Finite Group Databasebeta

← #11777743

Subgroups — #11777743

Order 1152.

373 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 373 conjugacy classes of subgroups — too many to draw. Only its 98 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×#4830C211yesmaximal, characteristic, complemented
2C4×#3717C211yesmaximal, characteristic
3#10590338C211yesmaximal, characteristic
4#10589349C211yesmaximal, characteristic
5C3×#4479C211yesmaximal, characteristic, complemented
6#10589083C211yesmaximal, characteristic, complemented
7#10589079C211yesmaximal, characteristic, complemented
8#70936—31nomaximal
9#68562—31nomaximal
10C4×C4⋊(C3⋊D3)C2211yescharacteristic
11C3×C4×Dic6C2211yescharacteristic
12C2×#3717C411yescharacteristic
13C2×#3712—21no—
14C32×C4⋊C8C2211yescharacteristic
15C4×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×Dic3⋊C8—21no—
17C3×C8×Dic3—21no—
18C3×C4⋊(C3⋊C8)C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63573C411yescharacteristic
22#63566C411yescharacteristic
23#63564C411yescharacteristic
24#63557—21no—
25C3×C8×Q8—31no—
26C4×#215—31no—
27#4838—31no—
28#4830D311yescharacteristic, complemented
29C4×#204—31no—
30#4688—31no—
31C3×#335—31no—
32#4535—31no—
33#4484—31no—
34#4483—31no—
35#4483—31no—
36#4483—61no—
37#4479D311yescharacteristic, complemented
38#4454—31no—
39#4454—31no—
40C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
41C2×C4×C3⋊D3—21no—
42C2×C3×Dic6C2×C411yescharacteristic
43C2×C32×C8—21no—
44C32×C42C2311yescharacteristic
45#3752—21no—
46C4⋊(C3⋊Dic3)C2×C411yescharacteristic
47C2×C3⋊(C3⋊C8)—21no—
48C3×C4⋊Dic3C2×C411yescharacteristic
49C3×Dic3⋊C4—21no—
50C3×C4×Dic3—21no—
51C2×C3×C3⋊C8—21no—
52C2×C3×C3⋊C8C2×C411yescharacteristic
53C2×C3×C3⋊C8C2×C411yescharacteristic
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56#3717—22no—
57#3712—42no—
58#1588—91noSylow
59C3×C4×Q8—31no—
60C2×#215—31no—
61C2×#204—31no—
62C2×ΓL(1,𝔽25)—61no—
63C2×ΓL(1,𝔽25)—61no—
64C4×D12—31no—
65C4×D12—31no—
66C4×D12—61no—
67C4×Dic6C2×D311yescharacteristic
68C3×C4⋊C8—21no—
69C3×C4⋊C8—31no—
70C3×C4⋊C8—61no—
71C3×C4⋊C8C2×D311yescharacteristic
72C3×C4⋊C8C2×D311yescharacteristic
73C3×C4×C8—31no—
74C3×C4×C8—31no—
75C3×C4×C8—61no—
76#835—31no—
77#821—31no—
78(C2×D3)⋊C8—61no—
79(C2×D3)⋊C8—61no—
80#817—31no—
81#816—31no—
82Dic3⋊C8—21no—
83C8×Dic3—21no—
84#809—31no—
85#808—31no—
86C4⋊(C3⋊C8)C2×D311yescharacteristic
87C4×C3⋊C8—31no—
88C4×C3⋊C8—31no—
89C4×C3⋊C8—61no—
90C4×C3⋊C8C2×D311yescharacteristic
91C4×C3⋊C8C2×D311yescharacteristic
92C22×C3⋊D3—21no—
93C2×C32×C4C22×C411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4#4111yescharacteristic
96C2×C3⋊Dic3—21no—
97C4⋊(C3⋊D3)—21no—
98C4⋊(C3⋊D3)—21no—
99C4×C3⋊D3—41no—
100C4×C3⋊D3—41no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C3×Dic6—21no—
104C3×Dic6M1612yescomplemented
105C32×C8—41no—
106C3⋊(C3⋊C8)—22no—
107C3×C3⋊C8—21no—
108C3×C3⋊C8—21no—
109C3×C3⋊C8—21no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C8×Q8—31no—
113C4×SD16—91no—
114#436—91no—
115#335—31no—
116Q8⋊C8—91no—
117D4⋊C8—91no—
118#322—91no—
119C2×C3×Q8—31no—
120C2×D12—31no—
121C2×D12—31no—
122C2×D12—61no—
123C2×C4×D3—61no—
124C2×C4×D3—61no—
125C2×C4×D3—121no—
126C2×Dic6C4×D311yescharacteristic
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C2×C3×C8—31no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—41no—
134C2×C3×C8—61no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—61no—
137C3×C42—21no—
138C3×C42—61no—
139C3×C42C22×D311yescharacteristic
140C3×C42C22×D311yescharacteristic
141#215—62no—
142(C2×D3)⋊C4—61no—
143(C2×D3)⋊C4—61no—
144(C2×D3)⋊C4—121no—
145C4⋊Dic3—31no—
146C4⋊Dic3—31no—
147C4⋊Dic3—61no—
148C4⋊Dic3C4×D311yescharacteristic
149Dic3⋊C4—21no—
150C4×Dic3—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—121no—
