Finite Group Databasebeta

← #11781024

Subgroups — #11781024

Order 1152.

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

normal subgroups only

This group has 381 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#10592779C211yesmaximal, characteristic, complemented
2C3×#5083C211yesmaximal, characteristic
3#10590329C211yesmaximal, characteristic
4#10590268C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589098C211yesmaximal, characteristic, complemented
7#10589096C211yesmaximal, characteristic, complemented
8#71036—31nomaximal
9#68442—31nomaximal
10C32×C4⋊Q8C2211yescharacteristic
11#64078C2211yescharacteristic
12C2×C3×#215—21no—
13C3×C4×D12C2211yescharacteristic
14C2×D3×C3⋊C8—21no—
15#63635—21no—
16#63635—21no—
17C4×C3⋊(C3⋊C8)C2211yescharacteristic
18C3×#835—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×C4⋊(C3⋊C8)C2211yescharacteristic
22#63573—21no—
23#63563—21no—
24(C2×D3)⋊(C3⋊C8)—21no—
25C3×#461—31no—
26#5083D311yescharacteristic
27#5080—31no—
28#5080—31no—
29#5080—61no—
30D4×C3⋊C8—31no—
31#5008—31no—
32#4842—31no—
33#4834—31no—
34C3×#332—31no—
35#4547D311yescharacteristic, complemented
36#4481—31no—
37#4481—31no—
38#4480—31no—
39#4480—31no—
40C2×C32×Q8—21no—
41C2×#634—21no—
42C2×C3×D12D411yescharacteristic
43C2×C3×C4×D3—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C42C2311yescharacteristic
47C4⋊(C3⋊Dic3)—21no—
48C4⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊C8)D411yescharacteristic
50C2×C3⋊(C3⋊C8)D411yescharacteristic
51C3×#215—42no—
52C3×(C2×D3)⋊C4—21no—
53C3×C4⋊Dic3D411yescharacteristic
54C2×C3×C3⋊C8—21no—
55C2×C3×C3⋊C8—21no—
56C2×C3×C3⋊C8—21no—
57D3×C3⋊C8—42no—
58#1671—91noSylow
59C2×C3×SD16—61no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8C2×D311yescharacteristic
62C3×C4⋊Q8C2×D311yescharacteristic
63C3×C4×D4—31no—
64C2×#215—21no—
65C22×C3⋊C8—61no—
66C2×D3×C8—61no—
67C4×D12C2×D311yescharacteristic
68#869—31no—
69#869—31no—
70#869—61no—
71C3×C4⋊C8—31no—
72C3×C4⋊C8—31no—
73C3×C4⋊C8—31no—
74C3×#103—61no—
75#835—21no—
76#832—61no—
77#830—61no—
78#821—61no—
79(C2×D3)⋊C8—61no—
80#817—61no—
81#808—61no—
82#808—61no—
83#808—61no—
84#808—61no—
85#808—61no—
86#808—62no—
87#808—62no—
88C4⋊(C3⋊C8)C2×D311yescharacteristic
89C4⋊(C3⋊C8)C2×D311yescharacteristic
90C4⋊(C3⋊C8)C2×D311yescharacteristic
91C4×C3⋊C8—31no—
92C4×C3⋊C8—31no—
93C4×C3⋊C8—61no—
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—41no—
102#634—81no—
103C2×C3×Dic3—21no—
104C3×D12—21no—
105C3×D12SD1612yes—
106C3×C4×D3—41no—
107C3×C4×D3—41no—
108C3⋊(C3⋊C8)—21no—
109C3⋊(C3⋊C8)SD1612yescomplemented
110C3×C3⋊C8—41no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8—41no—
113#498—91no—
114#476—91no—
115#461—31no—
116C8×D4—91no—
117#334—91no—
118#333—91no—
119#332—31no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—41no—
123C2×C3×D4—31no—
124C22×C3×C4—61no—
125C2×D12#6711yescharacteristic
126C2×C4×D3—21no—
127C2×Dic6—61no—
128C2×Dic6—61no—
129C2×Dic6—121no—
130C3×SD16—122no—
131C2×C3×C8—61no—
132C2×C3×C8—61no—
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×C22⋊C4—61no—
142C3×C42—21no—
143C3×C42C22×D311yescharacteristic
144C3×C42C22×D311yescharacteristic
145#215—42no—
146(C2×D3)⋊C4—21no—
147C4⋊Dic3—62no—
148C4⋊Dic3—61no—
149C4⋊Dic3—61no—
150C4⋊Dic3—61no—
151C4⋊Dic3—61no—
152C4⋊Dic3—121no—
153C4⋊Dic3#6711yescharacteristic
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—31no—
158C2×C3⋊C8—31no—
159C2×C3⋊C8—31no—
160C2×C3⋊C8—31no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8—61no—
163C2×C3⋊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—81no—
176C3×Dic3—41no—
177C2×SD16—61no—
178C22×C8—181no—
179C4⋊Q8—91no—
180C4⋊Q8D3211yescharacteristic
181C4×D4—31no—
182#106—181no—
183#106—181no—
