Finite Group Databasebeta

← C2×#21461128 #30610495

Subgroups — C2×#21461128 #30610495

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840781C211yesmaximal, characteristic
2C2×D61⋊(C4⋊C8)C211yesmaximal, characteristic
3#21461128C218yesmaximal, direct product factor
4C2×Q8⋊#92420C214yesmaximal, complemented
5C2×#16840317C211yesmaximal, characteristic
6C22×D61⋊C8—21no—
7#16840781—22no—
8#16840781C2214yes—
9C2×#14519844—21no—
10D61⋊(C4⋊C8)—22no—
11D61⋊(C4⋊C8)C2214yes—
12C2×C4⋊#92420—21no—
13C2×C4⋊#92420C2212yes—
14C2×#14519834—21no—
15C2×C4×#92420C2212yes—
16Q8⋊#92420C22116yescomplemented
17#16840317—22no—
18#16840317—22no—
19#16840317C414yes—
20#16840317C2214yes—
21C2×#14519777C411yescharacteristic
22C2×#14519768—21no—
23C2×#14519757—21no—
24C2×#14519753C412yes—
25C2×Q8×Dic61C2212yes—
26C2×#14519742—21no—
27C2×#14519735—21no—
28C2×C42×D61C2211yescharacteristic
29C2×C4⋊Q8×C61C411yescharacteristic
30C2×D61⋊C8—24no—
31C2×D61⋊C8—44no—
32C22×#92420—22no—
33C22×#92420—22no—
34#14519844—24no—
35C4⋊#92420—24no—
36C4⋊#92420—22no—
37C4⋊#92420C2314yes—
38#14519834—24no—
39C4×#92420—22no—
40C4×#92420C2314yes—
41C22×Dic122—21no—
42C22×C4×D61C2×C411yescharacteristic
43C22×C4×D61D411yescharacteristic
44C22×C4×D61D411yescharacteristic
45#14519777—22no—
46#14519777C2×C414yes—
47#14519768—24no—
48#14519768—24no—
49#14519757—24no—
50#14519757—24no—
51#14519753—24no—
52#14519753C2×C418yes—
53Q8×Dic61—22no—
54Q8×Dic61—22no—
55Q8×Dic61C2×C414yes—
56Q8×Dic61C2314yes—
57#14519742—24no—
58#14519742—24no—
59#14519735—24no—
60#14519735—24no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—22no—
63C2×C4⋊Dic61C2×C412yes—
64C2×#11758196—22no—
65C2×#11758198—22no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70C42×D61C2×C412yes—
71C42×D61D412yes—
72C42×D61D412yes—
73C42×D61C2312yes—
74C22×Q8×C61C2×C412yes—
75C4⋊Q8×C61—22no—
76C4⋊Q8×C61C2×C414yes—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C42×C61C2×C411yescharacteristic
80C23×D61C22⋊C411yescharacteristic
81C2×Q8×C61—22no—
82C2×Q8×C61—28no—
83C2×Q8×C61C22×C414yes—
84C22×C4×C61—22no—
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61C22×C411yescharacteristic
88C22×Dic61—22no—
89C22×Dic61C22⋊C411yescharacteristic
90C22×Dic61C2×D411yescharacteristic
91C22×Dic61C2×D411yescharacteristic
92C2×C4×D61—24no—
93C2×C4×D61—42no—
94C2×C4×D61C22⋊C412yes—
95C2×C4×D61C22⋊C412yes—
96C2×C4×D61C22⋊C412yes—
97C2×C4×D61SD1614yes—
98C2×C4×D61Q1614yes—
99C2×C4×D61C22×C412yes—
100C2×C4×D61C2×D412yes—
101C2×C4×D61C2×D412yes—
102C2×Dic122—24no—
103C2×Dic122—44no—
104C2×#92420—24no—
105C2×#92420—24no—
106C2×#92420—44no—
107C2×#92420—44no—
108D61⋊C8—48no—
109C4⋊C4×C61—24no—
110C4⋊C4×C61—24no—
111C42×C61C22⋊C412yes—
112C42×C61C22×C412yes—
113#11758198—28no—
114C4⋊Dic61—24no—
115C4⋊Dic61—22no—
116C4⋊Dic61—24no—
117C4⋊Dic61—24no—
118C4⋊Dic61C22×C414yes—
119#11758196—28no—
120C4×Dic61—24no—
121C4×Dic61—24no—
122C4×Dic61C22⋊C412yes—
