Finite Group Databasebeta

← C2×C4⋊C4×#1057 #16797117

Subgroups — C2×C4⋊C4×#1057 #16797117

Order 3328.

1438 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊#1057C211yesmaximal, characteristic
2C22×C4×#1057C212yesmaximal
3C22×C4×#1057C211yesmaximal, characteristic
4C4⋊C4×#1057C218yesmaximal, direct product factor
5C2×#11718290C212yesmaximal
6C2×C4⋊C4×Dic13C211yesmaximal, characteristic
7C23×#1057C2211yescharacteristic
8C23×#1057C2212yes—
9C23×#1057C2211yescharacteristic
10C22×C13×C4⋊C4C411yescharacteristic
11C22×C4⋊Dic13C411yescharacteristic
12C22×C4×Dic13C2212yes—
13C22×C4×Dic13C2211yescharacteristic
14C2×C4⋊#1057C418yescomplemented
15C2×C4⋊#1057C2214yes—
16C2×C4×#1057—28no—
17C2×C4×#1057C418yescomplemented
18C2×C4×#1057C2218yes—
19C2×C4×#1057C2214yes—
20C2×#88071C412yes—
21C4⋊C4×Dic13C414yes—
22C4⋊C4×Dic13C2214yes—
23#11718290C2218yes—
24C23×C4×C13C2×C412yes—
25C23×C4×C13C2×C411yescharacteristic
26C23×Dic13C2×C411yescharacteristic
27C23×Dic13C2×C412yes—
28C23×Dic13C2311yescharacteristic
29C22×#1057—24no—
30C22×#1057—22no—
31C22×#1057—22no—
32C22×#1057—24no—
33C22×#1057C2×C414yes—
34C22×#1057C2×C418yes—
35C22×#1057C2×C414yes—
36C22×#1057D414yes—
37C22×#1057Q814yes—
38C22×#1057C2312yes—
39C22×#1057C2314yes—
40C22×#1057C2312yes—
41C2×C13×C4⋊C4C818yescomplemented
42C2×C13×C4⋊C4C2×C414yes—
43C2×C4⋊Dic13C818yescomplemented
44C2×C4⋊Dic13C2×C414yes—
45C2×C4×Dic13—24no—
46C2×C4×Dic13—24no—
47C2×C4×Dic13C2×C414yes—
48C2×C4×Dic13C2×C412yes—
49C2×C4×Dic13C2×C414yes—
50C2×C4×Dic13C2×C414yes—
51C2×C4×Dic13C2314yes—
52C2×C4×Dic13C2312yes—
53C4⋊#1057C2×C4116yescomplemented
54C4×#1057—216no—
55C4×#1057C2×C4116yescomplemented
56#88071C2×C418yes—
57C2×C8×C4⋊C4—131nomaximal, Sylow
58C24×C13C22×C411yescharacteristic
59C22×C4×C13—24no—
60C22×C4×C13—24no—
61C22×C4×C13C4214yes—
62C22×C4×C13C2×C814yes—
63C22×C4×C13C2×C818yes—
64C22×C4×C13C2×C814yes—
65C22×C4×C13C22×C414yes—
66C22×C4×C13C22×C412yes—
67C22×Dic13—24no—
68C22×Dic13—22no—
69C22×Dic13—22no—
70C22×Dic13—24no—
71C22×Dic13C4214yes—
72C22×Dic13C2×C818yes—
73C22×Dic13C22×C412yes—
74C22×Dic13C22×C414yes—
75C22×Dic13C22×C411yescharacteristic
76C22×Dic13C22×C412yes—
77C22×Dic13C22×C412yes—
78C22×Dic13C2×D412yes—
79C22×Dic13C2×Q812yes—
80C22×Dic13#4112yes—
81C22×Dic13#4112yes—
82C22×Dic13C2411yescharacteristic
83C2×#1057—28no—
84C2×#1057—216no—
85C2×#1057—24no—
86C2×#1057—22no—
87C2×#1057—22no—
88C2×#1057—24no—
89C2×#1057C4⋊C4116yesdirect product factor
90C2×#1057C22×C414yes—
91C2×#1057C22×C418yes—
92C2×#1057C22×C414yes—
93C2×#1057C2×D414yes—
94C2×#1057C2×Q814yes—
95C13×C4⋊C4C2×C8116yescomplemented
96C4⋊Dic13C2×C8116yescomplemented
97C4×Dic13—28no—
98C4×Dic13—24no—
99C4×Dic13—24no—
100C4×Dic13C4218yes—
101C4×Dic13C22×C414yes—
102C4×Dic13C22×C414yes—
103C22×C4⋊C8—131no—
104C22×C4×C8—131no—
105C22×C4×C8—132no—
106C2×C4×C4⋊C4—131no—
107C8×C4⋊C4—138no—
108C2×#336—132no—
109C23×C13C2×C4212yes—
110C23×C13C4×D412yes—
111C23×C13C4×Q812yes—
112C23×C13C22×C812yes—
113C23×C13C22×C812yes—
114C23×C13#13112yes—
115C23×C13#13112yes—
116C23×C13C23×C411yescharacteristic
117C2×C4×C13—24no—