184#106—181no—
185C4⋊C8—31no—
186C4⋊C8—31no—
187C4⋊C8—31no—
188#103—61no—
189#102—181no—
190C22⋊C8—181no—
191C4×C8—91no—
192C23×C3—61no—
193C22×D3—21no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C3×Q8—41no—
198C3×Q8—81no—
199C3×Q8—81no—
200C3×D4—32no—
201C3×D4—61no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—61no—
209C2×C3×C4—121no—
210C2×C3×C4—121no—
211C2×C3×C4#23511yescharacteristic
212C2×C3×C4D3×D411yescharacteristic
213C2×C3×C4D3×D411yescharacteristic
214C2×C3×C4D3×D411yescharacteristic
215C2×C3×C4#23711yescharacteristic
216C2×C3×C4C2×#6711yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—122no—
219C2×Dic3—121no—
220C2×Dic3—121no—
221D12—21no—
222D12#21412yes—
223C4×D3—41no—
224C4×D3—41no—
225Dic6—242no—
226Dic6—241no—
227Dic6—241no—
228C3×C8—121no—
229C3×C8—121no—
230C3×C8—121no—
231C3⋊C8—32no—
232C3⋊C8—32no—
233C3⋊C8—41no—
234C3⋊C8—41no—
235C3⋊C8—41no—
236C3⋊C8—62no—
237C3⋊C8—62no—
238C3⋊C8—61no—
239C3⋊C8—61no—
240C2×C32#46111yescharacteristic
241C2×C32#46511yescharacteristic
242C2×C32#47111yescharacteristic
243C3×D3—42no—
244C2×Q8—21no—
245C2×Q8—181no—
246C2×D4—31no—
247C22×C4—61no—
248SD16—122no—
249C2×C8—61no—
250C2×C8—61no—
251C2×C8—61no—
252C2×C8—91no—
253C2×C8—91no—
254C2×C8—362no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—31no—
258C4⋊C4—181no—
259C4⋊C4—181no—
260C22⋊C4—61no—
261C42C2×D3211yescharacteristic
262C22×C3—21no—
263C22×C3—62no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3#88411yescharacteristic
267C22×C3#93911yescharacteristic
268C2×D3—22no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—22no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—82no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4—121no—
282C3×C4#91211yescharacteristic
283C3×C4D3×SD1611yescharacteristic
284C3×C4D3×SD1611yescharacteristic
285C3×C4D3×SD1611yescharacteristic
286C3×C4#91511yescharacteristic
287C3×C4#91811yescharacteristic
288C3×C4C2×#21411yescharacteristic
289C3×C4#95011yescharacteristic
290Dic3—41no—
291Dic3—242no—
292Dic3—241no—
293Dic3—241no—
294C32#167111yescharacteristic, complemented, Sylow
295C23—61no—
296Q8—41no—
297Q8—41no—
298Q8—721no—
299D4—32no—
300D4—61no—
301C2×C4—41no—
302C2×C4—61no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—361no—
306C2×C4#380411yescharacteristic
307C2×C4D3×#6711yescharacteristic
308C2×C4#381111yescharacteristic
309C8—92no—
310C8—121no—
311C8—121no—
312C8—121no—
313C8—181no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—122no—
318C2×C3#475411yescharacteristic
319C2×C3#477811yescharacteristic
320C2×C3#481811yescharacteristic
321C2×C3#504111yescharacteristic
322C2×C3#516211yescharacteristic
323C2×C3#520311yescharacteristic
324D3—42no—
325C22—62no—
326C22—121no—
327C22—121no—
328C22#6397111yescharacteristic
329C4—21no—
330C4—81no—
331C4—121no—
332C4—721no—
333C4D3×#21411yescharacteristic
334C4#6393311yescharacteristic
335C4#6394011yescharacteristic
336C4#6395111yescharacteristic
337C3—21no—
338C3#6844211yescharacteristic, complemented
339C3#7103611yescharacteristic, complemented
340C2—122no—
341C2#1059146011yescharacteristic
342C2#1059156911yescharacteristic
343C2#1059164311yescharacteristic
subgroupG/H
centerC22#63971
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
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
1. #10592779C2semidirect
5. C3×#4547C2semidirect
6. #10589098C2semidirect
7. #10589096C2semidirect
35. #4547D3semidirect
109. C3⋊(C3⋊C8)SD16semidirect
294. C32#1671semidirect
338. C3#68442semidirect
339. C3#71036semidirect

Maximal subgroups

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

subgroupindex
#105927792
C3×#50832
#105903292
#105902682
C3×#45472
#105890982
#105890962
#710363
#684423