Finite Group Databasebeta

← #11781846

Subgroups — #11781846

Order 1152.

352 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C3×#5083C211yesmaximal, characteristic
2C3×#5080C211yesmaximal, characteristic, complemented
3#10590337C211yesmaximal, characteristic
4#10590266C211yesmaximal, characteristic
5#10589413C211yesmaximal, characteristic, complemented
6#10589099C211yesmaximal, characteristic, complemented
7#10589095C211yesmaximal, characteristic, complemented
8#71345—31nomaximal
9#71040—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×C3×#215—21no—
12C3×C4×D12C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3711—21no—
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#835—21no—
18C3×#808—21no—
19C3×#808—21no—
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63574—21no—
23#63562—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#498—31no—
26C3×#461—31no—
27#5084—31no—
28#5083D311yescharacteristic
29#5080D311yescharacteristic, complemented
30#5012—31no—
31#5008—31no—
32#4688—31no—
33#4547—31no—
34#4547—31no—
35#4547—61no—
36#4546—31no—
37#4481—31no—
38#4479—31no—
39#4458—31no—
40C2×C32×Q8—21no—
41C2×C3×D12D411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47C2×C3⋊(C3⋊C8)—21no—
48C2×C3⋊(C3⋊C8)—21no—
49C3×#215—42no—
50C3×(C2×D3)⋊C4—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3D411yescharacteristic
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8D411yescharacteristic
56C2×C3×C3⋊C8D411yescharacteristic
57#3711—42no—
58#1691—91noSylow
59C2×C3×SD16—61no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—31no—
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×C4⋊Q8C2×D311yescharacteristic
64C3×C4×D4—31no—
65C2×#215—21no—
66C2×#208—61no—
67C2×ΓL(1,𝔽25)—61no—
68C4×D12C2×D311yescharacteristic
69#869C2×D311yescharacteristic
70C3×#106—61no—
71C3×#106—61no—
72C3×C4⋊C8—31no—
73C3×#103—61no—
74C3×C4×C8—31no—
75#835—21no—
76#832—61no—
77#830—61no—
78(C2×D3)⋊C8—61no—
79#809—61no—
80#808—21no—
81#808—21no—
82#808—61no—
83#808—61no—
84C4⋊(C3⋊C8)—31no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4×C3⋊C8C2×D311yescharacteristic
92C22×C3×D3—21no—
93C32×Q8—41no—
94C32×Q8—41no—
95C2×C32×C4—41no—
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C2×C3×Dic3—21no—
100C2×C3×Dic3—41no—
101C3×D12—21no—
102C3×D12—21no—
103C3×C4×D3—41no—
104C3×C4×D3—41no—
105C3×Dic6—81no—
106C3⋊(C3⋊C8)—41no—
107C3⋊(C3⋊C8)—41no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110C3×C3⋊C8SD1612yescomplemented
111#498—31no—
112#476—91no—
113#461—31no—
114#436—91no—
115#335—91no—
116#333—91no—
117#332—91no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—41no—
121C2×C3×Q8—61no—
122C2×C3×D4—31no—
123C22×C3×C4—61no—
124C2×D12#6711yescharacteristic
125C2×C4×D3—21no—
126C2×Dic6—21no—
127C3×SD16—122no—
128C2×C3×C8—31no—
129C2×C3×C8—31no—
130C2×C3×C8—61no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—61no—
140C3×C22⋊C4—61no—
141C3×C42—21no—
142C3×C42C22×D311yescharacteristic
143C3×C42C22×D311yescharacteristic
144#215—42no—
145(C2×D3)⋊C4—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3#6711yescharacteristic
149#208—122no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—61no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—121no—
156C2×C3⋊C8—121no—
157C2×C3⋊C8#6711yescharacteristic
158C2×C3⋊C8#6711yescharacteristic
159ΓL(1,𝔽25)—122no—
160C22×C32#12111yescharacteristic
161C2×C3×D3—41no—
162C2×C3×D3—41no—
163C2×C3×D3—41no—
164C32×C4—21no—
165C32×C4—81no—
166C32×C4C2×SD1611yescharacteristic
167C32×C4AGL(1,ℤ/8)11yescharacteristic
168C32×C4#13711yescharacteristic
169C32×C4#13711yescharacteristic
170C3×Dic3—41no—
171C3×Dic3—81no—
172C2×SD16—61no—
173C2×M16—181no—
174C4⋊Q8—31no—
175C4⋊Q8D3211yescharacteristic