118C2×C4×C13—24no—
119C2×C4×C13—216no—
120C2×C4×C13C4×C8116yescomplemented
121C2×C4×C13C2×C4214yes—
122C2×C4×C13C22×C818yes—
123C2×C4×C13C22×C814yes—
124C2×C4×C13C22×C814yes—
125C2×Dic13—24no—
126C2×Dic13—22no—
127C2×Dic13—22no—
128C2×Dic13—24no—
129C2×Dic13—28no—
130C2×Dic13—216no—
131C2×Dic13C2×C4214yes—
132C2×Dic13C2×C4212yes—
133C2×Dic13C2×C4⋊C414yes—
134C2×Dic13#11714yes—
135C2×Dic13C4×D414yes—
136C2×Dic13C4×D412yes—
137C2×Dic13C4×Q814yes—
138C2×Dic13C4×Q812yes—
139C2×Dic13C22×C818yes—
140C2×Dic13C23×C411yescharacteristic
141C2×Dic13C23×C411yescharacteristic
142C2×Dic13C22×D411yescharacteristic
143C2×Dic13C22×Q811yescharacteristic
144C2×Dic13C2×#4111yescharacteristic
145C2×Dic13C2×#4111yescharacteristic
146#1057—24no—
147#1057—28no—
148#1057C2×C4⋊C418yesdirect product factor
149C23×C8—131no—
150C23×C8—132no—
151C23×C8—131no—
152C22×C4⋊C4—131no—
153C22×C4⋊C4#26111yescharacteristic
154C22×C42—131no—
155C22×C42—132no—
156C2×C4⋊C8—138no—
157C2×C4⋊C8—134no—
158C2×C4×C8—138no—
159C2×C4×C8—134no—
160C2×C4×C8—138no—
161C2×C4×C8—268no—
162C4×C4⋊C4—134no—
163C4×C4⋊C4—134no—
164C2×#95—132no—
165#336—138no—
166C22×C13C4×C4⋊C414yes—
167C22×C13C2×C4×C814yes—
168C22×C13#40514yes—
169C22×C13C8×D414yes—
170C22×C13C8×D414yes—
171C22×C13C8×Q814yes—
172C22×C13C8×Q814yes—
173C22×C13C22×C4211yescharacteristic
174C22×C13C2×C4×D411yescharacteristic
175C22×C13C2×C4×Q811yescharacteristic
176C22×C13C23×C811yescharacteristic
177C22×C13C23×C811yescharacteristic
178C22×C13C2×#13111yescharacteristic
179C22×C13C2×#13111yescharacteristic
180C4×C13—28no—
181C4×C13C2×C4×C818yescomplemented
182Dic13—24no—
183Dic13—28no—
184Dic13C4×C4⋊C414yes—
185Dic13C22×C4⋊C411yescharacteristic
186Dic13C2×#11711yescharacteristic
187Dic13C2×C4×D411yescharacteristic
188Dic13C2×C4×Q811yescharacteristic
189C23×C4—131no—
190C23×C4—132no—
191C23×C4—131no—
192C23×C4C2×#26112yes—
193C23×C4C2×#26111yescharacteristic
194C22×C8—132no—
195C22×C8—134no—
196C22×C8—134no—
197C22×C8—134no—
198C22×C8—132no—
199C22×C8—134no—
200C22×C8—134no—
201C22×C8—138no—
202C22×C8—264no—
203C22×C8—262no—
204C22×C8—264no—
205C22×C8—262no—
206C2×C4⋊C4—138no—
207C2×C4⋊C4—134no—
208C2×C4⋊C4#105718yesdirect product factor
209C2×C4⋊C4C2×#26114yes—
210C2×C42—134no—
211C2×C42—134no—
212C2×C42—132no—
213C2×C42—132no—
214C2×C42—134no—
215C2×C42—134no—
216C2×C42—264no—
217C2×C42—264no—
218C4⋊C8—1316no—
219C4×C8—1316no—
220C4×C8—2616no—
221#95—138no—
222C2×C13C8×C4⋊C418yescomplemented
223C2×C13C2×C4×C4⋊C411yescharacteristic
224C2×C13C22×C4×C811yescharacteristic
225C2×C13C2×#40511yescharacteristic
226C2×C13C2×C8×D411yescharacteristic
227C2×C13C2×C8×D411yescharacteristic
228C2×C13C2×C8×Q811yescharacteristic
229C2×C13C2×C8×Q811yescharacteristic
230C24C22×#26111yescharacteristic
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—132no—
234C22×C4—132no—
235C22×C4—132no—
236C22×C4—132no—
237C22×C4—132no—
238C22×C4—132no—
239C22×C4—131no—
240C22×C4—131no—
241C22×C4—138no—
242C22×C4—134no—
243C22×C4—132no—
244C22×C4—134no—
245C22×C4—262no—
246C22×C4—264no—
247C22×C4—262no—
248C22×C4—264no—
249C22×C4D13⋊C814yes—
