Finite Group Databasebeta

← #11780955

Subgroups — #11780955

Order 1152.

376 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 92 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×#5092C211yesmaximal, characteristic, complemented
2C3×#5087C211yesmaximal, characteristic
3C3⋊C8×D12C211yesmaximal, characteristic
4#10590269C211yesmaximal, characteristic
5#10589413C211yesmaximal, characteristic, complemented
6#10589099C211yesmaximal, characteristic, complemented
7#10589092C211yesmaximal, characteristic, complemented
8#71305—31nomaximal
9#71101—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C2×C3×#216—21no—
12C2×C3×#216—21no—
13C3×C4×D12C2211yescharacteristic
14C3×#869C2211yescharacteristic
15C2×D3×C3⋊C8—21no—
16C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×#809—21no—
19C3×#807—21no—
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63576—21no—
23#63562—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#494—31no—
26C3×#474—31no—
27#5092D311yescharacteristic, complemented
28#5087—31no—
29#5087D311yescharacteristic
30D4×C3⋊C8—31no—
31#5007—31no—
32C8×D12—31no—
33#4547—31no—
34#4547—31no—
35#4547—31no—
36#4547—61no—
37#4486—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×#216—42no—
50C3×#216—42no—
51C3×(C2×D3)⋊C4—21no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3—21no—
54C3×C4⋊Dic3D411yescharacteristic
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8D411yescharacteristic
57C2×C3×C3⋊C8D411yescharacteristic
58D3×C3⋊C8—42no—
59#1669—91noSylow
60C2×C3×Q16—61no—
61C2×C3×Q16—61no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—31no—
64C3×C4⋊Q8C2×D311yescharacteristic
65C3×C4⋊Q8C2×D311yescharacteristic
66C3×C4×D4—31no—
67C2×#216—21no—
68C2×#216—21no—
69C22×C3⋊C8—61no—
70C2×D3×C8—61no—
71C4×D12C2×D311yescharacteristic
72#869C2×D311yescharacteristic
73C3×#107—61no—
74C3×C4⋊C8—31no—
75C3×#102—61no—
76C3×C4×C8—31no—
77#832—61no—
78#830—61no—
79(C2×D3)⋊C8—61no—
80#809—21no—
81#809—61no—
82#807—21no—
83#807—61no—
84#807—61no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—31no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—61no—
90C4⋊(C3⋊C8)—61no—
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3×D3—21no—
94C32×Q8—41no—
95C32×Q8—41no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4#4111yescharacteristic
100C2×C3×Dic3—21no—
101C2×C3×Dic3—41no—
102C3×D12—21no—
103C3×D12Q1612yescomplemented
104C3×C4×D3—41no—
105C3×C4×D3—41no—
106C3×Dic6—41no—
107C3×Dic6—41no—
108C3⋊(C3⋊C8)—41no—
109C3⋊(C3⋊C8)—41no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8D812yes—
113#494—31no—
114#474—31no—
115#474—91no—
116C8×D4—91no—
117#335—91no—
118#332—91no—
119#332—91no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—41no—
123C2×C3×Q8—61no—
124C2×C3×D4—31no—
125C22×C3×C4—61no—
126C2×D12#6711yescharacteristic
127C2×C4×D3—21no—
128C2×Dic6—21no—
129C3×Q16—122no—
130C3×Q16—122no—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—61no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—61no—
142C3×C4⋊C4—61no—
143C3×C22⋊C4—61no—
144C3×C42—21no—
145C3×C42C22×D311yescharacteristic
146C3×C42C22×D311yescharacteristic
147#216—42no—
148#216—42no—
149(C2×D3)⋊C4—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3#6711yescharacteristic
153C2×C3⋊C8—21no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—61no—
156C2×C3⋊C8—61no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8—121no—
159C2×C3⋊C8—121no—
160C2×C3⋊C8—122no—
161C2×C3⋊C8#6711yescharacteristic
162C2×C3⋊C8#6711yescharacteristic
163D3×C8—122no—
164C22×C32#12111yescharacteristic
165C2×C3×D3—22no—
166C2×C3×D3—41no—
167C2×C3×D3—41no—
168C32×C4—21no—
169C32×C4—81no—
170C32×C4C2×D811yescharacteristic
171C32×C4C2×Q1611yescharacteristic
172C32×C4#13511yescharacteristic
173C32×C4AGL(1,ℤ/8)11yescharacteristic
174C3×Dic3—41no—