176C4×D4—31no—
177#106—61no—
178#106—61no—
179#106—181no—
180C4⋊C8—31no—
181C4⋊C8—91no—
182C4⋊C8—91no—
183#103—61no—
184#102—181no—
185C22⋊C8—181no—
186C4×C8—31no—
187C23×C3—61no—
188C22×D3—21no—
189C3×Q8—41no—
190C3×Q8—41no—
191C3×Q8—41no—
192C3×Q8—41no—
193C3×Q8—81no—
194C3×Q8—81no—
195C3×Q8—241no—
196C3×D4—61no—
197C3×D4—61no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—42no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—61no—
205C2×C3×C4—121no—
206C2×C3×C4—121no—
207C2×C3×C4—121no—
208C2×C3×C4D3×D411yescharacteristic
209C2×C3×C4D3×D411yescharacteristic
210C2×C3×C4#23711yescharacteristic
211C2×C3×C4#23711yescharacteristic
212C2×C3×C4C2×#6711yescharacteristic
213C2×C3×C4C2×#6711yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—41no—
216D12—21no—
217D12—21no—
218C4×D3—41no—
219C4×D3—41no—
220Dic6—81no—
221C3×C8—32no—
222C3×C8—61no—
223C3×C8—121no—
224C3⋊C8—21no—
225C3⋊C8—41no—
226C3⋊C8—121no—
227C3⋊C8—121no—
228C3⋊C8—121no—
229C3⋊C8—121no—
230C3⋊C8—241no—
231C3⋊C8—241no—
232C3⋊C8#21512yescomplemented
233C2×C32#46111yescharacteristic
234C2×C32#46811yescharacteristic
235C2×C32#47111yescharacteristic
236C3×D3—81no—
237C2×Q8—21no—
238C2×Q8—61no—
239C2×D4—31no—
240C22×C4—61no—
241SD16—122no—
242M16—362no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—61no—
246C2×C8—181no—
247C2×C8—181no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—31no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C22⋊C4—61no—
254C42C2×D3211yescharacteristic
255C22×C3—21no—
256C22×C3—121no—
257C22×C3—121no—
258C22×C3—121no—
259C22×C3#93811yescharacteristic
260C22×C3#93911yescharacteristic
261C2×D3—41no—
262C2×D3—41no—
263C2×D3—41no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—82no—
272C3×C4—81no—
273C3×C4—81no—
274C3×C4—121no—
275C3×C4—241no—
276C3×C4D3×SD1611yescharacteristic
277C3×C4#91411yescharacteristic
278C3×C4#91511yescharacteristic
279C3×C4#91511yescharacteristic
280C3×C4#93211yescharacteristic
281C3×C4C2×#21511yescharacteristic
282C3×C4#94211yescharacteristic
283C3×C4#95111yescharacteristic
284Dic3—41no—
285Dic3—81no—
286C32#169111yescharacteristic, complemented, Sylow
287C23—61no—
288Q8—41no—
289Q8—41no—
290Q8—241no—
291D4—61no—
292D4—61no—
293C2×C4—41no—
294C2×C4—61no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4#380511yescharacteristic
299C2×C4D3×#6711yescharacteristic
300C2×C4D3×#6711yescharacteristic
301C8—32no—
302C8—61no—
303C8—121no—
304C8—361no—
305C8—361no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—241no—
310C2×C3#504211yescharacteristic
311C2×C3#508311yescharacteristic
312C2×C3#516211yescharacteristic
313C2×C3#517411yescharacteristic
314C2×C3#517711yescharacteristic
315C2×C3#520311yescharacteristic
316D3—81no—
317C22—121no—
318C22—121no—
319C22—121no—
320C22#6397011yescharacteristic
321C4—21no—
322C4—81no—
323C4—121no—
324C4—241no—
325C4#6392911yescharacteristic
326C4#6393411yescharacteristic
327C4D3×#21511yescharacteristic
328C4#6394611yescharacteristic
329C3—21no—
330C3#7104011yescharacteristic, complemented
331C3#7134511yescharacteristic, complemented
332C2—241no—
333C2#1059150211yescharacteristic
334C2#1059156411yescharacteristic
335C2#1059164211yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1691
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×#5080C2semidirect
5. #10589413C2semidirect
6. #10589099C2semidirect
7. #10589095C2semidirect
29. #5080D3semidirect
110. C3×C3⋊C8SD16semidirect
232. C3⋊C8#215semidirect
286. C32#1691semidirect
330. C3#71040semidirect
331. C3#71345semidirect

Maximal subgroups

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

subgroupindex
C3×#50832
C3×#50802
#105903372
#105902662
#105894132
#105890992
#105890952
#713453
#710403