Finite Group Databasebeta

← C2×#21461080 #30609962

Subgroups — C2×#21461080 #30609962

Order 15616.

1650 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 1650 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840782C212yesmaximal, complemented
2#21461080C218yesmaximal, direct product factor
3C2×#16840606C214yesmaximal, complemented
4C22×#14519739C211yesmaximal, characteristic
5#16840782—24no—
6#16840782C2218yescomplemented
7C2×#14519844—22no—
8C2×C22⋊#92420C2214yes—
9C2×#14519832—22no—
10#16840606C22116yescomplemented
11C23×D244C411yescharacteristic
12C23×C4×D61C412yes—
13C2×#14519739—22no—
14C2×#14519739—22no—
15C2×#14519739—24no—
16C2×#14519739C414yes—
17C2×#14519739C2214yes—
18C22×#11758198C2212yes—
19C22×C4×Dic61C2211yescharacteristic
20C22×C4⋊C4×C61C411yescharacteristic
21C22×#92420—24no—
22#14519844—28no—
23C22⋊#92420—24no—
24C22⋊#92420C2318yes—
25#14519832—28no—
26C24×D61C2×C412yes—
27C22×D244—28no—
28C22×D244—22no—
29C22×D244—22no—
30C22×D244C2×C414yes—
31C23×Dic61C2×C411yescharacteristic
32C23×Dic61C2311yescharacteristic
33C22×C4×D61—24no—
34C22×C4×D61—22no—
35C22×C4×D61—22no—
36C22×C4×D61—48no—
37C22×C4×D61C2×C418yes—
38#14519739—28no—
39#14519739—28no—
40#14519739—28no—
41#14519739—28no—
42C2×#11758198—24no—
43C2×#11758198—24no—
44C2×#11758198C2×C414yes—
45C2×#11758198C2314yes—
46C2×C4×Dic61C2×C412yes—
47C2×C4×Dic61D412yes—
48C2×C4×Dic61D412yes—
49C2×C4×Dic61D414yes—
50C2×C4×Dic61C2312yes—
51C23×C4×C61C2×C412yes—
52C23×C4×C61C2×C411yescharacteristic
53C2×C4⋊C4×C61—22no—
54C2×C4⋊C4×C61—22no—
55C2×C4⋊C4×C61C2×C414yes—
56C24×C61C22×C411yescharacteristic
57C23×D61—22no—
58C23×D61—22no—
59C23×D61—24no—
60C23×D61—44no—
61C23×D61—44no—
62C23×D61M1618yes—
63C23×D61C22×C414yes—
64C22×C4×C61—24no—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—44no—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C22⋊C412yes—
70C22×C4×C61C22×C412yes—
71C22×C4×C61C22×C414yes—
72C22×Dic61—24no—
73C22×Dic61—42no—
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C22⋊C412yes—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22×C412yes—
78C22×Dic61C22×C411yescharacteristic
79C22×Dic61C2×D412yes—
80C22×Dic61C2×D412yes—
81C22×Dic61C2411yescharacteristic
82C2×D244—216no—
83C2×D244—216no—
84C2×D244—216no—
85C2×D244—28no—
86C2×C4×D61—28no—
87C2×C4×D61—28no—
88C2×C4×D61—48no—
89C2×C4×D61—416no—
90C2×C4×D61—48no—
91C2×C4×D61—416no—
92C2×#92420—28no—
93C2×#92420—48no—
94C4⋊C4×C61—28no—
95#11758198—28no—
96#11758198—28no—
97C4×Dic61C22⋊C414yes—
98C4×Dic61C22⋊C414yes—
99C4×Dic61C2×D414yes—
100C4×Dic61C2×D414yes—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61C2×C22⋊C411yescharacteristic
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61C2×M1614yes—
106C23×C61C23×C411yescharacteristic
107C22×D61—24no—
108C22×D61—24no—
109C22×D61—28no—
110C22×D61—28no—
111C22×D61—44no—
112C22×D61—44no—
113C22×D61—416no—
114C22×D61—44no—
115C22×D61—44no—
116C22×D61—44no—
117C22×D61—416no—
118C22×D61—44no—
119C22×D61C2×M1618yes—
120C2×C4×C61—24no—
121C2×C4×C61—24no—
122C2×C4×C61—24no—
123C2×C4×C61—24no—
124C2×C4×C61—44no—
125C2×C4×C61—44no—
126C2×C4×C61—44no—
127C2×C4×C61C2×C22⋊C414yes—
128C2×Dic61—24no—
129C2×Dic61—22no—
130C2×Dic61—24no—
131C2×Dic61—24no—
132C2×Dic61—41no—
133C2×Dic61—42no—
134C2×Dic61—41no—
135C2×Dic61#10014yes—
136C2×Dic61C2×C22⋊C412yes—
137C2×Dic61C2×C22⋊C412yes—
138C2×Dic61C2×C22⋊C412yes—
139C2×Dic61C22×D412yes—
140D244—232no—
141C4×D61—416no—
142C4×D61—416no—
143#92420—48no—
144C2×#1466—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—21no—
149C22×C61—28no—
150C22×C61#40712yes—
151C22×C61#40712yes—
152C22×C61#41012yes—
153C22×C61C2×#10012yes—
154C22×C61C22×C22⋊C411yescharacteristic
155C22×C61C22×M1612yes—
156C2×D61—28no—
157C2×D61—48no—
158C2×D61—416no—
159C2×D61—44no—
160C2×D61—44no—
161C2×D61—416no—
162C2×D61—48no—
163C4×C61—24no—
164C4×C61—44no—
165Dic61—22no—
166Dic61—22no—
