Finite Group Databasebeta

← C2×Q8×#63218 #17898708

Subgroups — C2×Q8×#63218 #17898708

Order 4352.

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×D17C211yesmaximal, characteristic
2Q8×#63218C2116yesmaximal, direct product factor
3C2×#11762939C213yesmaximal, complemented
4C2×#11762936C213yesmaximal, complemented
5C2×Q8×#3622C211yesmaximal, characteristic, complemented
6C2×#11762831C213yesmaximal
7C2×C4×#63218C213yesmaximal
8C2×C8×Q8×C17C211yesmaximal, characteristic, complemented
9C22×Q8×D17C411yescharacteristic
10C22×#63218C2213yes—
11C22×#63218C2211yescharacteristic
12C2×Q8×Dic17C411yescharacteristic
13C2×C4×Dic34C2213yes—
14C2×#10587573C413yes—
15C4×Q8×D17C418yes—
16C4×Q8×D17C2218yes—
17C2×C4⋊C4×D17C413yes—
18C2×C42×D17C2213yes—
19#11762939C22124yescomplemented
20#11762936C22124yescomplemented
21Q8×#3622C2218yescomplemented
22#11762831C22124yes—
23C2×C4⋊#3622C2213yes—
24C2×#10587485C2216yes—
25C4×#63218C22124yes—
26C2×#10587486C2213yes—
27C2×#10587486C2213yes—
28C2×C4×#3622C2213yes—
29C2×C4×Q8×C17C2211yescharacteristic
30C8×Q8×C17C2218yescomplemented
31C2×C17×C4⋊C8C2213yes—
32C2×C4×C8×C17C2213yes—
33C22×Q8×C17C2×C411yescharacteristic
34C2×Q8×D17—28no—
35C2×Q8×D17C2×C418yes—
36C22×C4×D17C2×C413yes—
37C22×C4×D17C2×C413yes—
38C22×C4×D17C2311yescharacteristic
39C22×Dic34C2×C413yes—
40C22×C8×C17C2313yes—
41C22×C8×C17C2311yescharacteristic
42C4×Q8×C17C2×C414yes—
43C4×Q8×C17C2314yes—
44C2×C4⋊C4×C17C2×C413yes—
45C2×C42×C17C2313yes—
46Q8×Dic17C2×C418yes—
47C2×C4⋊Dic17C2×C413yes—
48C2×#63225C2×C416yes—
49C2×C4×Dic17C2×C413yes—
50C2×C4×Dic17C2×C413yes—
51C2×C4×Dic17C2313yes—
52C22×#3622C2313yes—
53C22×#3622C2311yescharacteristic
54C2×#63218—212no—
55C2×#63218Q818yesdirect product factor
56C2×#63218C23112yes—
57C2×#63218C2314yes—
58C4⋊C4×D17C2×C4124yes—
59#10587573C2×C4124yes—
60C42×D17C2×C4112yes—
61C42×D17C23112yes—
62C4×Dic34C2×C4112yes—
63C4×Dic34C23112yes—
64C17×C4⋊C8C23112yes—
65C4×C8×C17C23112yes—
66#10587486C23112yes—
67#10587486C23112yes—
68#10587485C23124yes—
69C4⋊#3622C23112yes—
70C4×#3622C23112yes—
71C23×D17C22×C411yescharacteristic
72C2×Q8×C17M1618yescomplemented
73C2×Q8×C17C22×C414yes—
74C22×C4×C17C22×C413yes—
75C22×C4×C17C22×C413yes—
76C22×C4×C17C2411yescharacteristic
77C22×Dic17C22×C413yes—
78C22×Dic17C22×C413yes—
79C22×Dic17C22×C411yescharacteristic
80Q8×D17—232no—
81C2×C4×D17—212no—
82C2×C4×D17—46no—
83C2×C4×D17C22×C4112yes—
84C2×C4×D17C22×C4112yes—
85C2×C4×D17C22×C412yes—
86C2×C4×D17C2×Q814yes—
87C2×C4×D17#4114yes—
88C2×C4×D17C2412yes—
89C2×Dic34—212no—
90C2×Dic34C22×C4112yes—
91C2×C8×C17—26no—
92C2×C8×C17C2×Q814yescomplemented
93C2×C8×C17C2416yes—
94C2×C8×C17C2412yes—
95C4⋊C4×C17C22×C4112yes—
96C42×C17C22×C416yes—
97C42×C17C2416yes—
98C4⋊Dic17C22×C4112yes—
99#63225C22×C4124yes—
100C4×Dic17C22×C416yes—
101C4×Dic17C22×C4112yes—
102C4×Dic17C22×C4112yes—
103C4×Dic17C2416yes—
104C2×#3622—26no—
105C2×#3622C2×Q814yescomplemented
