Finite Group Databasebeta

← #11781025

Subgroups — #11781025

Order 1152.

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

normal subgroups only

This group has 392 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#10592782C211yesmaximal, characteristic
2C3×#5080C211yesmaximal, characteristic, complemented
3#10590327C211yesmaximal, characteristic
4#10590300C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589097C211yesmaximal, characteristic, complemented
7#10589095C211yesmaximal, characteristic, complemented
8#71344—31nomaximal
9#68442—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×#3755—21no—
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3710—21no—
15#63662—21no—
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#808—21no—
18C3×#808—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63573—21no—
23#63564—21no—
24#63557—21no—
25C3×#498—31no—
26#5084—31no—
27#5083—31no—
28#5083—31no—
29#5083—61no—
30#5080D311yescharacteristic, complemented
31#4842—31no—
32#4834—31no—
33C8×D12—31no—
34C3×#332—31no—
35#4547D311yescharacteristic, complemented
36#4546—31no—
37#4481—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#3755—42no—
48#3752—21no—
49C4⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3⋊(C3⋊C8)—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8D411yescharacteristic
56C2×C3×C3⋊C8D411yescharacteristic
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×#215—61no—
65C2×#215—62no—
66C2×D3×C8—61no—
67C2×D3×C8—61no—
68C4×D12—31no—
69C4×D12—31no—
70C4×D12—61no—
71#869C2×D311yescharacteristic
72C3×#106—61no—
73C3×#106—61no—
74C3×C4⋊C8—31no—
75C3×C4⋊C8—31no—
76C3×C4×C8—31no—
77#835—61no—
78#835—61no—
79#835—62no—
80#821—61no—
81(C2×D3)⋊C8—61no—
82(C2×D3)⋊C8—61no—
83#817—61no—
84#809—61no—
85#808—21no—
86#808—21no—
87#808—61no—
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—41no—
96C32×Q8—41no—
97C2×C32×C4—41no—
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4#4111yescharacteristic
101C2×C3⋊Dic3—21no—
102C4⋊(C3⋊D3)—21no—
103C4⋊(C3⋊D3)SD1612yes—
104C4×C3⋊D3—41no—
105C4×C3⋊D3—41no—
106C2×C3×Dic3—41no—
107C3×Dic6—81no—
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—91no—
119#332—31no—
120C2×C3×Q8—22no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
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—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—41no—
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#215—122no—
148#215—124no—
149(C2×D3)⋊C4—61no—
150(C2×D3)⋊C4—61no—
151(C2×D3)⋊C4—121no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—61no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—62no—
160C2×C3⋊C8—61no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8#6711yescharacteristic
163C2×C3⋊C8#6711yescharacteristic
164D3×C8—122no—
165D3×C8—122no—
166C22×C32#12111yescharacteristic
167C2×C3⋊D3—22no—
168C2×C3⋊D3—41no—
169C2×C3⋊D3—41no—
170C32×C4—21no—
171C32×C4—81no—
172C32×C4C2×SD1611yescharacteristic
173C32×C4C2×SD1611yescharacteristic
174C32×C4#13511yescharacteristic
175C32×C4#13711yescharacteristic
176C3⋊Dic3—41no—
177C3×Dic3—81no—
178C2×SD16—181no—
179C22×C8—181no—
180C4⋊Q8—31no—
181C4⋊Q8D3211yescharacteristic
182C4×D4—91no—
183#106—61no—
184#106—61no—
185#106—181no—
186C4⋊C8—31no—
187C4⋊C8—31no—
188C4⋊C8—91no—
189#103—181no—
190#102—181no—
191C22⋊C8—181no—
192C4×C8—31no—
193C22×D3—61no—
194C22×D3—61no—
195C22×D3—121no—
196C3×Q8—42no—
197C3×Q8—42no—
198C3×Q8—41no—
