Finite Group Databasebeta

← #11781034

Subgroups — #11781034

Order 1152.

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

normal subgroups only

This group has 386 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
1#10592783C211yesmaximal, characteristic
2C3×#5080C211yesmaximal, characteristic, complemented
3#10590379C211yesmaximal, characteristic
4#10590327C211yesmaximal, characteristic
5C3×#4546C211yesmaximal, characteristic, complemented
6#10589097C211yesmaximal, characteristic, complemented
7#10589092C211yesmaximal, characteristic, complemented
8#71344—31nomaximal
9#68496—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C4×C4⋊(C3⋊D3)C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×#3717—21no—
14C2×#3710—21no—
15#63636—21no—
16#63635—21no—
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×#808—21no—
19C3×#808—21no—
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×C4×C3⋊C8C2211yescharacteristic
23#63566—21no—
24#63557—21no—
25C3×#498—31no—
26#5084—31no—
27#5084—31no—
28#5084—61no—
29#5083—31no—
30#5080D311yescharacteristic, complemented
31#4841—31no—
32#4834—31no—
33C8×D12—31no—
34C3×#333—31no—
35#4547—31no—
36#4546D311yescharacteristic, complemented
37#4486—31no—
38#4480—31no—
39#4457—31no—
40C2×C32×Q8—21no—
41C2×C4⋊(C3⋊D3)D411yescharacteristic
42C2×C4×C3⋊D3—21no—
43C2×C3×Dic6—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47#3752—21no—
48C4⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3⋊(C3⋊C8)—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C2×C3×C3⋊C8—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3717—42no—
57#3710—42no—
58#1671—91noSylow
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—31no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C2×#215—61no—
64C2×#204—61no—
65C2×D3×C8—61no—
66C2×D3×C8—61no—
67C4×D12—31no—
68C4×D12—31no—
69C4×D12—61no—
70#869C2×D311yescharacteristic
71C3×#106—61no—
72C3×#106—61no—
73C3×C4⋊C8—31no—
74C3×C4⋊C8—31no—
75C3×C4×C8—31no—
76#835—61no—
77(C2×D3)⋊C8—61no—
78(C2×D3)⋊C8—61no—
79#816—61no—
80#809—61no—
81#809—61no—
82#809—62no—
83#808—21no—
84#808—21no—
85#808—61no—
86#808—61no—
87#808—62no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—61no—
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4⋊(C3⋊C8)C2×D311yescharacteristic
93C4×C3⋊C8C2×D311yescharacteristic
94C22×C3⋊D3—21no—
95C32×Q8—81no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4#4111yescharacteristic
100C2×C3⋊Dic3—21no—
101C4⋊(C3⋊D3)—21no—
102C4⋊(C3⋊D3)SD1612yes—
103C4×C3⋊D3—41no—
104C4×C3⋊D3—41no—
105C2×C3×Dic3—41no—
106C3×Dic6—41no—
107C3×Dic6—41no—
108C3⋊(C3⋊C8)—41no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—41no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8SD1612yescomplemented
113#498—31no—
114#476—91no—
115#461—91no—
116C8×D4—91no—
117#334—91no—
118#333—31no—
119#332—91no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—41no—
123C2×C3×Q8—61no—
124C2×D12—31no—
125C2×D12—31no—
126C2×D12—61no—
127C2×C4×D3—61no—
128C2×C4×D3—61no—
129C2×C4×D3—121no—
130C2×Dic6—21no—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—61no—
134C2×C3×C8—61no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—61no—
142C3×C4⋊C4—61no—
143C3×C42—21no—
144C3×C42C22×D311yescharacteristic
145C3×C42C22×D311yescharacteristic
146#215—122no—
147(C2×D3)⋊C4—61no—
148(C2×D3)⋊C4—61no—
149(C2×D3)⋊C4—121no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3—61no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—62no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—61no—
160C2×C3⋊C8#6711yescharacteristic
161C2×C3⋊C8#6711yescharacteristic
162#204—122no—
163D3×C8—122no—
164D3×C8—122no—
165C22×C32#12111yescharacteristic
166C2×C3⋊D3—22no—
167C2×C3⋊D3—41no—
168C2×C3⋊D3—41no—
169C32×C4—21no—
170C32×C4—81no—
171C32×C4C2×SD1611yescharacteristic
172C32×C4C2×SD1611yescharacteristic
173C32×C4#13511yescharacteristic
174C32×C4#13711yescharacteristic
175C3⋊Dic3—41no—
176C3×Dic3—81no—
177C2×SD16—181no—