106C2×#3622C2416yes—
107C2×#3622C2412yes—
108#63218—224no—
109#63218C2×Q8116yesdirect product factor
110C2×Q8×M16—171nomaximal, Sylow
111C23×C17C23×C411yescharacteristic
112C22×D17—22no—
113C22×D17—22no—
114C22×D17C4×Q814yes—
115C22×D17C23×C412yes—
116Q8×C17C2×M16116yescomplemented
117C2×C4×C17—26no—
118C2×C4×C17C2×M16112yes—
119C2×C4×C17C23×C416yes—
120C2×C4×C17C23×C411yescharacteristic
121C2×C4×C17C23×C416yes—
122C2×C4×C17C22×Q812yes—
123C2×C4×C17C2×#4112yes—
124C2×C4×C17C2511yescharacteristic
125C2×Dic17—26no—
126C2×Dic17—26no—
127C2×Dic17C4×Q814yes—
128C2×Dic17C23×C416yes—
129C2×Dic17C23×C416yes—
130C2×Dic17C23×C412yes—
131C4×D17—224no—
132C4×D17—412no—
133C4×D17C4×Q818yes—
134C4×D17C22×Q814yes—
135C4×D17C2×#4114yes—
136Dic34—224no—
137C8×C17—26no—
138C8×C17C22×Q814yescomplemented
139#3622—26no—
140#3622C22×Q814yescomplemented
141C22×C4×Q8—171no—
142Q8×M16—1716no—
143C2×#446—173no—
144C2×#446—173no—
145C2×C8×Q8—171no—
146C2×C8×Q8D1711yescharacteristic, complemented
147C2×#423—173no—
148C2×C4×M16—173no—
149C22×C17C2×C4×Q812yes—
150C22×C17C22×M1612yes—
151C22×C17#56812yes—
152C22×C17C24×C411yescharacteristic
153C2×D17—28no—
154C2×D17—22no—
155C2×D17—22no—
156C2×D17C2×C4×Q814yes—
157C4×C17—26no—
158C4×C17C2×C4×Q812yes—
159C4×C17C22×M16112yes—
160C4×C17C23×Q811yescharacteristic
161C4×C17C22×#4111yescharacteristic
162Dic17—26no—
163Dic17—26no—
164Dic17C2×C4×Q814yes—
165C23×Q8—171no—
166C22×M16—171no—
167C22×M16—173no—
168C2×C4×Q8—171no—
169C2×C4×Q8—178no—
170C2×C4×Q8—178no—
171C2×C4×Q8—173no—
172C2×C4×Q8—173no—
173C2×C4×Q8C2×D1711yescharacteristic
174C22×C4⋊C4—173no—
175C22×C42—173no—
176#446—1724no—
177#446—1724no—
178C8×Q8—178no—
179C8×Q8C2×D1718yescomplemented
180#423—1724no—
181C2×C4⋊C8—176no—
182C2×C4⋊C8—173no—
183C2×C4⋊C8C2×D1713yes—
184C4×M16—1724no—
185C2×#97—173no—
186C2×#97—173no—
187C2×C4×C8—173no—
188C2×C4×C8C2×D1713yes—
189C2×C17Q8×M1614yescomplemented
190C2×C17C22×C4×Q811yescharacteristic
191C2×C17C23×M1611yescharacteristic
192C2×C17C2×#56811yescharacteristic
193D17—24no—
194C22×Q8—178no—
195C22×Q8—173no—
196C22×Q8—348no—
197C22×Q8C4×D1711yescharacteristic
198C23×C4—171no—
199C23×C4—173no—
200C23×C4—173no—
201C2×M16—178no—
202C2×M16—174no—
203C2×M16—1712no—
204C2×M16—3412no—
205C22×C8—171no—
206C22×C8—173no—
207C22×C8C22×D1713yes—
208C22×C8C22×D1711yescharacteristic
209C4×Q8—178no—
210C4×Q8—1724no—
211C4×Q8—1712no—
212C4×Q8—1712no—
213C4×Q8C4×D1714yes—
214C4×Q8C22×D1714yes—
215C2×C4⋊C4—176no—
216C2×C4⋊C4—173no—
217C2×C4⋊C4—1724no—
218C2×C4⋊C4C4×D1713yes—
219C2×C42—173no—
220C2×C42—173no—
221C2×C42—173no—
222C2×C42—1712no—
223C2×C42—1712no—
224C2×C42C22×D1713yes—
225C4⋊C8—1724no—
226C4⋊C8—1712no—
227C4⋊C8C22×D17112yes—
228#97—1712no—
229#97—1712no—
230C4×C8—1712no—
231C4×C8C22×D17112yes—
232C17C2×Q8×M1611yescharacteristic, complemented, Sylow
233C24—171no—
234C2×Q8—1712no—
235C2×Q8—3432no—
236C2×Q8—3412no—
237C2×Q8#6321818yesdirect product factor
