Finite Group Databasebeta

← #11781531

Subgroups — #11781531

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×#5087C211yesmaximal, characteristic
2C3×#5079C211yesmaximal, characteristic, complemented
3C3⋊C8×D12C211yesmaximal, characteristic
4#10590266C211yesmaximal, characteristic
5#10589413C211yesmaximal, characteristic, complemented
6#10589099C211yesmaximal, characteristic, complemented
7#10589095C211yesmaximal, characteristic, complemented
8#71233—31nomaximal
9#71103—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11C3×C4×D12C2211yescharacteristic
12C3×#869C2211yescharacteristic
13C2×D3×C3⋊C8—21no—
14C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#835—21no—
17C3×#835—21no—
18C3×#809—21no—
19C3×#807—21no—
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4×C3⋊C8C2211yescharacteristic
22#63574—21no—
23#63562—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#487—31no—
26C3×#474—31no—
27#5087D311yescharacteristic
28#5084—31no—
29#5079D311yescharacteristic, complemented
30D4×C3⋊C8—31no—
31#5008—31no—
32C8×D12—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×(C2×D3)⋊C4—21no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D411yescharacteristic
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56D3×C3⋊C8—42no—
57#1683—91noSylow
58C3×C4⋊Q8—21no—
59C3×C4⋊Q8—31no—
60C3×C4⋊Q8C2×D311yescharacteristic
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4×D4—31no—
63C22×C3⋊C8—61no—
64C2×D3×C8—61no—
65C4×D12C2×D311yescharacteristic
66#869C2×D311yescharacteristic
67C3×#107—61no—
68C3×C4⋊C8—31no—
69C3×#103—61no—
70C3×#103—61no—
71C3×#102—61no—
72C3×C4×C8—31no—
73#835—21no—
74#835—21no—
75#832—61no—
76#830—61no—
77(C2×D3)⋊C8—61no—
78#809—21no—
79#809—61no—
80#808—61no—
81#808—61no—
82#807—21no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—31no—
85C4⋊(C3⋊C8)—31no—
86C4⋊(C3⋊C8)—31no—
87C4⋊(C3⋊C8)—61no—
88C4⋊(C3⋊C8)—61no—
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4×C3⋊C8C2×D311yescharacteristic
91C22×C3×D3—21no—
92C32×Q8—41no—
93C32×Q8—41no—
94C2×C32×C4—41no—
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4#4111yescharacteristic
98C2×C3×Dic3—21no—
99C2×C3×Dic3—41no—
100C3×D12—21no—
101C3×D12Q1612yes—
102C3×C4×D3—41no—
103C3×C4×D3—41no—
104C3×Dic6—81no—
105C3⋊(C3⋊C8)—41no—
106C3⋊(C3⋊C8)—41no—
107C3×C3⋊C8—21no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—41no—
110#487—31no—
111#476—91no—
112#474—31no—
113C8×D4—91no—
114#335—91no—
115#333—91no—
116#332—91no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—41no—
120C2×C3×Q8—61no—
121C2×C3×D4—31no—
122C22×C3×C4—61no—
123C2×D12#6711yescharacteristic
124C2×C4×D3—21no—
125C2×Dic6—21no—
126C2×C3×C8—31no—
127C2×C3×C8—31no—
128C2×C3×C8—61no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—61no—
137C3×C4⋊C4—61no—
138C3×C22⋊C4—61no—
139C3×C42—21no—
140C3×C42C22×D311yescharacteristic
141C3×C42C22×D311yescharacteristic
142(C2×D3)⋊C4—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3#6711yescharacteristic
146C2×C3⋊C8—21no—
147C2×C3⋊C8—61no—
148C2×C3⋊C8—61no—
149C2×C3⋊C8—61no—
150C2×C3⋊C8—61no—
151C2×C3⋊C8—121no—
152C2×C3⋊C8—121no—
153C2×C3⋊C8—122no—
154C2×C3⋊C8#6711yescharacteristic
155C2×C3⋊C8#6711yescharacteristic
156D3×C8—122no—
157C22×C32#12111yescharacteristic
158C2×C3×D3—22no—
159C2×C3×D3—41no—
160C2×C3×D3—41no—
161C32×C4—21no—
162C32×C4—81no—
163C32×C4C2×Q1611yescharacteristic
164C32×C4#13511yescharacteristic
165C32×C4#13511yescharacteristic
166C32×C4AGL(1,ℤ/8)11yescharacteristic
167C3×Dic3—41no—
168C3×Dic3—81no—
169C22×C8—181no—
170C4⋊Q8—31no—
171C4⋊Q8D3211yescharacteristic
172C4×D4—31no—
173#107—61no—
174#106—181no—
175C4⋊C8—31no—
176C4⋊C8—91no—