178C22×C8—181no—
179C4⋊Q8—31no—
180C4⋊Q8D3211yescharacteristic
181C4×D4—91no—
182#106—61no—
183#106—61no—
184#106—181no—
185C4⋊C8—31no—
186C4⋊C8—31no—
187C4⋊C8—91no—
188#103—181no—
189#102—181no—
190C22⋊C8—181no—
191C4×C8—31no—
192C22×D3—61no—
193C22×D3—61no—
194C22×D3—121no—
195C3×Q8—81no—
196C3×Q8—81no—
197C3×Q8—82no—
198C3×Q8—121no—
199C3×Q8—121no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—42no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—121no—
207C2×C3×C4#23511yescharacteristic
208C2×C3×C4D3×D411yescharacteristic
209C2×C3×C4D3×D411yescharacteristic
210C2×C3×C4D3×D411yescharacteristic
211C2×C3×C4#23711yescharacteristic
212C2×C3×C4C2×#6711yescharacteristic
213C2×Dic3—41no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—121no—
217D12—32no—
218D12—32no—
219D12—62no—
220D12—61no—
221D12—61no—
222D12—121no—
223C4×D3—121no—
224C4×D3—121no—
225C4×D3—121no—
226C4×D3—121no—
227C4×D3—241no—
228C4×D3—241no—
229Dic6—41no—
230Dic6—41no—
231C3×C8—32no—
232C3×C8—61no—
233C3×C8—121no—
234C3×C8—121no—
235C3⋊C8—21no—
236C3⋊C8—41no—
237C3⋊C8—41no—
238C3⋊C8—122no—
239C3⋊C8—121no—
240C3⋊C8—121no—
241C3⋊C8#21512yescomplemented
242C2×C32#46111yescharacteristic
243C2×C32#46511yescharacteristic
244C2×C32#47111yescharacteristic
245C3⋊D3—42no—
246C2×Q8—21no—
247C2×Q8—61no—
248C2×D4—91no—
249C22×C4—181no—
250SD16—362no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—61no—
254C2×C8—61no—
255C2×C8—181no—
256C2×C8—362no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—91no—
262C22⋊C4—181no—
263C42C2×D3211yescharacteristic
264C22×C3—21no—
265C22×C3#88411yescharacteristic
266C22×C3#93811yescharacteristic
267C2×D3—62no—
268C2×D3—62no—
269C2×D3—122no—
270C2×D3—121no—
271C2×D3—121no—
272C2×D3—121no—
273C2×D3—121no—
274C2×D3—241no—
275C2×D3—241no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—82no—
284C3×C4—81no—
285C3×C4—81no—
286C3×C4—241no—
287C3×C4D3×SD1611yescharacteristic
288C3×C4D3×SD1611yescharacteristic
289C3×C4D3×SD1611yescharacteristic
290C3×C4#91611yescharacteristic
291C3×C4#91811yescharacteristic
292C3×C4#91911yescharacteristic
293C3×C4C2×#21511yescharacteristic
294C3×C4#94211yescharacteristic
295Dic3—81no—
296Dic3—121no—
297Dic3—121no—
298Dic3—241no—
299C32#167111yescharacteristic, complemented, Sylow
300C23—181no—
301Q8—81no—
302Q8—121no—
303Q8—121no—
304D4—92no—
305D4—181no—
306C2×C4—41no—
307C2×C4—121no—
308C2×C4—181no—
309C2×C4—361no—
310C2×C4—361no—
311C2×C4#380411yescharacteristic
312C2×C4D3×#6711yescharacteristic
313C2×C4#381111yescharacteristic
314C8—32no—
315C8—61no—
316C8—121no—
317C8—121no—
318C8—361no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3#474811yescharacteristic
323C2×C3#480711yescharacteristic
324C2×C3#481811yescharacteristic
325C2×C3#508311yescharacteristic
326C2×C3#517111yescharacteristic
327C2×C3#517711yescharacteristic
328D3—122no—
329D3—122no—
330D3—242no—
331C22—182no—
332C22—361no—
333C22—361no—
334C22#6395811yescharacteristic
335C4—21no—
336C4—81no—
337C4—241no—
338C4—361no—
339C4D3×#21511yescharacteristic
340C4#6394011yescharacteristic
341C4#6394811yescharacteristic
342C4#6395011yescharacteristic
343C3—21no—
344C3#6849611yescharacteristic, complemented
345C3#7134411yescharacteristic, complemented
346C2—362no—
347C2#1059140711yescharacteristic
348C2#1059156511yescharacteristic
349C2#1059159111yescharacteristic
subgroupG/H
centerC22#63958
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C4⋊(C3⋊D3)D4
hypercenterC4⋊Q8D32
hypocenterC32#1671
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. C3×#4546C2semidirect
6. #10589097C2semidirect
7. #10589092C2semidirect
30. #5080D3semidirect
36. #4546D3semidirect
112. C3×C3⋊C8SD16semidirect
241. C3⋊C8#215semidirect
299. C32#1671semidirect
344. C3#68496semidirect
345. C3#71344semidirect

Maximal subgroups

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

subgroupindex
#105927832
C3×#50802
#105903792
#105903272
C3×#45462
#105890972
#105890922
#713443
#684963