Finite Group Databasebeta

← #11722876

Subgroups — #11722876

Order 864.

342 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C4⋊#1113C211yesmaximal, characteristic, complemented
2C3×#3796C211yesmaximal, characteristic, complemented
3#88793C211yesmaximal, characteristic, complemented
4#88789C211yesmaximal, characteristic, complemented
5#88787C211yesmaximal, characteristic, complemented
6C3×#3717C211yesmaximal, characteristic, complemented
7C3×#3714C211yesmaximal, characteristic, complemented
8#63939—31nomaximal
9#63925—31nomaximal
10#63798—31nomaximal
11C2×#1113—21no—
12C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×C4×C3⋊D3C2211yescharacteristic
15C32×D12C2211yescharacteristic, complemented
16C32×Dic6C2211yescharacteristic, complemented
17#6295—21no—
18C3×#626—21no—
19C3×D3×Dic3—21no—
20C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C32×C3⋊C8C2211yescharacteristic, complemented
22C3×#239—31no—
23C3×#237—31no—
24#3802—31no—
25D3×D12—31no—
26D3×D12—31no—
27#3796D311yescharacteristic, complemented
28#3755—31no—
29#3753—31no—
30#3743—31no—
31C3×#215—31no—
32C3×#213—31no—
33C3×D24—31no—
34C3×#204—31no—
35#3717D311yescharacteristic, complemented
36#3715—31no—
37#3715—31no—
38#3715—61no—
39#3714D311yescharacteristic, complemented
40#3713—31no—
41#3713—31no—
42#3713—61no—
43#3711—31no—
44#3711—31no—
45#3711—61no—
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3—21no—
48C2×C3×C3⋊D3D411yescharacteristic
49C2×C32×D3—21no—
50#1113—42no—
51C33×C4C2311yescharacteristic
52C3×C3⋊Dic3D411yescharacteristic
53C32×Dic3—21no—
54#949—91no—
55#914—91no—
56#911—91no—
57C2×D32—61no—
58C2×D32—61no—
59C2×D32—61no—
60C32×Q8—31no—
61C32×D4—31no—
62C4⋊(C3⋊D3)C2×D311yescharacteristic
63C4⋊(C3⋊D3)C2×D311yescharacteristic
64C4×C3⋊D3C2×D311yescharacteristic
65C3×#67—61no—
66C2×C3×Dic3—61no—
67C3×D12—21no—
68C3×D12—31no—
69C3×D12—31no—
70C3×D12—31no—
71C3×D12—31no—
72C3×D12—61no—
73C3×D12—61no—
74C3×D12—61no—
75C3×D12C2×D311yescharacteristic, complemented
76C3×D12C2×D311yescharacteristic, complemented
77C3×C4×D3—31no—
78C3×C4×D3—31no—
79C3×C4×D3—61no—
80C3×C4×D3—61no—
81C3×Dic6—21no—
82C3×Dic6C2×D311yescharacteristic, complemented
83C3×Dic6C2×D311yescharacteristic, complemented
84#626—21no—
85#626—61no—
86#626—61no—
87#625—61no—
88D3×Dic3—21no—
89C32×C8—31no—
90C3⋊(C3⋊C8)—31no—
91C3⋊(C3⋊C8)—31no—
92C3⋊(C3⋊C8)—31no—
93C3⋊(C3⋊C8)—61no—
94C3⋊(C3⋊C8)—61no—
95C3⋊(C3⋊C8)—61no—
96C3⋊(C3⋊C8)—121no—
97C3×C3⋊C8—21no—
98C3×C3⋊C8C2×D311yescharacteristic, complemented
99C3×C3⋊C8C2×D311yescharacteristic, complemented
100C2×C33C2×D411yescharacteristic
101C3×C3⋊D3—21no—
102C3×C3⋊D3—41no—
103C3×C3⋊D3—41no—
104C32×D3—41no—
105C3×#41—91no—
106#239—31no—
107#237—31no—
108D3×D4—91no—
109D3×D4—91no—
110C2×D12—91no—
111C3×SD16—91no—
112C3×D8—91no—
113#215—31no—
114#215—91no—
115#215—91no—
116#215—181no—
117#214—91no—
118#213—31no—
119#213—91no—
120#213—91no—
121#213—91no—
122#213—181no—
123#208—91no—
124D24—31no—
125#204—31no—
126ΓL(1,𝔽25)—91no—
127ΓL(1,𝔽25)—91no—
128ΓL(1,𝔽25)—181no—
129C22×C32—61no—
130C2×C3⋊D3—21no—
131C2×C3⋊D3—21no—
132C2×C3⋊D3#6711yescharacteristic
133C2×C3×D3—21no—
134C2×C3×D3—21no—
135C2×C3×D3—31no—
136C2×C3×D3—31no—
137C2×C3×D3—41no—
138C2×C3×D3—61no—
139C2×C3×D3—61no—
140C2×C3×D3—61no—
141C2×C3×D3—61no—
142C2×C3×D3—61no—
143C2×C3×D3—121no—
144C2×C3×D3—121no—
145D32—122no—
146D32—122no—
147D32—122no—
148C32×C4—21no—
149C32×C4—21no—
150C32×C4—21no—
151C32×C4—41no—
152C32×C4—61no—
153C32×C4C22×D311yescharacteristic
154C32×C4C22×D311yescharacteristic
155C32×C4C22×D311yescharacteristic
156C3⋊Dic3#6711yescharacteristic
157C3×Dic3—21no—
158C3×Dic3—21no—
159C3×Dic3—31no—
160C3×Dic3—31no—
161C3×Dic3—41no—
162C3×Dic3—61no—
163AGL(1,ℤ/8)—271noSylow
164C33AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
165C22×D3—181no—
166C22×D3—181no—
167C22×D3—181no—
168C3×Q8—31no—
169C3×Q8—31no—
170C3×Q8—61no—
