q8×C61C2×M16116yescomplemented
116C2×C4×C61—26no—
117C2×C4×C61C2×M16112yes—
118C2×C4×C61C23×C416yes—
119C2×C4×C61C23×C411yescharacteristic
120C2×C4×C61C23×C416yes—
121C2×C4×C61C22×Q812yes—
122C2×C4×C61C2×#4112yes—
123C2×C4×C61C2511yescharacteristic
124C2×Dic61—26no—
125C2×Dic61—26no—
126C2×Dic61C4×Q814yes—
127C2×Dic61C23×C416yes—
128C2×Dic61C23×C416yes—
129C2×Dic61C23×C412yes—
130C4×D61—224no—
131C4×D61—412no—
132C4×D61C4×Q818yes—
133C4×D61C22×Q814yes—
134C4×D61C2×#4114yes—
135Dic122—224no—
136C8×C61—26no—
137C8×C61C22×Q814yescomplemented
138#92418—26no—
139#92418C22×Q814yescomplemented
140C2×Q8×M16—611nomaximal, Sylow
141C22×C61C2×C4×Q812yes—
142C22×C61C22×M1612yes—
143C22×C61#56812yes—
144C22×C61C24×C411yescharacteristic
145C2×D61—28no—
146C2×D61—22no—
147C2×D61—22no—
148C2×D61C2×C4×Q814yes—
149C4×C61—26no—
150C4×C61C2×C4×Q812yes—
151C4×C61C22×M16112yes—
152C4×C61C23×Q811yescharacteristic
153C4×C61C22×#4111yescharacteristic
154Dic61—26no—
155Dic61—26no—
156Dic61C2×C4×Q814yes—
157C22×C4×Q8—611no—
158Q8×M16—6116no—
159C2×#446—613no—
160C2×#446—613no—
161C2×C8×Q8—611no—
162C2×C8×Q8D6111yescharacteristic, complemented
163C2×#423—613no—
164C2×C4×M16—613no—
165C2×C61Q8×M1614yescomplemented
166C2×C61C22×C4×Q811yescharacteristic
167C2×C61C23×M1611yescharacteristic
168C2×C61C2×#56811yescharacteristic
169D61—24no—
170C23×Q8—611no—
171C22×M16—611no—
172C22×M16—613no—
173C2×C4×Q8—611no—
174C2×C4×Q8—618no—
175C2×C4×Q8—618no—
176C2×C4×Q8—613no—
177C2×C4×Q8—613no—
178C2×C4×Q8C2×D6111yescharacteristic
179C22×C4⋊C4—613no—
180C22×C42—613no—
181#446—6124no—
182#446—6124no—
183C8×Q8—618no—
184C8×Q8C2×D6118yescomplemented
185#423—6124no—
186C2×C4⋊C8—616no—
187C2×C4⋊C8—613no—
188C2×C4⋊C8C2×D6113yes—
189C4×M16—6124no—
190C2×#97—613no—
191C2×#97—613no—
192C2×C4×C8—613no—
193C2×C4×C8C2×D6113yes—
194C61C2×Q8×M1611yescharacteristic, complemented, Sylow
195C22×Q8—618no—
196C22×Q8—613no—
197C22×Q8—1228no—
198C22×Q8C4×D6111yescharacteristic
199C23×C4—611no—
200C23×C4—613no—
201C23×C4—613no—
202C2×M16—618no—
203C2×M16—614no—
204C2×M16—6112no—
205C2×M16—12212no—
206C22×C8—611no—
207C22×C8—613no—
208C22×C8C22×D6113yes—
209C22×C8C22×D6111yescharacteristic
210C4×Q8—618no—
211C4×Q8—6124no—
212C4×Q8—6112no—
213C4×Q8—6112no—
214C4×Q8C4×D6114yes—
215C4×Q8C22×D6114yes—
216C2×C4⋊C4—616no—
217C2×C4⋊C4—613no—
218C2×C4⋊C4—6124no—
219C2×C4⋊C4C4×D6113yes—
220C2×C42—613no—
221C2×C42—613no—
222C2×C42—613no—
223C2×C42—6112no—
224C2×C42—6112no—
225C2×C42C22×D6113yes—
226C4⋊C8—6124no—
227C4⋊C8—6112no—
228C4⋊C8C22×D61112yes—
229#97—6112no—
230#97—6112no—
231C4×C8—6112no—
232C4×C8C22×D61112yes—
233C24—611no—
234C2×Q8—6112no—
235C2×Q8—12232no—
236C2×Q8—12212no—
237C2×Q8#1175818918yesdirect product factor
238C2×Q8C2×C4×D6114yes—
239C22×C4—611no—
240C22×C4—612no—
241C22×C4—614no—
242C22×C4—614no—
243C22×C4—612no—
244C22×C4—613no—