123C4×Dic61C22⋊C412yes—
124C4×Dic61C22×C411yescharacteristic
125C4×Dic61C2×D411yescharacteristic
126C4×Dic61C2×D411yescharacteristic
127C4×Dic61C2×D412yes—
128C4×Dic61C2×D412yes—
129C4×Dic61C2411yescharacteristic
130C23×C61C2×C22⋊C411yescharacteristic
131C22×D61—22no—
132C22×D61—22no—
133C22×D61#10314yes—
134C22×D61C2×C22⋊C412yes—
135Q8×C61—28no—
136Q8×C61—44no—
137C2×C4×C61—22no—
138C2×C4×C61—24no—
139C2×C4×C61—44no—
140C2×C4×C61#10314yes—
141C2×C4×C61C2×C22⋊C412yes—
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×C4×C61C2×C22⋊C412yes—
144C2×C4×C61#13114yes—
145C2×C4×C61C23×C411yescharacteristic
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—24no—
149C2×Dic61—44no—
150C2×Dic61C2×C22⋊C411yescharacteristic
151C2×Dic61C2×C22⋊C412yes—
152C2×Dic61C2×C22⋊C411yescharacteristic
153C2×Dic61C2×SD1612yes—
154C2×Dic61C2×Q1612yes—
155C2×Dic61C22×D411yescharacteristic
156C2×Dic61C22×D411yescharacteristic
157C4×D61—24no—
158C4×D61—24no—
159C4×D61—42no—
160C4×D61—42no—
161C4×D61#10318yes—
162C4×D61C2×SD1614yes—
163C4×D61C2×Q1614yes—
164Dic122—48no—
165#92420—44no—
166#92420—44no—
167C2×#1481—611nomaximal, Sylow
168C22×C61#40812yes—
169C22×C61C2×#10312yes—
170C22×C61#42112yes—
171C22×C61C22×C22⋊C411yescharacteristic
172C2×D61—22no—
173C2×D61—22no—
174C2×D61—28no—
175C2×D61C2×#10314yes—
176C4×C61—22no—
177C4×C61—44no—
178C4×C61#40812yes—
179C4×C61C2×#10314yes—
180C4×C61C2×#13112yes—
181Dic61—22no—
182Dic61—21no—
183Dic61—21no—
184Dic61—44no—
185Dic61C2×#10312yes—
186Dic61C22×SD1611yescharacteristic
187Dic61C22×Q1611yescharacteristic
188C2×#533—611no—
189C2×#431—611no—
190C22×C4⋊C8—611no—
191#1481—618no—
192C2×Q8⋊C8—614no—
193C2×C61#148114yescomplemented
194C2×C61C2×#40811yescharacteristic
195C2×C61C22×#10311yescharacteristic
196C2×C61C2×#42111yescharacteristic
197D61—24no—
198C23×C8—1221no—
199#533—614no—
200#533—614no—
201#533—1222no—
202#533—1222no—
203C2×C4⋊Q8—611no—
204C2×C4⋊Q8#692311yescharacteristic
205C2×#125—1221no—
206C2×C22⋊Q8—612no—
207C2×C22⋊Q8—1221no—
208C2×C4×Q8—612no—
209C2×C4×Q8—1221no—
210C2×#117—1221no—
211C22×C42—611no—
212#431—614no—
213#431—1222no—
214C2×C4⋊C8—612no—
215C2×C4⋊C8—614no—
216C2×C4⋊C8—1221no—
217C2×C4⋊C8—1222no—
218C2×C4⋊C8—1221no—
219C2×C22⋊C8—1221no—
220C2×C4×C8—612no—
221Q8⋊C8—6116no—
222C61C2×#148111yescharacteristic, complemented, Sylow
223C22×Q8—1221no—
224C22×Q8C2×#692312yes—
225C23×C4—611no—
226C23×C4—611no—
227C23×C4—611no—
228C22×C8—1222no—
229C22×C8—1222no—
230C22×C8—1224no—
231C22×C8—2444no—
232C4⋊Q8—22no—
233C4⋊Q8—614no—
234C4⋊Q8—1222no—
235C4⋊Q8C2×#692314yes—
236#125—1224no—
237#125—1224no—
238C22⋊Q8—618no—
239C22⋊Q8—1224no—
240C22⋊Q8—1224no—
241C22⋊Q8—1224no—
242C4×Q8—614no—
243C4×Q8—614no—
244C4×Q8—1224no—
245C4×Q8—1224no—
246C4×Q8—1222no—
247C4×Q8—1222no—
248#117—1224no—
249#117—1224no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—612no—
253C2×C4⋊C4—1222no—
254C2×C4⋊C4—1222no—