167Dic61—22no—
168Dic61C2×#10014yes—
169C22×C4×D4—611no—
170C2×#432—612no—
171#1466—618no—
172C2×#323—614no—
173C2×C61—22no—
174C2×C61—22no—
175C2×C61#146614yescomplemented
176C2×C61C2×#40711yescharacteristic
177C2×C61C2×#41011yescharacteristic
178C2×C61C22×#10011yescharacteristic
179D61—48no—
180C23×D4—611no—
181C24×C4—612no—
182C2×C4×D4—614no—
183C2×C4×D4—614no—
184C2×C4×D4—1222no—
185C2×C4×D4—1222no—
186C2×C4×D4—1224no—
187C22×C4⋊C4#692311yescharacteristic
188C22×C22⋊C4—612no—
189C22×C42—611no—
190#432—618no—
191#432—1224no—
192C2×C4⋊C8—1222no—
193C2×C22⋊C8—614no—
194C2×#97—1222no—
195#323—6116no—
196C61C2×#146611yescharacteristic, complemented, Sylow
197C25—612no—
198C22×D4—614no—
199C22×D4—1228no—
200C22×D4—1222no—
201C22×D4—1222no—
202C23×C4—611no—
203C23×C4—611no—
204C23×C4—618no—
205C23×C4—1224no—
206C23×C4—1222no—
207C23×C4—1222no—
208C23×C4—2448no—
209C23×C4C2×#692312yes—
210C23×C4C2×#692311yescharacteristic
211C22×C8—1224no—
212C4×D4—1228no—
213C4×D4—1228no—
214C4×D4—1228no—
215C4×D4—1228no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C2×#692314yes—
219C2×C22⋊C4—614no—
220C2×C22⋊C4—614no—
221C2×C22⋊C4—1224no—
222C2×C22⋊C4—1224no—
223C2×C42—612no—
224C2×C42—612no—
225C2×C42—614no—
226C2×C42—612no—
227C2×C42—612no—
228C4⋊C8—1228no—
229C22⋊C8—618no—
230C22⋊C8—1224no—
231#97—1228no—
232C24—618no—
233C24—614no—
234C24—1222no—
235C24—1222no—
236C24—1224no—
237C24—2444no—
238C24—2444no—
239C24C22×#692311yescharacteristic
240C2×D4—12216no—
241C2×D4—12216no—
242C2×D4—12216no—
243C2×D4—1228no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—44no—
248C22×C4—611no—
249C22×C4—612no—
250C22×C4—612no—
251C22×C4—611no—
252C22×C4—612no—
253C22×C4—612no—
254C22×C4—611no—
255C22×C4—611no—
256C22×C4—1228no—
257C22×C4—1228no—
258C22×C4—1224no—
259C22×C4—2448no—
260C22×C4—24416no—
261C22×C4—2448no—
262C22×C4—24416no—
263C22×C4—2442no—
264C22×C4C22⋊#692312yes—
265C22×C4C22⋊#692312yes—
266C22×C4C22×#692312yes—
267C22×C4C22×#692314yes—
268C2×C8—1228no—
269C2×C8—2448no—
270C4⋊C4—28no—
271C22⋊C4—1228no—
272C22⋊C4—1228no—
273C42—614no—
274C42—614no—
275C42—614no—
276C42—614no—
277C23—22no—
278C23—22no—
279C23—618no—
280C23—1228no—
281C23—1228no—
282C23—1224no—
283C23—1224no—
284C23—2444no—
285C23—24416no—
286C23—2444no—
287C23—2444no—
288C23—24416no—
289C23—2444no—
290C23—2444no—
291C23—2444no—
292C23C2×C22⋊#692311yescharacteristic
293C23C2×C22⋊#692311yescharacteristic
294C23D61⋊M1614yes—
295C23C23×#692311yescharacteristic
296D4—12232no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—614no—
305C2×C4—612no—
306C2×C4—612no—
307C2×C4—612no—
308C2×C4—612no—
309C2×C4—1224no—
310C2×C4—1224no—
311C2×C4—1224no—
312C2×C4—1222no—
313C2×C4—24416no—
314C2×C4—24416no—
315C2×C4—2441no—
316C2×C4—2442no—
317C2×C4—2441no—
318C2×C4C2×C22⋊#692314yes—
319C8—2448no—
320C22—28no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—1228no—
326C22—2448no—
327C22—24416no—
328C22—2448no—
329C22—24416no—
330C22—2444no—
331C22—2444no—
332C22#1684073412yes—
333C22#1684073412yes—
334C22#1684074312yes—
335C22D61⋊#10012yes—
336C22C22×C22⋊#692311yescharacteristic
337C22C2×D61⋊M1612yes—
338C4—24no—
339C4—44no—
340C4—614no—
341C4—1222no—
342C4—1222no—
343C4—1222no—
344C2—22no—
345C2—22no—
346C2—2448no—
347C2#2146108014yesdirect product factor
348C2C2×#1684073411yescharacteristic
349C2C2×#1684074311yescharacteristic
350C2C2×D61⋊#10011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D244C4
cosocleC22×Dic61C24
hypercenterC22×C4⋊C4#6923
hypocenterC61C2×#1466
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16840782C2semidirect
2. #21461080C2direct (N × H)
3. C2×#16840606C2semidirect
6. #16840782C22semidirect
10. #16840606C22semidirect
175. C2×C61#1466semidirect
196. C61C2×#1466semidirect
347. C2#21461080direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168407822
#214610802
C2×#168406062
C22×#145197392
C2×#146661