238C2×Q8C2×C4×D1714yes—
239C22×C4—171no—
240C22×C4—172no—
241C22×C4—174no—
242C22×C4—174no—
243C22×C4—172no—
244C22×C4—173no—
245C22×C4—173no—
246C22×C4—1712no—
247C22×C4—1712no—
248C22×C4—3412no—
249C22×C4—686no—
250C22×C4C2×C4×D1713yes—
251C22×C4C2×C4×D1713yes—
252C22×C4C23×D1711yescharacteristic
253M16—1716no—
254M16—3424no—
255C2×C8—26no—
256C2×C8—174no—
257C2×C8—172no—
258C2×C8—176no—
259C2×C8—346no—
260C2×C8Q8×D1714yescomplemented
261C2×C8C23×D1716yes—
262C2×C8C23×D1712yes—
263C4⋊C4—1724no—
264C4⋊C4—1712no—
265C4⋊C4C2×C4×D17112yes—
266C42—1712no—
267C42—1712no—
268C42—176no—
269C42—176no—
270C42C2×C4×D1716yes—
271C42C23×D1716yes—
272C23—174no—
273C23—172no—
274C23—342no—
275C23—342no—
276C23C22×C4×D1711yescharacteristic
277Q8—3424no—
278Q8C2×#63218116yesdirect product factor
279C2×C4—26no—
280C2×C4—174no—
281C2×C4—172no—
282C2×C4—178no—
283C2×C4—174no—
284C2×C4—174no—
285C2×C4—176no—
286C2×C4—176no—
287C2×C4—346no—
288C2×C4—346no—
289C2×C4—3424no—
290C2×C4—6812no—
291C2×C4C2×#63218112yes—
292C2×C4C22×C4×D1716yes—
293C2×C4C22×C4×D1711yescharacteristic
294C2×C4C22×C4×D1716yes—
295C2×C4C2×Q8×D1712yes—
296C2×C4#41×D1712yes—
297C2×C4C24×D1711yescharacteristic
298C8—26no—
299C8—174no—
300C8—346no—
301C8C2×Q8×D1714yescomplemented
302C22—174no—
303C22—348no—
304C22—342no—
305C22—342no—
306C22C4×Q8×D1712yes—
307C22C22×#6321812yes—
308C22#1176357412yes—
309C22C23×C4×D1711yescharacteristic
310C4—26no—
311C4—174no—
312C4—346no—
313C4—346no—
314C4C4×Q8×D1712yes—
315C4C22×#63218112yes—
316C4C22×Q8×D1711yescharacteristic
317C4C2×#41×D1711yescharacteristic
318C2—344no—
319C2Q8×#6321814yesdirect product factor
320C2C2×C4×Q8×D1711yescharacteristic
321C2C23×#6321811yescharacteristic
322C2C2×#1176357411yescharacteristic
subgroupG/H
centerC22×C4C23×D17
commutatorC22×C17C24×C4
FrattiniC2×C4C24×D17
FittingC2×C8×Q8×C17C2
socleC23×C17C23×C4
prime radicalC23×D17C22×C4
cosocleC2×C4×C17C25
hypercenterC2×C8×Q8D17
hypocenterC17C2×Q8×M16
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. Q8×#63218C2direct (N × H)
3. C2×#11762939C2semidirect
4. C2×#11762936C2semidirect
5. C2×Q8×#3622C2semidirect
8. C2×C8×Q8×C17C2semidirect
19. #11762939C22semidirect
20. #11762936C22semidirect
21. Q8×#3622C22semidirect
30. C8×Q8×C17C22semidirect
55. C2×#63218Q8direct (N × H)
72. C2×Q8×C17M16semidirect
92. C2×C8×C17C2×Q8semidirect
105. C2×#3622C2×Q8semidirect
109. #63218C2×Q8direct (N × H)
116. Q8×C17C2×M16semidirect
138. C8×C17C22×Q8semidirect
140. #3622C22×Q8semidirect
146. C2×C8×Q8D17semidirect
179. C8×Q8C2×D17semidirect
189. C2×C17Q8×M16semidirect
232. C17C2×Q8×M16semidirect
237. C2×Q8#63218direct (N × H)
260. C2×C8Q8×D17semidirect
278. Q8C2×#63218direct (N × H)
301. C8C2×Q8×D17semidirect
319. C2Q8×#63218direct (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×D172
Q8×#632182
C2×#117629392
C2×#117629362
C2×Q8×#36222
C2×#117628312
C2×C4×#632182
C2×C8×Q8×C172
C2×Q8×M1617