255C2×C4⋊C4—1221no—
256C2×C22⋊C4—1222no—
257C2×C42—611no—
258C2×C42—611no—
259C2×C42—611no—
260C2×C42—612no—
261C2×C42—612no—
262C2×C42—612no—
263C2×C42—612no—
264C2×C42—1222no—
265C2×C42C2×#692311yescharacteristic
266C4⋊C8—614no—
267C4⋊C8—1222no—
268C4⋊C8—1224no—
269C4⋊C8—1224no—
270C22⋊C8—1224no—
271C4×C8—614no—
272C4×C8—1222no—
273C24—611no—
274C2×Q8—22no—
275C2×Q8—28no—
276C2×Q8—1224no—
277C2×Q8—2444no—
278C2×Q8C22×#692314yes—
279C22×C4—22no—
280C22×C4—611no—
281C22×C4—614no—
282C22×C4—614no—
283C22×C4—612no—
284C22×C4—612no—
285C22×C4—611no—
286C22×C4—611no—
287C22×C4—612no—
288C22×C4—612no—
289C22×C4—612no—
290C22×C4—612no—
291C22×C4—1224no—
292C22×C4—1222no—
293C22×C4—2442no—
294C22×C4C22⋊#692311yescharacteristic
295C22×C4C22⋊#692311yescharacteristic
296C22×C4C22×#692311yescharacteristic
297C2×C8—1224no—
298C2×C8—1224no—
299C2×C8—2448no—
300C2×C8—2444no—
301C2×C8—2444no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—614no—
305C4⋊C4—1228no—
306C4⋊C4—1224no—
307C4⋊C4—1224no—
308C4⋊C4—1222no—
309C4⋊C4—1224no—
310C22⋊C4—1228no—
311C42—611no—
312C42—612no—
313C42—611no—
314C42—612no—
315C42—612no—
316C42—612no—
317C42—611no—
318C42—611no—
319C42—1224no—
320C42—1224no—
321C42C22⋊#692312yes—
322C42C22×#692312yes—
323C23—614no—
324C23—612no—
325C23—1222no—
326C23—1222no—
327C23C2×C22⋊#692311yescharacteristic
328Q8—28no—
329Q8—44no—
330Q8—2448no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—44no—
334C2×C4—612no—
335C2×C4—612no—
336C2×C4—611no—
337C2×C4—611no—
338C2×C4—618no—
339C2×C4—614no—
340C2×C4—614no—
341C2×C4—612no—
342C2×C4—611no—
343C2×C4—611no—
344C2×C4—1222no—
345C2×C4—1222no—
346C2×C4—1224no—
347C2×C4—1224no—
348C2×C4—1224no—
349C2×C4—2442no—
350C2×C4—2442no—
351C2×C4—2444no—
352C2×C4#1451984014yes—
353C2×C4C2×C22⋊#692312yes—
354C2×C4C2×C22⋊#692311yescharacteristic
355C2×C4C2×C22⋊#692312yes—
356C2×C4#1451986414yes—
357C2×C4C23×#692311yescharacteristic
358C8—2444no—
359C8—2444no—
360C22—614no—
361C22—1222no—
362C22—1228no—
363C22—1222no—
364C22#1684073612yes—
365C22C2×#1451984012yes—
366C22#1684076112yes—
367C22C22×C22⋊#692311yescharacteristic
368C4—22no—
369C4—44no—
370C4—611no—
371C4—611no—
372C4—612no—
373C4—1222no—
374C4—1221no—
375C4—1221no—
376C4—2444no—
377C4#1684073612yes—
378C4C2×#1451984014yes—
379C4C2×#1451986412yes—
380C2—1224no—
381C2#2146112814yesdirect product factor
382C2C2×#1684073611yescharacteristic
383C2C22×#1451984011yescharacteristic
384C2C2×#1684076111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC4×Dic61C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61C2×#1481
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21461128C2direct (N × H)
4. C2×Q8⋊#92420C2semidirect
16. Q8⋊#92420C22semidirect
193. C2×C61#1481semidirect
222. C61C2×#1481semidirect
381. C2#21461128direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407812
C2×D61⋊(C4⋊C8)2
#214611282
C2×Q8⋊#924202
C2×#168403172
C2×#148161