Finite Group Databasebeta

← C2×#73111 #11712644

Subgroups — C2×#73111 #11712644

Order 768.

721 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C22×C8×A4C211yesmaximal, characteristic
2C2×C4⋊SL(2,ℤ/4)C212yesmaximal
3#73111C214yesmaximal, direct product factor
4C2×#1808—31nomaximal, Sylow
5C22×C4×A4C2211yescharacteristic
6C22×SL(2,ℤ/4)—22no—
7C2×C8×A4C414yescomplemented
8C2×C8×A4C2212yes—
9C4⋊SL(2,ℤ/4)—22no—
10C4⋊SL(2,ℤ/4)C2214yes—
11C2×#818—41nomaximal
12C24×C8D311yescharacteristic
13C22×#107—31no—
14C22×#107—31no—
15C2×#380—32no—
16#1808—34no—
17#1808—34no—
18C2×#340—32no—
19C23×A4D411yescharacteristic
20C2×C4×A4C2×C412yes—
21C2×C4×A4D411yescharacteristic
22C2×C4×A4Q812yes—
23C2×C4×A4C2311yescharacteristic
24C2×SL(2,ℤ/4)—24no—
25C2×SL(2,ℤ/4)—44no—
26C22×C3×C8—41no—
27C2×C4⋊Dic3—42no—
28C8×A4C2×C414yescomplemented
29#818—44no—
30C24×C4C2×D311yescharacteristic
31C23×C8—31no—
32C23×C8—32no—
33C23×C8—31no—
34C23×C8—34no—
35C23×C8Dic314yescomplemented
36C23×C8C2×D312yes—
37C22×C4⋊C4—32no—
38C22×C4⋊C4—32no—
39C22×C22⋊C4—62no—
40C2×#107—34no—
41C2×#107—34no—
42C2×#107—64no—
43C2×#107—61no—
44C2×#107—62no—
45C2×#107—61no—
46#380—34no—
47#380—34no—
48#380—62no—
49#380—62no—
50C2×#95—62no—
51#340—38no—
52C22×A4C4⋊C412yes—
53C22×A4D812yes—
54C22×A4Q1612yes—
55C22×A4C2×D411yescharacteristic
56C22×C3×C4—41no—
57C22×Dic3—82no—
58C4×A4C4⋊C412yes—
59C4×A4C22×C411yescharacteristic
60C4×A4C2×Q811yescharacteristic
61SL(2,ℤ/4)—44no—
62C2×C3×C8—44no—
63C2×C3×C8—42no—
64C4⋊Dic3—44no—
65C4⋊Dic3—82no—
66C25D1211yescharacteristic
67C23×C4—31no—
68C23×C4—32no—
69C23×C4—31no—
70C23×C4—32no—
71C23×C4—31no—
72C23×C4—31no—
73C23×C4—62no—
74C23×C4—62no—
75C23×C4Dic612yes—
76C23×C4D1211yescharacteristic
77C23×C4C2×Dic312yes—
78C23×C4C22×D311yescharacteristic
79C22×C8—31no—
80C22×C8—34no—
81C22×C8—34no—
82C22×C8—32no—
83C22×C8—32no—
84C22×C8—34no—
85C22×C8—38no—
86C22×C8—34no—
87C22×C8—32no—
88C22×C8—64no—
89C22×C8—62no—
90C22×C8—61no—
91C22×C8C2×Dic314yescomplemented
92C22×C8S411yescharacteristic
93C2×C4⋊C4—34no—
94C2×C4⋊C4—34no—
95C2×C4⋊C4—62no—
96C2×C4⋊C4—64no—
97C2×C4⋊C4—62no—
98C2×C4⋊C4—64no—
99C2×C4⋊C4—62no—
100C2×C4⋊C4—62no—
101C2×C4⋊C4—62no—
102C2×C4⋊C4—62no—
103C2×C22⋊C4—64no—
104C2×C22⋊C4—64no—
105C2×C22⋊C4—124no—
106C2×C22⋊C4—124no—
107#107—68no—
108#107—64no—
109#107—64no—
110#95—64no—
111#95—64no—
112C23×C3—41no—
113C2×A4#10714yescomplemented
114C2×A4C2×C4⋊C411yescharacteristic
115C2×A4C2×D811yescharacteristic
116C2×A4C2×Q1611yescharacteristic
117C2×C3×C4—42no—
118C2×C3×C4—41no—
119C2×C3×C4—42no—
120C2×C3×C4—41no—
121C2×Dic3—84no—
122C2×Dic3—164no—
123C3×C8—44no—
124C24—31no—
125C24—31no—
126C24—32no—
127C24—32no—
128C24—32no—
129C24D2412yes—
130C24Dic1212yes—
131C24C4⋊Dic312yes—
132C24C2×D1211yescharacteristic
133C22×C4—31no—
134C22×C4—32no—
135C22×C4—32no—
136C22×C4—31no—
137C22×C4—31no—
138C22×C4—32no—
139C22×C4—32no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—32no—
145C22×C4—31no—
146C22×C4—32no—
147C22×C4—32no—
148C22×C4—32no—
149C22×C4—32no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—61no—
154C22×C4—61no—
155C22×C4—62no—
156C22×C4—61no—
157C22×C4—62no—
158C22×C4—64no—
159C22×C4—64no—
160C22×C4—124no—
161C22×C4—124no—