171C3×D4—31no—
172C3×D4—31no—
173C3×D4—61no—
174C3×D4—91no—
175C3×D4—91no—
176C3×D4—181no—
177C2×C3×C4—91no—
178C2×C3×C4—181no—
179#67—61no—
180#67—181no—
181#67—181no—
182C2×Dic3—61no—
183D12—31no—
184D12—31no—
185D12—31no—
186D12—31no—
187D12—61no—
188D12—61no—
189D12—61no—
190D12—181no—
191D12D3211yescharacteristic, complemented
192C4×D3—31no—
193C4×D3—31no—
194C4×D3—61no—
195C4×D3—61no—
196Dic6D3211yescharacteristic, complemented
197C3×C8—31no—
198C3×C8—31no—
199C3×C8—61no—
200C3⋊C8—91no—
201C3⋊C8—91no—
202C3⋊C8—91no—
203C3⋊C8—181no—
204C3⋊C8—181no—
205C3⋊C8—181no—
206C3⋊C8—361no—
207C3⋊C8D3211yescharacteristic, complemented
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—41no—
212C2×C32—121no—
213C2×C32D3×D411yescharacteristic
214C2×C32D3×D411yescharacteristic
215C2×C32C2×#6711yescharacteristic
216C3⋊D3—21no—
217C3⋊D3—41no—
218C3⋊D3—41no—
219C3×D3—41no—
220C3×D3—41no—
221C3×D3—61no—
222C3×D3—61no—
223C3×D3—81no—
224C3×D3—121no—
225C3×D3—121no—
226C3×D3—121no—
227C3×D3—121no—
228C3×D3—121no—
229C3×D3—241no—
230C3×D3—241no—
231#41—91no—
232C2×D4—271no—
233SD16—91no—
234SD16—271no—
235D8—91no—
236D8—271no—
237M16—271no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3—91no—
241C22×C3—121no—
242C22×C3—181no—
243C22×C3—181no—
244C2×D3—21no—
245C2×D3—31no—
246C2×D3—31no—
247C2×D3—61no—
248C2×D3—61no—
249C2×D3—61no—
250C2×D3—61no—
251C2×D3—61no—
252C2×D3—121no—
253C2×D3—121no—
254C2×D3—362no—
255C2×D3—362no—
256C2×D3—362no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—91no—
264C3×C4—121no—
265C3×C4C2×D3211yescharacteristic
266C3×C4C2×D3211yescharacteristic
267C3×C4C2×D3211yescharacteristic
268Dic3—21no—
269Dic3—31no—
270Dic3—31no—
271Dic3—61no—
272C32—21no—
273C32—21no—
274C32—21no—
275C32—41no—
276C32#91111yescharacteristic, complemented
277C32#91411yescharacteristic, complemented
278C32#94911yescharacteristic, complemented
279C23—541no—
280Q8—31no—
281D4—31no—
282D4—91no—
283D4—91no—
284D4—181no—
285D4—541no—
286C2×C4—91no—
287C2×C4—181no—
288C8—31no—
289C8—271no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—41no—
294C2×C3—121no—
295C2×C3—121no—
296C2×C3—181no—
297C2×C3—241no—
298C2×C3—361no—
299C2×C3—361no—
300C2×C3#380211yescharacteristic
301C2×C3D3×#6711yescharacteristic
302C2×C3D3×#6711yescharacteristic
303D3—41no—
304D3—61no—
305D3—61no—
306D3—121no—
307D3—121no—
308D3—121no—
309D3—121no—
310D3—121no—
311D3—241no—
312D3—241no—
313C22—61no—
314C22—91no—
315C22—181no—
316C22—181no—
317C22—1082no—
318C4—61no—
319C4—91no—
320C4D3311yescharacteristic
321C3—21no—
322C3—21no—
323C3—21no—
324C3—41no—
325C3#6379811yescharacteristic, complemented
326C3#6392511yescharacteristic, complemented
327C3#6393911yescharacteristic, complemented
328C2—121no—
329C2—181no—
330C2—361no—
331C2—361no—
332C2#8895311yescharacteristic
subgroupG/H
centerC2#88953
commutator, Fitting, cosocleC33×C4C23
Frattini, hypercenterC4D33
socleC2×C33C2×D4
hypocenterC33AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4⋊#1113C2semidirect
2. C3×#3796C2semidirect
3. #88793C2semidirect
4. #88789C2semidirect
5. #88787C2semidirect
6. C3×#3717C2semidirect
7. C3×#3714C2semidirect
15. C32×D12C22semidirect
16. C32×Dic6C22semidirect
20. C3⋊(C3⋊(C3⋊C8))C22semidirect
21. C32×C3⋊C8C22semidirect
27. #3796D3semidirect
35. #3717D3semidirect
39. #3714D3semidirect
75. C3×D12C2×D3semidirect
76. C3×D12C2×D3semidirect
82. C3×Dic6C2×D3semidirect
83. C3×Dic6C2×D3semidirect
98. C3×C3⋊C8C2×D3semidirect
99. C3×C3⋊C8C2×D3semidirect
164. C33AGL(1,ℤ/8)semidirect
191. D12D32semidirect
196. Dic6D32semidirect
207. C3⋊C8D32semidirect
276. C32#911semidirect
277. C32#914semidirect
278. C32#949semidirect
325. C3#63798semidirect
326. C3#63925semidirect
327. C3#63939semidirect

Maximal subgroups

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

subgroupindex
C4⋊#11132
C3×#37962
#887932
#887892
#887872
C3×#37172
C3×#37142
#639393
#639253
#637983