175C3×Dic3—81no—
176C2×Q16—61no—
177C2×Q16—61no—
178C22×C8—181no—
179C4⋊Q8—31no—
180C4⋊Q8D3211yescharacteristic
181C4×D4—31no—
182#107—61no—
183#107—181no—
184C4⋊C8—31no—
185C4⋊C8—91no—
186C4⋊C8—91no—
187#102—61no—
188#102—181no—
189C22⋊C8—181no—
190C4×C8—31no—
191C23×C3—61no—
192C22×D3—21no—
193C3×Q8—41no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—81no—
198C3×Q8—81no—
199C3×Q8—121no—
200C3×Q8—121no—
201C3×D4—32no—
202C3×D4—61no—
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—61no—
210C2×C3×C4—121no—
211C2×C3×C4—121no—
212C2×C3×C4—121no—
213C2×C3×C4D3×D411yescharacteristic
214C2×C3×C4D3×D411yescharacteristic
215C2×C3×C4#23711yescharacteristic
216C2×C3×C4#23711yescharacteristic
217C2×C3×C4C2×#6711yescharacteristic
218C2×C3×C4C2×#6711yescharacteristic
219C2×Dic3—21no—
220C2×Dic3—41no—
221D12—21no—
222D12#21612yescomplemented
223C4×D3—41no—
224C4×D3—41no—
225Dic6—41no—
226Dic6—41no—
227C3×C8—32no—
228C3×C8—61no—
229C3×C8—121no—
230C3⋊C8—21no—
231C3⋊C8—41no—
232C3⋊C8—121no—
233C3⋊C8—121no—
234C3⋊C8—121no—
235C3⋊C8—121no—
236C3⋊C8—241no—
237C3⋊C8—241no—
238C3⋊C8#21312yes—
239C2×C32#45911yescharacteristic
240C2×C32#46711yescharacteristic
241C2×C32#47311yescharacteristic
242C3×D3—42no—
243C2×Q8—21no—
244C2×Q8—61no—
245C2×D4—31no—
246C22×C4—61no—
247Q16—122no—
248Q16—122no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—61no—
252C2×C8—181no—
253C2×C8—181no—
254C2×C8—362no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—31no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C22⋊C4—61no—
261C42C2×D3211yescharacteristic
262C22×C3—21no—
263C22×C3—62no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3#93811yescharacteristic
267C22×C3#93911yescharacteristic
268C2×D3—22no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—41no—
278C3×C4—82no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4—121no—
282C3×C4—241no—
283C3×C4D3×D811yescharacteristic
284C3×C4#91111yescharacteristic
285C3×C4D3×Q1611yescharacteristic
286C3×C4#91911yescharacteristic
287C3×C4C2×#21311yescharacteristic
288C3×C4#93211yescharacteristic
289C3×C4C2×#21611yescharacteristic
290C3×C4#95011yescharacteristic
291Dic3—41no—
292Dic3—81no—
293C32#166911yescharacteristic, complemented, Sylow
294C23—61no—
295Q8—41no—
296Q8—41no—
297Q8—121no—
298Q8—121no—
299D4—32no—
300D4—61no—
301C2×C4—41no—
302C2×C4—61no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4#380511yescharacteristic
307C2×C4D3×#6711yescharacteristic
308C2×C4D3×#6711yescharacteristic
309C8—32no—
310C8—61no—
311C8—121no—
312C8—361no—
313C8—361no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—122no—
318C2×C3#505111yescharacteristic
319C2×C3#507211yescharacteristic
320C2×C3#515011yescharacteristic
321C2×C3#518811yescharacteristic
322C2×C3#519211yescharacteristic
323C2×C3#520411yescharacteristic
324D3—42no—
325C22—62no—
326C22—121no—
327C22—121no—
328C22#6397011yescharacteristic
329C4—21no—
330C4—81no—
331C4—121no—
332C4—241no—
333C4D3×#21311yescharacteristic
334C4#6393211yescharacteristic
335C4D3×#21611yescharacteristic
336C4#6394911yescharacteristic
337C3—21no—
338C3#7110111yescharacteristic, complemented
339C3#7130511yescharacteristic, complemented
340C2—122no—
341C2#1059142811yescharacteristic
342C2#1059163511yescharacteristic
343C2#1059164111yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1669
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
1. C3×#5092C2semidirect
5. #10589413C2semidirect
6. #10589099C2semidirect
7. #10589092C2semidirect
27. #5092D3semidirect
103. C3×D12Q16semidirect
222. D12#216semidirect
293. C32#1669semidirect
338. C3#71101semidirect
339. C3#71305semidirect

Maximal subgroups

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

subgroupindex
C3×#50922
C3×#50872
C3⋊C8×D122
#105902692
#105894132
#105890992
#105890922
#713053
#711013