Finite Group Databasebeta

← C2×#10589183 #11796464

Subgroups — C2×#10589183 #11796464

Order 1152.

685 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 116 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×#64156C211yesmaximal, characteristic
2C2×C3×#836C212yesmaximal, complemented
3#10589183C214yesmaximal, direct product factor
4C2×#4477—32nomaximal
5C22×C32×Q8C2211yescharacteristic
6C22×C4⋊(C3⋊D3)C411yescharacteristic
7#64156—22no—
8#64156C412yes—
9#64156C2212yes—
10C2×#3752—22no—
11C2×C4×C3⋊Dic3C411yescharacteristic
12C2×C3×#208—22no—
13C3×#836—22no—
14C3×#836C2214yescomplemented
15C2×#947—32no—
16C2×#947—61no—
17C2×C3×#101—32no—
18C2×#836D312yescomplemented
19#4477—38no—
20C23×C3⋊D3—21no—
21C2×C32×Q8—41no—
22C2×C32×Q8—41no—
23C2×C32×Q8C2×C411yescharacteristic
24C2×C32×Q8D412yes—
25C2×C32×Q8C2311yescharacteristic
26C22×C32×C4C2×C411yescharacteristic
27C22×C32×C4D412yes—
28C22×C3⋊Dic3—21no—
29C2×C4⋊(C3⋊D3)—21no—
30C2×C4⋊(C3⋊D3)—41no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C4⋊(C3⋊D3)C2×C412yes—
33#3752—24no—
34#3752—24no—
35C4×C3⋊Dic3—21no—
36C4×C3⋊Dic3C2×C412yes—
37C3×#208—24no—
38C3×#208—42no—
39C2×C3×C3⋊C8—42no—
40C2×#355—91noSylow
41C22×C3×Q8—21no—
42C22×C3×Q8C2×D312yes—
43C22×D12—32no—
44C22×D12—61no—
45C2×C3×M16—62no—
46#947—34no—
47#947—34no—
48#947—62no—
49#947—62no—
50#947—64no—
51#947—62no—
52#947—62no—
53C2×(C2×D3)⋊C4—64no—
54C2×(C2×D3)⋊C4—62no—
55C2×(C2×D3)⋊C4—62no—
56C2×C4×Dic3—32no—
57C2×C4×Dic3—61no—
58C2×#208—22no—
59C3×#101—34no—
60C3×#101—62no—
61#836—22no—
62#836C2×D314yescomplemented
63C23×C32C22⋊C411yescharacteristic
64C22×C3⋊D3—22no—
65C22×C3⋊D3—44no—
66C22×C3⋊D3—41no—
67C22×C3⋊D3—41no—
68C32×Q8—41no—
69C32×Q8—42no—
70C32×Q8—41no—
71C2×C32×C4—21no—
72C2×C32×C4—21no—
73C2×C32×C4—22no—
74C2×C32×C4—22no—
75C2×C32×C4C22⋊C411yescharacteristic
76C2×C32×C4C22⋊C412yes—
77C2×C32×C4C22×C411yescharacteristic
78C2×C32×C4C2×D412yes—
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3—42no—
81C4⋊(C3⋊D3)—42no—
82C4⋊(C3⋊D3)—42no—
83C3×C3⋊C8—44no—
84C2×#124—91no—
85C2×#101—32no—
86#355—94no—
87C23×D3—62no—
88C23×D3—62no—
89C2×C3×Q8—21no—
90C2×C3×Q8—22no—
91C2×C3×Q8—21no—
92C2×C3×Q8—42no—
93C2×C3×Q8—42no—
94C2×C3×Q8—42no—
95C2×C3×Q8—42no—
96C2×C3×Q8C4×D312yes—
97C2×C3×Q8D1212yes—
98C2×C3×Q8#6712yes—
99C2×C3×Q8C22×D312yes—
100C22×C3×C4—21no—
101C22×C3×C4—22no—
102C22×C3×C4C4×D312yes—
103C22×C3×C4D1212yes—
104C22×C3×C4#6712yes—
105C22×Dic3—62no—
106C22×Dic3—62no—
107C2×D12—34no—
108C2×D12—62no—
109C2×D12—62no—
110C2×D12—62no—
111C2×D12—122no—
112C2×D12—122no—
113C2×D12—122no—
114C2×D12—122no—
115C3×M16—64no—
116C3×M16—122no—
117C2×C3×C8—122no—
118(C2×D3)⋊C4—68no—
119(C2×D3)⋊C4—68no—
120(C2×D3)⋊C4—64no—
121(C2×D3)⋊C4—64no—
122(C2×D3)⋊C4—64no—
123(C2×D3)⋊C4—64no—
124C4×Dic3—34no—
125C4×Dic3—62no—
126C4×Dic3—62no—
127C4×Dic3—62no—
128#208—24no—
129#208—42no—
130C2×C3⋊C8—42no—
131C22×C32—22no—
132C22×C32#9911yescharacteristic
133C22×C32#9911yescharacteristic
134C22×C32C2×C22⋊C411yescharacteristic
135C2×C3⋊D3—44no—
136C2×C3⋊D3—42no—
137C2×C3⋊D3—82no—
138C2×C3⋊D3—82no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—22no—
142C32×C4—22no—
143C3⋊Dic3—42no—
144C22×Q8D3211yescharacteristic
145C22×D4—91no—
146C2×M16—62no—
147#124—92no—
148#124—92no—
149#124—182no—
