Finite Group Databasebeta

← #11796865

Subgroups — #11796865

Order 1152.

414 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 67 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×#63822C211yesmaximal, characteristic
2C2×#63773C211yesmaximal, characteristic, complemented
3#10591138C211yesmaximal, characteristic, complemented
4#10590064C214yesmaximal, complemented
5C2×C3×#235—21no—
6C22×C3⋊(C3⋊C8)C2211yescharacteristic
7C3×#926—21no—
8C2×#3718C411yescharacteristic, complemented
9C2×#3715—21no—
10#63822—21no—
11#63822—21no—
12#63822C412yescomplemented
13#63822C2212yes—
14C2×#3713C411yescharacteristic, complemented
15#63773—21no—
16#63773C2212yescomplemented
17C2×#3708—21no—
18#63576—21no—
19#63574C2212yescomplemented
20C2×#950—61no—
21#4998—61no—
22C22×C32×C4D411yescharacteristic
23C2×C3×#67—41no—
24C3×#235—22no—
25C3×#235—22no—
26C3×#235—41no—
27C3×#235—41no—
28C2×C3×D12—21no—
29C2×C3×C4×D3—41no—
30C2×C3×Dic6—21no—
31C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
32C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
33C2×C3⋊(C3⋊C8)D411yescharacteristic
34C2×C3⋊(C3⋊C8)D411yescharacteristic
35C2×C3⋊(C3⋊C8)D411yescharacteristic
36C2×C3⋊(C3⋊C8)C2311yescharacteristic
37C3×#217—41no—
38C3×C4⋊Dic3—22no—
39C3×C4×Dic3—41no—
40#3718—21no—
41#3718C2×C412yes—
42#3715—22no—
43#3715—22no—
44#3713—21no—
45#3713C2×C412yes—
46#3708—22no—
47#2145—91nomaximal, Sylow
48C2×C3×#41—61no—
49C2×#235—21no—
50C3×#117—61no—
51#950—62no—
52#950—62no—
53#950—62no—
54#950—62no—
55C2×#216—61no—
56C2×#215—61no—
57C2×#214—61no—
58C2×#213—61no—
59#926—21no—
60C22×C3⋊C8—61no—
61C22×C3⋊C8—61no—
62#808—62no—
63#807—62no—
64C23×C32D811yescharacteristic
65C22×C3×D3—41no—
66C2×C32×C4C22⋊C411yescharacteristic
67C2×C32×C4C22⋊C411yescharacteristic
68C2×C32×C4D811yescharacteristic
69C2×C32×C4SD1611yescharacteristic
70C2×C32×C4SD1611yescharacteristic
71C2×C32×C4C2×D411yescharacteristic
72C3×#67—42no—
73C3×#67—42no—
74C2×C3×Dic3—41no—
75C2×C3×Dic3—42no—
76C3×D12—22no—
77C3×D12—41no—
78C3×C4×D3—42no—
79C3×C4×D3—42no—
80C3×Dic6—22no—
81C3×Dic6—41no—
82C3⋊(C3⋊C8)C22⋊C411yescharacteristic
83C3⋊(C3⋊C8)C22⋊C411yescharacteristic
84C3⋊(C3⋊C8)C22×C411yescharacteristic
85C3⋊(C3⋊C8)C2×D411yescharacteristic
86C2×#135—91no—
87C2×M32—91no—
88#427—91no—
89#360—94no—
90C3×#41—62no—
91C3×#41—62no—
92C3×#41—121no—
93C3×#41—121no—
94C2×C3×Q8—61no—
95C2×C3×D4—61no—
96C2×C3×D4—121no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—121no—
100C2×#67—41no—
101#235—22no—
102#235—22no—
103#235—41no—
104#235—41no—
105C2×D12—21no—
106C2×C4×D3—41no—
107C2×Dic6—21no—
108C3×C4⋊C4—62no—
109C3×C22⋊C4—121no—
110C3×C42—121no—
111#217—41no—
112#216—62no—
113#216—62no—
114#215—62no—
115#215—62no—
116#214—62no—
117#214—62no—
118#213—62no—
119#213—62no—
120C4⋊Dic3—22no—
121C4×Dic3—41no—
122C2×C3⋊C8—61no—
123C2×C3⋊C8—61no—
124C2×C3⋊C8—61no—
125C2×C3⋊C8—61no—
126C2×C3⋊C8—61no—
127C2×C3⋊C8—61no—
128C2×C3⋊C8—61no—
129C2×C3⋊C8—61no—
130C2×C3⋊C8—61no—
131C2×C3⋊C8—61no—
132C2×C3⋊C8—61no—
133C2×C3⋊C8—61no—
134C22×C32—41no—
135C22×C32#10211yescharacteristic
136C22×C32#10211yescharacteristic
137C22×C32C2×D811yescharacteristic
138C2×C3×D3—42no—
139C2×C3×D3—82no—
140C32×C4#10211yescharacteristic
141C32×C4#10211yescharacteristic