199C3×Q8—41no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—241no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—121no—
210C2×C3×C4#23511yescharacteristic
211C2×C3×C4D3×D411yescharacteristic
212C2×C3×C4D3×D411yescharacteristic
213C2×C3×C4D3×D411yescharacteristic
214C2×C3×C4#23711yescharacteristic
215C2×C3×C4C2×#6711yescharacteristic
216C2×Dic3—41no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—121no—
220D12—32no—
221D12—32no—
222D12—62no—
223D12—61no—
224D12—61no—
225D12—121no—
226C4×D3—121no—
227C4×D3—121no—
228C4×D3—121no—
229C4×D3—121no—
230C4×D3—241no—
231C4×D3—241no—
232Dic6—81no—
233C3×C8—32no—
234C3×C8—61no—
235C3×C8—121no—
236C3×C8—121no—
237C3⋊C8—21no—
238C3⋊C8—41no—
239C3⋊C8—41no—
240C3⋊C8—122no—
241C3⋊C8—121no—
242C3⋊C8—121no—
243C3⋊C8#21512yescomplemented
244C2×C32#46111yescharacteristic
245C2×C32#46511yescharacteristic
246C2×C32#47111yescharacteristic
247C3⋊D3—42no—
248C2×Q8—21no—
249C2×Q8—61no—
250C2×D4—91no—
251C22×C4—181no—
252SD16—362no—
253C2×C8—31no—
254C2×C8—31no—
255C2×C8—61no—
256C2×C8—61no—
257C2×C8—181no—
258C2×C8—362no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4—91no—
264C22⋊C4—181no—
265C42C2×D3211yescharacteristic
266C22×C3—21no—
267C22×C3#88411yescharacteristic
268C22×C3#93811yescharacteristic
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—122no—
272C2×D3—121no—
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—121no—
276C2×D3—241no—
277C2×D3—241no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—82no—
286C3×C4—81no—
287C3×C4—81no—
288C3×C4—241no—
289C3×C4#91211yescharacteristic
290C3×C4D3×SD1611yescharacteristic
291C3×C4D3×SD1611yescharacteristic
292C3×C4D3×SD1611yescharacteristic
293C3×C4#91611yescharacteristic
294C3×C4#91811yescharacteristic
295C3×C4C2×#21511yescharacteristic
296C3×C4#94211yescharacteristic
297Dic3—81no—
298Dic3—121no—
299Dic3—121no—
300Dic3—241no—
301C32#167111yescharacteristic, complemented, Sylow
302C23—181no—
303Q8—41no—
304Q8—41no—
305Q8—241no—
306D4—92no—
307D4—181no—
308C2×C4—41no—
309C2×C4—121no—
310C2×C4—181no—
311C2×C4—361no—
312C2×C4—361no—
313C2×C4#380411yescharacteristic
314C2×C4D3×#6711yescharacteristic
315C2×C4#381111yescharacteristic
316C8—32no—
317C8—61no—
318C8—121no—
319C8—121no—
320C8—361no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3#475411yescharacteristic
325C2×C3#477811yescharacteristic
326C2×C3#481811yescharacteristic
327C2×C3#508311yescharacteristic
328C2×C3#517111yescharacteristic
329C2×C3#517711yescharacteristic
330D3—122no—
331D3—122no—
332D3—242no—
333C22—182no—
334C22—361no—
335C22—361no—
336C22#6395811yescharacteristic
337C4—21no—
338C4—81no—
339C4—241no—
340C4—361no—
341C4#6393011yescharacteristic
342C4#6393611yescharacteristic
343C4D3×#21511yescharacteristic
344C4#6394811yescharacteristic
345C3—21no—
346C3#6844211yescharacteristic, complemented
347C3#7134411yescharacteristic, complemented
348C2—362no—
349C2#1059146511yescharacteristic
350C2#1059154211yescharacteristic
351C2#1059156511yescharacteristic
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×#4547C2semidirect
6. #10589097C2semidirect
7. #10589095C2semidirect
30. #5080D3semidirect
35. #4547D3semidirect
112. C3×C3⋊C8SD16semidirect
243. C3⋊C8#215semidirect
301. C32#1671semidirect
346. C3#68442semidirect
347. C3#71344semidirect

Maximal subgroups

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

subgroupindex
#105927822
C3×#50802
#105903272
#105903002
C3×#45472
#105890972
#105890952
#713443
#684423