150C2×C22⋊C4—182no—
151C2×C42—91no—
152#101—34no—
153#101—62no—
154C23×C3—21no—
155C23×C3(C2×D3)⋊C412yes—
156C22×D3—64no—
157C22×D3—64no—
158C22×D3—122no—
159C22×D3—122no—
160C22×D3—128no—
161C22×D3—128no—
162C22×D3—122no—
163C22×D3—122no—
164C3×Q8—42no—
165C3×Q8—42no—
166C3×Q8—42no—
167C3×Q8—42no—
168C3×Q8—42no—
169C3×Q8—44no—
170C3×Q8—42no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—21no—
180C2×C3×C4—22no—
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4(C2×D3)⋊C412yes—
186C2×C3×C4(C2×D3)⋊C412yes—
187C2×C3×C4(C2×D3)⋊C412yes—
188C2×C3×C4C2×C4×D312yes—
189C2×C3×C4C2×D1212yes—
190C2×C3×C4C2×#6712yes—
191C2×Dic3—64no—
192C2×Dic3—64no—
193C2×Dic3—124no—
194C2×Dic3—124no—
195D12—124no—
196D12—124no—
197D12—124no—
198D12—124no—
199C3×C8—124no—
200C3⋊C8—44no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32#35512yescomplemented
204C2×C32C2×#9911yescharacteristic
205C3⋊D3—82no—
206C24—181no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8#62411yescharacteristic
210C2×Q8#62612yes—
211C2×Q8C2×D3211yescharacteristic
212C2×D4—92no—
213C2×D4—181no—
214C2×D4—361no—
215C2×D4—361no—
216C22×C4—181no—
217C22×C4#62411yescharacteristic
218C22×C4#62612yes—
219M16—64no—
220M16—122no—
221C2×C8—122no—
222C22⋊C4—184no—
223C22⋊C4—184no—
224C42—92no—
225C42—181no—
226C22×C3—24no—
227C22×C3—24no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3#80612yes—
232C22×C3#80612yes—
233C22×C3C2×(C2×D3)⋊C412yes—
234C2×D3—124no—
235C2×D3—128no—
236C2×D3—128no—
237C2×D3—124no—
238C2×D3—244no—
239C2×D3—244no—
240C2×D3—244no—
241C2×D3—244no—
242C3×C4—22no—
243C3×C4—22no—
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—42no—
251C3×C4—42no—
252Dic3—124no—
253Dic3—124no—
254C32C2×#35511yescharacteristic, complemented, Sylow
255C23—182no—
256C23—364no—
257C23—361no—
258C23—361no—
259C23#372211yescharacteristic
260Q8—41no—
261Q8—42no—
262Q8—41no—
263D4—362no—
264D4—362no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—182no—
270C2×C4—362no—
271C2×C4#372211yescharacteristic
272C2×C4#372212yes—
273C2×C4C2×#62411yescharacteristic
274C2×C4C2×#62612yes—
275C8—124no—
276C2×C3—22no—
277C2×C3—22no—
278C2×C3—22no—
279C2×C3—22no—
280C2×C3—21no—
281C2×C3—22no—
282C2×C3#447714yescomplemented
283C2×C3C2×#80612yes—
284D3—244no—
285D3—244no—
286C22—22no—
287C22—364no—
288C22—362no—
289C22—722no—
290C22—722no—
291C22#6358111yescharacteristic
292C22#6358111yescharacteristic
293C22C2×#372211yescharacteristic
294C4—22no—
295C4—21no—
296C4—22no—
297C4—21no—
298C4—362no—
299C3—21no—
300C3C2×#447712yescomplemented
301C2—21no—
302C2—21no—
303C2—722no—
304C2#1058918312yesdirect product factor
305C2C2×#6358111yescharacteristic
subgroupG/H
centerC22#63581
commutatorC2×C32×C4C22×C4
FrattiniC2×Q8C2×D32
FittingC22×C32×Q8C22
socleC22×C32#99
prime radicalC22×C4⋊(C3⋊D3)C4
cosocleC2×C32×Q8C23
hypercenterC22×Q8D32
hypocenterC32C2×#355
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#836C2semidirect
3. #10589183C2direct (N × H)
14. C3×#836C22semidirect
18. C2×#836D3semidirect
62. #836C2×D3semidirect
203. C2×C32#355semidirect
254. C32C2×#355semidirect
282. C2×C3#4477semidirect
300. C3C2×#4477semidirect
304. C2#10589183direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#641562
C2×C3×#8362
#105891832
C2×#44773