Finite Group Databasebeta

← C2×#72987 #11711815

Subgroups — C2×#72987 #11711815

Order 768.

528 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 37 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×#376⋊C3C211yesmaximal, characteristic, complemented
2#72987C214yesmaximal, direct product factor
3C2×#4622C211yesmaximal, characteristic
4C2×#4621C211yesmaximal, characteristic, complemented
5C2×#1500—31nomaximal, Sylow
6C2×C4×S4—41nomaximal
7#376⋊C3C2212yescomplemented
8C2×#95⋊C3C2211yescharacteristic
9#4622C2212yes—
10#4621C2212yescomplemented
11C2×#529—31no—
12C2×#431—31no—
13C2×#376D311yescharacteristic, complemented
14#1500—34no—
15#1500—34no—
16C2×#328—31no—
17C2×#328—31no—
18C2×#327—31no—
19C2×#327—31no—
20C22×S4—41no—
21C2×C4×A4—41no—
22C2×SL(2,ℤ/4)—41no—
23C4×S4—44no—
24#95⋊C3C2311yescharacteristic
25#529—34no—
26#529—34no—
27#529—62no—
28#529—62no—
29C2×#126—61no—
30C2×#125—31no—
31C2×#124—31no—
32C2×#123—61no—
33C2×C22⋊Q8—31no—
34C2×#121—31no—
35C2×C4×Q8—31no—
36C2×C4×D4—31no—
37C2×C4×D4—61no—
38C2×#117—61no—
39C22×C42—31no—
40#431—34no—
41#431—62no—
42C2×C4⋊C8—61no—
43C2×C22⋊C8—31no—
44C2×C22⋊C8—31no—
45C2×C4×C8—61no—
46#376—32no—
47#376C2×D312yescomplemented
48C2×#95—31no—
49C2×#95C2×D311yescharacteristic
50#328—34no—
51#328—32no—
52#328—32no—
53#327—34no—
54#327—32no—
55#327—32no—
56C22×A4—41no—
57C2×S4—42no—
58C2×S4—44no—
59C2×C4×D3—161no—
60C4×A4—42no—
61SL(2,ℤ/4)—42no—
62C22×Q8—31no—
63C22×D4—31no—
64C22×D4—61no—
65C23×C4—31no—
66C23×C4—31no—
67C23×C4—61no—
68C23×C4S411yescharacteristic
69C22×C8—61no—
70C22×C8—61no—
71#126—64no—
72#126—64no—
73#125—34no—
74#125—62no—
75#124—34no—
76#124—62no—
77#123—64no—
78#123—64no—
79C22⋊Q8—32no—
80C22⋊Q8—32no—
81C22⋊Q8—61no—
82C22⋊Q8—61no—
83#121—32no—
84#121—32no—
85#121—61no—
86#121—61no—
87C4×Q8—34no—
88C4×Q8—62no—
89C4×D4—34no—
90C4×D4—62no—
91C4×D4—64no—
92C4×D4—64no—
93#117—64no—
94#117—64no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—61no—
98C2×C4⋊C4—61no—
99C2×C4⋊C4—61no—
100C2×C4⋊C4—61no—
101C2×C22⋊C4—61no—
102C2×C22⋊C4—61no—
103C2×C22⋊C4—61no—
104C2×C22⋊C4—61no—
105C2×C22⋊C4—61no—
106C2×C42—31no—
107C2×C42—31no—
108C2×C42—31no—
109C2×C42—31no—
110C2×C42—32no—
111C2×C42—32no—
112C2×C42—32no—
113C2×C42—32no—
114C2×C42—61no—
115C4⋊C8—64no—
116C22⋊C8—32no—
117C22⋊C8—32no—
118C22⋊C8—61no—
119C22⋊C8—61no—
120C4×C8—64no—
121#95—32no—
122#95—31no—
123#95—61no—
124#95C22×D311yescharacteristic
125C22×D3—161no—
126C2×A4—42no—
127C2×A4—41no—
128S4—44no—
129C2×C3×C4—161no—
130C2×Dic3—161no—
131C4×D3—164no—
132C24—61no—
133C24C2×S411yescharacteristic
134C2×Q8—32no—
135C2×Q8—61no—
136C2×Q8—122no—
137C2×D4—32no—
138C2×D4—61no—
139C2×D4—62no—
140C2×D4—62no—
141C2×D4—122no—