245C22×C4—613no—
246C22×C4—6112no—
247C22×C4—6112no—
248C22×C4—12212no—
249C22×C4—2446no—
250C22×C4C2×C4×D6113yes—
251C22×C4C2×C4×D6113yes—
252C22×C4C23×D6111yescharacteristic
253M16—6116no—
254M16—12224no—
255C2×C8—26no—
256C2×C8—614no—
257C2×C8—612no—
258C2×C8—616no—
259C2×C8—1226no—
260C2×C8Q8×D6114yescomplemented
261C2×C8C23×D6116yes—
262C2×C8C23×D6112yes—
263C4⋊C4—6124no—
264C4⋊C4—6112no—
265C4⋊C4C2×C4×D61112yes—
266C42—6112no—
267C42—6112no—
268C42—616no—
269C42—616no—
270C42C2×C4×D6116yes—
271C42C23×D6116yes—
272C23—614no—
273C23—612no—
274C23—1222no—
275C23—1222no—
276C23C22×C4×D6111yescharacteristic
277Q8—12224no—
278Q8C2×#11758189116yesdirect product factor
279C2×C4—26no—
280C2×C4—614no—
281C2×C4—612no—
282C2×C4—618no—
283C2×C4—614no—
284C2×C4—614no—
285C2×C4—616no—
286C2×C4—616no—
287C2×C4—1226no—
288C2×C4—1226no—
289C2×C4—12224no—
290C2×C4—24412no—
291C2×C4C2×#11758189112yes—
292C2×C4C22×C4×D6116yes—
293C2×C4C22×C4×D6111yescharacteristic
294C2×C4C22×C4×D6116yes—
295C2×C4C2×Q8×D6112yes—
296C2×C4#41×D6112yes—
297C2×C4C24×D6111yescharacteristic
298C8—26no—
299C8—614no—
300C8—1226no—
301C8C2×Q8×D6114yescomplemented
302C22—614no—
303C22—1222no—
304C22—1222no—
305C22—1228no—
306C22C4×Q8×D6112yes—
307C22C22×#1175818912yes—
308C22#1684052712yes—
309C22C23×C4×D6111yescharacteristic
310C4—26no—
311C4—614no—
312C4—1226no—
313C4—1226no—
314C4C4×Q8×D6112yes—
315C4C22×#11758189112yes—
316C4C22×Q8×D6111yescharacteristic
317C4C2×#41×D6111yescharacteristic
318C2—1224no—
319C2Q8×#1175818914yesdirect product factor
320C2C2×C4×Q8×D6111yescharacteristic
321C2C23×#1175818911yescharacteristic
322C2C2×#1684052711yescharacteristic
subgroupG/H
centerC22×C4C23×D61
commutatorC22×C61C24×C4
FrattiniC2×C4C24×D61
FittingC2×C8×Q8×C61C2
socleC23×C61C23×C4
prime radicalC23×D61C22×C4
cosocleC2×C4×C61C25
hypercenterC2×C8×Q8D61
hypocenterC61C2×Q8×M16
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. Q8×#11758189C2direct (N × H)
3. C2×#16839892C2semidirect
4. C2×#16839889C2semidirect
5. C2×Q8×#92418C2semidirect
8. C2×C8×Q8×C61C2semidirect
19. #16839892C22semidirect
20. #16839889C22semidirect
21. Q8×#92418C22semidirect
30. C8×Q8×C61C22semidirect
40. C2×#11758189Q8direct (N × H)
72. C2×Q8×C61M16semidirect
92. C2×C8×C61C2×Q8semidirect
105. C2×#92418C2×Q8semidirect
109. #11758189C2×Q8direct (N × H)
115. Q8×C61C2×M16semidirect
137. C8×C61C22×Q8semidirect
139. #92418C22×Q8semidirect
162. C2×C8×Q8D61semidirect
165. C2×C61Q8×M16semidirect
184. C8×Q8C2×D61semidirect
194. C61C2×Q8×M16semidirect
237. C2×Q8#11758189direct (N × H)
260. C2×C8Q8×D61semidirect
278. Q8C2×#11758189direct (N × H)
301. C8C2×Q8×D61semidirect
319. C2Q8×#11758189direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D612
Q8×#117581892
C2×#168398922
C2×#168398892
C2×Q8×#924182
C2×#168397842
C2×C4×#117581892
C2×C8×Q8×C612
C2×Q8×M1661