142C32×C4C2×C22⋊C411yescharacteristic
143C32×C4C2×SD1611yescharacteristic
144C3×Dic3—42no—
145C3×Dic3—82no—
146C2×#41—61no—
147#135—92no—
148#135—92no—
149#135—181no—
150#135—181no—
151C2×Q16—91no—
152C2×SD16—181no—
153C2×D8—91no—
154C22×C8—91no—
155#117—61no—
156M32—92no—
157M32—181no—
158C2×C16—181no—
159#107—181no—
160#106—92no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—121no—
164C22×D3—41no—
165C3×Q8—62no—
166C3×Q8—121no—
167C3×D4—62no—
168C3×D4—121no—
169C3×D4—122no—
170C3×D4—122no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—122no—
184C2×C3×C4—122no—
185C2×C3×C4—121no—
186C2×C3×C4—122no—
187#67—42no—
188#67—42no—
189C2×Dic3—41no—
190C2×Dic3—42no—
191D12—22no—
192D12—41no—
193C4×D3—42no—
194C4×D3—42no—
195Dic6—22no—
196Dic6—41no—
197C3⋊C8—61no—
198C3⋊C8—61no—
199C3⋊C8—61no—
200C3⋊C8—61no—
201C3⋊C8—61no—
202C3⋊C8—61no—
203C3⋊C8—61no—
204C3⋊C8—61no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32C2×#10211yescharacteristic
209C3×D3—82no—
210#41—62no—
211#41—62no—
212#41—121no—
213#41—121no—
214C2×Q8—61no—
215C2×D4—61no—
216C2×D4—121no—
217C22×C4—121no—
218C22×C4GO+(4,𝔽2)11yescharacteristic
219Q16—92no—
220Q16—181no—
221SD16—182no—
222SD16—182no—
223D8—92no—
224D8—181no—
225C2×C8—91no—
226C2×C8—91no—
227C2×C8—91no—
228C2×C8—91no—
229C2×C8—91no—
230C2×C8—91no—
231C4⋊C4—62no—
232C22⋊C4—121no—
233C42—121no—
234C16—182no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—42no—
242C22×C3—42no—
243C22×C3—122no—
244C22×C3—242no—
245C2×D3—42no—
246C2×D3—82no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—122no—
256C3×C4—242no—
257Dic3—42no—
258Dic3—82no—
259C32#214511yescharacteristic, complemented, Sylow
260C23—121no—
261C23#377411yescharacteristic
262Q8—62no—
263Q8—121no—
264D4—62no—
265D4—122no—
266D4—122no—
267D4—121no—
268C2×C4—121no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4D32⋊C411yescharacteristic
273C2×C4D32⋊C411yescharacteristic
274C2×C4#377411yescharacteristic
275C2×C4#377511yescharacteristic
276C2×C4#377511yescharacteristic
277C2×C4C2×GO+(4,𝔽2)11yescharacteristic
278C8—91no—
279C8—91no—
280C8—91no—
281C8—91no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—242no—
291D3—82no—
292C22—41no—
293C22—122no—
294C22—242no—
295C22#6373911yescharacteristic
296C22#6373911yescharacteristic
297C22C2×#377411yescharacteristic
298C4—122no—
299C4—242no—
300C4#6373911yescharacteristic
301C4#6373911yescharacteristic
302C4C2×D32⋊C411yescharacteristic
303C4C2×#377511yescharacteristic
304C3—21no—
305C3—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—242no—
310C2C2×#6373911yescharacteristic
subgroupG/H
centerC4#63739
commutatorC3⋊(C3⋊C8)C22×C4
FrattiniC2×C4C2×GO+(4,𝔽2)
FittingC22×C32×C4D4
socleC2×C32C2×#102
prime radicalC2×#63822C2
cosocleC2×C3⋊(C3⋊C8)C23
hypercenterC22×C4GO+(4,𝔽2)
hypocenterC32#2145
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#63773C2semidirect
3. #10591138C2semidirect
4. #10590064C2semidirect
8. C2×#3718C4semidirect
12. #63822C4semidirect
14. C2×#3713C4semidirect
16. #63773C22semidirect
19. #63574C22semidirect
259. C32#2145semidirect

Maximal subgroups

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

subgroupindex
C2×#638222
C2×#637732
#105911382
#105900642
#21459