162C22×C4—124no—
163C22×C4—122no—
164C22×C4—124no—
165C22×C4—124no—
166C22×C4—122no—
167C22×C4C4⋊Dic312yes—
168C22×C4C2×Dic611yescharacteristic
169C22×C4C22×Dic311yescharacteristic
170C22×C4C2×S411yescharacteristic
171C2×C8—34no—
172C2×C8—32no—
173C2×C8—34no—
174C2×C8—34no—
175C2×C8—38no—
176C2×C8—34no—
177C2×C8—62no—
178C2×C8—64no—
179C2×C8SL(2,ℤ/4)14yescomplemented
180C2×C8C2×S412yes—
181C4⋊C4—64no—
182C4⋊C4—64no—
183C4⋊C4—64no—
184C4⋊C4—64no—
185C4⋊C4—122no—
186C4⋊C4—124no—
187C4⋊C4—122no—
188C22⋊C4—128no—
189C22⋊C4—124no—
190C22⋊C4—124no—
191C22×C3—42no—
192C22×C3—41no—
193C22×C3—42no—
194C22×C3—42no—
195A4C2×#10711yescharacteristic, complemented
196C3×C4—41no—
197C3×C4—42no—
198C3×C4—41no—
199Dic3—164no—
200C23—32no—
201C23—32no—
202C23—31no—
203C23—31no—
204C23—32no—
205C23—32no—
206C23—32no—
207C23—32no—
208C23—32no—
209C23—31no—
210C23—32no—
211C23—31no—
212C23—34no—
213C23—32no—
214C23—34no—
215C23—34no—
216C23—31no—
217C23—61no—
218C23—64no—
219C23—61no—
220C23—61no—
221C23#81814yescomplemented
222C23C2×D2411yescharacteristic
223C23C2×Dic1211yescharacteristic
224C23C2×C4⋊Dic311yescharacteristic
225C23C4⋊S411yescharacteristic
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—32no—
229C2×C4—32no—
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—31no—
242C2×C4—31no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—62no—
246C2×C4—62no—
247C2×C4—124no—
248C2×C4—124no—
249C2×C4—124no—
250C2×C4—124no—
251C2×C4—124no—
252C2×C4—124no—
253C2×C4—248no—
254C2×C4—244no—
255C2×C4—244no—
256C2×C4A4⋊Q812yes—
257C2×C4C4⋊S411yescharacteristic
258C2×C4C2×SL(2,ℤ/4)12yes—
259C2×C4C22×S411yescharacteristic
260C8—34no—
261C8C2×SL(2,ℤ/4)14yescomplemented
262C2×C3—44no—
263C2×C3—41no—
264C2×C3—41no—
265C2×C3—41no—
266C22—34no—
267C22—31no—
268C22—31no—
269C22—31no—
270C22—34no—
271C22—34no—
272C22—34no—
273C22—34no—
274C22—31no—
275C22—31no—
276C22—32no—
277C22—31no—
278C22—31no—
279C22—32no—
280C22—31no—
281C22—31no—
282C22—32no—
283C22—62no—
284C22—61no—
285C22—62no—
286C22—62no—
287C22C2×#81811yescharacteristic, complemented
288C22#539812yes—
289C22#540212yes—
290C22C4⋊SL(2,ℤ/4)12yes—
291C22C2×C4⋊S411yescharacteristic
292C4—31no—
293C4—31no—
294C4—32no—
295C4—244no—
296C4—244no—
297C4C4⋊SL(2,ℤ/4)12yes—
298C4C2×A4⋊Q811yescharacteristic
299C4C22×SL(2,ℤ/4)11yescharacteristic
300C3—41noSylow
301C2—31no—
302C2—31no—
303C2—34no—
304C2—31no—
305C2—31no—
306C2#7311114yesdirect product factor
307C2C2×#539811yescharacteristic
308C2C2×#540211yescharacteristic
309C2C2×C4⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC23C4⋊S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC24×C8D3
socleC25D12
prime radicalC23×A4D4
cosocleC2×C4×A4C23
hypercenterC22×C8S4
hypocenterA4C2×#107
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. #73111C2direct (N × H)
7. C2×C8×A4C4semidirect
28. C8×A4C2×C4semidirect
35. C23×C8Dic3semidirect
91. C22×C8C2×Dic3semidirect
113. C2×A4#107semidirect
179. C2×C8SL(2,ℤ/4)semidirect
195. A4C2×#107semidirect
221. C23#818semidirect
261. C8C2×SL(2,ℤ/4)semidirect
287. C22C2×#818semidirect
306. C2#73111direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C8×A42
C2×C4⋊SL(2,ℤ/4)2
#731112
C2×#18083
C2×#8184