177C4⋊C8—91no—
178#103—61no—
179#103—61no—
180#102—61no—
181#102—181no—
182C22⋊C8—181no—
183C4×C8—31no—
184C23×C3—61no—
185C22×D3—21no—
186C3×Q8—41no—
187C3×Q8—41no—
188C3×Q8—41no—
189C3×Q8—41no—
190C3×Q8—81no—
191C3×Q8—81no—
192C3×Q8—241no—
193C3×D4—32no—
194C3×D4—61no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—42no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—61no—
202C2×C3×C4—121no—
203C2×C3×C4—121no—
204C2×C3×C4—121no—
205C2×C3×C4D3×D411yescharacteristic
206C2×C3×C4D3×D411yescharacteristic
207C2×C3×C4#23711yescharacteristic
208C2×C3×C4#23711yescharacteristic
209C2×C3×C4C2×#6711yescharacteristic
210C2×C3×C4C2×#6711yescharacteristic
211C2×Dic3—21no—
212C2×Dic3—41no—
213D12—21no—
214D12#21612yes—
215C4×D3—41no—
216C4×D3—41no—
217Dic6—81no—
218C3×C8—61no—
219C3×C8—61no—
220C3×C8—121no—
221C3⋊C8—21no—
222C3⋊C8—21no—
223C3⋊C8—41no—
224C3⋊C8—121no—
225C3⋊C8—121no—
226C3⋊C8—121no—
227C3⋊C8—121no—
228C3⋊C8—241no—
229C3⋊C8—241no—
230C2×C32#46311yescharacteristic
231C2×C32#46711yescharacteristic
232C2×C32#47111yescharacteristic
233C3×D3—42no—
234C2×Q8—21no—
235C2×Q8—61no—
236C2×D4—31no—
237C22×C4—61no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—61no—
241C2×C8—181no—
242C2×C8—181no—
243C2×C8—362no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—31no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C22⋊C4—61no—
250C42C2×D3211yescharacteristic
251C22×C3—21no—
252C22×C3—62no—
253C22×C3—121no—
254C22×C3—121no—
255C22×C3#93811yescharacteristic
256C22×C3#93911yescharacteristic
257C2×D3—22no—
258C2×D3—41no—
259C2×D3—41no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—41no—
267C3×C4—82no—
268C3×C4—81no—
269C3×C4—81no—
270C3×C4—121no—
271C3×C4—241no—
272C3×C4#91111yescharacteristic
273C3×C4#91211yescharacteristic
274C3×C4D3×Q1611yescharacteristic
275C3×C4#91911yescharacteristic
276C3×C4C2×#21611yescharacteristic
277C3×C4#94911yescharacteristic
278C3×C4#95011yescharacteristic
279C3×C4#95011yescharacteristic
280Dic3—41no—
281Dic3—81no—
282C32#168311yescharacteristic, complemented, Sylow
283C23—61no—
284Q8—41no—
285Q8—41no—
286Q8—241no—
287D4—32no—
288D4—61no—
289C2×C4—41no—
290C2×C4—61no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4#380511yescharacteristic
295C2×C4D3×#6711yescharacteristic
296C2×C4D3×#6711yescharacteristic
297C8—61no—
298C8—61no—
299C8—121no—
300C8—361no—
301C8—361no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—122no—
306C2×C3#505111yescharacteristic
307C2×C3#505711yescharacteristic
308C2×C3#515311yescharacteristic
309C2×C3#517711yescharacteristic
310C2×C3#518811yescharacteristic
311C2×C3#520311yescharacteristic
312D3—42no—
313C22—62no—
314C22—121no—
315C22—121no—
316C22#6397011yescharacteristic
317C4—21no—
318C4—81no—
319C4—121no—
320C4—241no—
321C4#6392511yescharacteristic
322C4#6393311yescharacteristic
323C4D3×#21611yescharacteristic
324C4#6394911yescharacteristic
325C3—21no—
326C3#7110311yescharacteristic, complemented
327C3#7123311yescharacteristic, complemented
328C2—122no—
329C2#1059143111yescharacteristic
330C2#1059163511yescharacteristic
331C2#1059164211yescharacteristic
subgroupG/H
centerC22#63970
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊Q8D32
hypocenterC32#1683
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×#5079C2semidirect
5. #10589413C2semidirect
6. #10589099C2semidirect
7. #10589095C2semidirect
29. #5079D3semidirect
282. C32#1683semidirect
326. C3#71103semidirect
327. C3#71233semidirect

Maximal subgroups

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

subgroupindex
C3×#50872
C3×#50792
C3⋊C8×D122
#105902662
#105894132
#105890992
#105890952
#712333
#711033