250C22×C4C4×#26114yes—
251C22×C4C2×#105718yes—
252C22×C4C2×#105714yes—
253C22×C4C22×#26114yes—
254C22×C4C22×#26112yes—
255C2×C8—134no—
256C2×C8—134no—
257C2×C8—1316no—
258C2×C8—134no—
259C2×C8—134no—
260C2×C8—138no—
261C2×C8—264no—
262C2×C8—2616no—
263C2×C8—268no—
264C2×C8—262no—
265C2×C8—262no—
266C2×C8—264no—
267C4⋊C4—1316no—
268C4⋊C4C2×#1057116yesdirect product factor
269C42—134no—
270C42—134no—
271C42—138no—
272C42—264no—
273C42—264no—
274C42—268no—
275C13C2×C8×C4⋊C411yescharacteristic, complemented, Sylow
276C23C2×D13⋊C812yes—
277C23C2×C4×#26112yes—
278C23#8823812yes—
279C23D4×#26112yes—
280C23#8824012yes—
281C23Q8×#26112yes—
282C23C22×#105712yes—
283C23C23×#26111yescharacteristic
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—216no—
287C2×C4—134no—
288C2×C4—134no—
289C2×C4—134no—
290C2×C4—134no—
291C2×C4—131no—
292C2×C4—131no—
293C2×C4—132no—
294C2×C4—131no—
295C2×C4—131no—
296C2×C4—132no—
297C2×C4—131no—
298C2×C4—131no—
299C2×C4—132no—
300C2×C4—138no—
301C2×C4—134no—
302C2×C4—268no—
303C2×C4—2616no—
304C2×C4—262no—
305C2×C4—264no—
306C2×C4—264no—
307C2×C4—262no—
308C2×C4C4×#1057116yescomplemented
309C2×C4C2×D13⋊C814yes—
310C2×C4C2×C4×#26114yes—
311C2×C4C22×#105718yes—
312C2×C4C22×#105714yes—
313C8—138no—
314C8—264no—
315C8—268no—
316C22#1171907214yes—
317C22#1171907314yes—
318C22#1171907414yes—
319C22C4⋊C4×#26114yes—
320C22C2×C4×#105714yes—
321C22D4×#105714yes—
322C22Q8×#105714yes—
323C22C22×D13⋊C811yescharacteristic
324C22C22×C4×#26111yescharacteristic
325C22C2×#8823811yescharacteristic
326C22C2×D4×#26111yescharacteristic
327C22C2×#8824011yescharacteristic
328C22C2×Q8×#26111yescharacteristic
329C22C23×#105711yescharacteristic
330C4—28no—
331C4—131no—
332C4—131no—
333C4—131no—
334C4—131no—
335C4—134no—
336C4—268no—
337C4—264no—
338C4C2×C4×#105718yescomplemented
339C2C4⋊C4×#105718yesdirect product factor
340C2C2×C4⋊C4×#26111yescharacteristic
341C2C22×C4×#105711yescharacteristic
342C2C2×#1171907211yescharacteristic
343C2C2×D4×#105711yescharacteristic
344C2C2×#1171907311yescharacteristic
345C2C2×Q8×#105711yescharacteristic
346C2C2×#1171907411yescharacteristic
subgroupG/H
centerC24C22×#261
commutatorC2×C13C22×C4×C8
FrattiniC23C23×#261
FittingC22×C13×C4⋊C4C4
socle, prime radicalC24×C13C22×C4
cosocleC22×Dic13C24
hypercenterC22×C4⋊C4#261
hypocenterC13C2×C8×C4⋊C4
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×#1057C2direct (N × H)
14. C2×C4⋊#1057C4semidirect
17. C2×C4×#1057C4semidirect
41. C2×C13×C4⋊C4C8semidirect
43. C2×C4⋊Dic13C8semidirect
53. C4⋊#1057C2×C4semidirect
55. C4×#1057C2×C4semidirect
89. C2×#1057C4⋊C4direct (N × H)
95. C13×C4⋊C4C2×C8semidirect
96. C4⋊Dic13C2×C8semidirect
120. C2×C4×C13C4×C8semidirect
148. #1057C2×C4⋊C4direct (N × H)
181. C4×C13C2×C4×C8semidirect
208. C2×C4⋊C4#1057direct (N × H)
222. C2×C13C8×C4⋊C4semidirect
268. C4⋊C4C2×#1057direct (N × H)
275. C13C2×C8×C4⋊C4semidirect
308. C2×C4C4×#1057semidirect
338. C4C2×C4×#1057semidirect
339. C2C4⋊C4×#1057direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊#10572
C22×C4×#10572
C4⋊C4×#10572
C2×#117182902
C2×C4⋊C4×Dic132
C2×C8×C4⋊C413