142C2×D4—122no—
143C2×D4—122no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—31no—
151C22×C4—32no—
152C22×C4—32no—
153C22×C4—31no—
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—64no—
158C22×C4—62no—
159C22×C4—62no—
160C22×C4—61no—
161C22×C4—61no—
162C22×C4—61no—
163C22×C4—121no—
164C22×C4—122no—
165C22×C4—122no—
166C22×C4—121no—
167C22×C4C2×S412yes—
168C2×C8—62no—
169C2×C8—62no—
170C2×C8—122no—
171C2×C8—122no—
172C4⋊C4—32no—
173C4⋊C4—32no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4—64no—
179C4⋊C4—64no—
180C4⋊C4—64no—
181C22⋊C4—62no—
182C22⋊C4—62no—
183C22⋊C4—64no—
184C22⋊C4—64no—
185C22⋊C4—64no—
186C22⋊C4—62no—
187C22⋊C4—62no—
188C42—32no—
189C42—32no—
190C42—32no—
191C42—32no—
192C42—32no—
193C42—32no—
194C42—32no—
195C42—32no—
196C42—64no—
197C22×C3—161no—
198C2×D3—162no—
199C2×D3—164no—
200A4—41no—
201C3×C4—162no—
202Dic3—162no—
203C23—31no—
204C23—31no—
205C23—32no—
206C23—62no—
207C23—121no—
208C23—122no—
209C23—122no—
210C23—121no—
211C23#98512yes—
212C23C22×S411yescharacteristic
213Q8—122no—
214Q8—122no—
215D4—122no—
216D4—122no—
217D4—122no—
218D4—124no—
219D4—122no—
220C2×C4—31no—
221C2×C4—32no—
222C2×C4—32no—
223C2×C4—32no—
224C2×C4—31no—
225C2×C4—32no—
226C2×C4—32no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—31no—
230C2×C4—61no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—64no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—124no—
242C2×C4—121no—
243C2×C4—121no—
244C2×C4—121no—
245C2×C4—121no—
246C2×C4—121no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—121no—
250C2×C4—124no—
251C2×C4#85711yescharacteristic
252C8—122no—
253C8—122no—
254C2×C3—162no—
255C2×C3—161no—
256D3—164no—
257C22—31no—
258C22—32no—
259C22—32no—
260C22—32no—
261C22—31no—
262C22—31no—
263C22—61no—
264C22—122no—
265C22—122no—
266C22—122no—
267C22—242no—
268C22—242no—
269C22C2×#85711yescharacteristic
270C22C2×#98511yescharacteristic
271C4—32no—
272C4—61no—
273C4—62no—
274C4—62no—
275C4—61no—
276C4—61no—
277C4—61no—
278C4—122no—
279C4—122no—
280C4—122no—
281C4C2×#85712yes—
282C3—161noSylow
283C2—31no—
284C2—31no—
285C2—32no—
286C2—242no—
287C2#7298712yesdirect product factor
288C2C22×#85711yescharacteristic
subgroupG/H
center, hypercenterC2×C4#857
commutator, cosocle, hypocenter#95⋊C3C23
FrattiniC23C22×S4
FittingC2×#376D3
socleC24C2×S4
prime radicalC2×#4622C2
LCSG ▷ #95⋊C3
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#376⋊C3C2semidirect
2. #72987C2direct (N × H)
4. C2×#4621C2semidirect
7. #376⋊C3C22semidirect
10. #4621C22semidirect
13. C2×#376D3semidirect
47. #376C2×D3semidirect
287. C2#72987direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#376⋊C32
#729872
C2×#46222
C2×#46212
C2×#15003
C2×C4×S44