155C2×C3⋊C8C4×D311yescharacteristic
156C2×C3⋊C8C4×D311yescharacteristic
157C2×C3⋊C8#6711yescharacteristic
158C2×C3⋊C8#6711yescharacteristic
159#204—62no—
160ΓL(1,𝔽25)—122no—
161ΓL(1,𝔽25)—122no—
162C22×C32C4×D411yescharacteristic
163C2×C3⋊D3—41no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—41no—
166C32×C4—21no—
167C32×C4C2×M1611yescharacteristic
168C32×C4#13111yescharacteristic
169C32×C4AGL(1,ℤ/8)11yescharacteristic
170C32×C4#13711yescharacteristic
171C3⋊Dic3—41no—
172C3×Dic3—22no—
173C3×Dic3—41no—
174C2×SD16—91no—
175C2×M16—181no—
176C4×Q8—31no—
177C4×D4—91no—
178#106—91no—
179C4⋊C8—31no—
180C4⋊C8—61no—
181C4⋊C8D3211yescharacteristic
182#103—91no—
183#102—91no—
184C22⋊C8—181no—
185C4×C8—31no—
186C4×C8—31no—
187C4×C8—61no—
188C4×C8—91no—
189C22×D3—61no—
190C22×D3—61no—
191C22×D3—121no—
192C3×Q8—32no—
193C3×Q8—61no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4C2×C4×D311yescharacteristic
200C2×C3×C4C2×C4×D311yescharacteristic
201C2×C3×C4#23511yescharacteristic
202C2×C3×C4D3×D411yescharacteristic
203C2×C3×C4#23911yescharacteristic
204C2×C3×C4C2×#6711yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—121no—
210D12—62no—
211D12—61no—
212D12—61no—
213D12—61no—
214D12—61no—
215D12—121no—
216C4×D3—121no—
217C4×D3—121no—
218C4×D3—121no—
219C4×D3—121no—
220C4×D3—241no—
221C4×D3—241no—
222Dic6—21no—
223Dic6ΓL(1,𝔽25)12yescomplemented
224C3×C8—41no—
225C3×C8—41no—
226C3×C8—61no—
227C3×C8—61no—
228C3×C8—61no—
229C3×C8—61no—
230C3×C8—81no—
231C3×C8—121no—
232C3⋊C8—21no—
233C3⋊C8—21no—
234C3⋊C8—21no—
235C3⋊C8—21no—
236C3⋊C8—41no—
237C3⋊C8—62no—
238C3⋊C8—62no—
239C3⋊C8—122no—
240C2×C32#43511yescharacteristic
241C2×C32#44011yescharacteristic
242C2×C32#44311yescharacteristic
243C3⋊D3—81no—
244C2×Q8—31no—
245C2×D4—91no—
246C22×C4—181no—
247SD16—182no—
248M16—362no—
249C2×C8—21no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—31no—
254C2×C8—61no—
255C2×C8—181no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—91no—
259C22⋊C4—181no—
260C42—61no—
261C42C2×D3211yescharacteristic
262C22×C3—21no—
263C22×C3#89311yescharacteristic
264C22×C3C4×#6711yescharacteristic
265C2×D3—122no—
266C2×D3—121no—
267C2×D3—121no—
268C2×D3—121no—
269C2×D3—121no—
270C2×D3—121no—
271C2×D3—121no—
272C2×D3—241no—
273C2×D3—241no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—62no—
282C3×C4—121no—
283C3×C4C2×ΓL(1,𝔽25)11yescharacteristic
284C3×C4#90111yescharacteristic
285C3×C4#90111yescharacteristic
286C3×C4D3×M1611yescharacteristic
287C3×C4#91111yescharacteristic
288C3×C4#91811yescharacteristic
289C3×C4#94211yescharacteristic
290C3×C4#94911yescharacteristic
291Dic3—22no—
292Dic3—41no—
293Dic3—121no—
294Dic3—121no—
295Dic3—241no—
296C32#158811yescharacteristic, complemented, Sylow
297C23—181no—
298Q8—32no—
299Q8—61no—
300D4—181no—
301D4—181no—
302C2×C4—61no—
303C2×C4—61no—
304C2×C4—181no—
305C2×C4—361no—
306C2×C4—361no—
307C2×C4#379911yescharacteristic
308C2×C4C4×D3211yescharacteristic
309C2×C4D3×#6711yescharacteristic
310C8—41no—
311C8—61no—
312C8—61no—
313C8—61no—
314C8—61no—
315C8—121no—
316C8—182no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3#483611yescharacteristic
321C2×C3#488911yescharacteristic
322C2×C3#489511yescharacteristic
323C2×C3#502611yescharacteristic
324C2×C3#511111yescharacteristic
325C2×C3#514411yescharacteristic
326D3—242no—
327D3—241no—
328D3—241no—
329C22—361no—
330C22—361no—
331C22—361no—
332C22#6390411yescharacteristic
333C4—21no—
334C4—62no—
335C4—121no—
336C4—361no—
337C4D3×ΓL(1,𝔽25)11yescharacteristic
338C4#6380311yescharacteristic
339C4#6392511yescharacteristic
340C4#6394811yescharacteristic
341C3—21no—
342C3#6856211yescharacteristic, complemented
343C3#7093611yescharacteristic, complemented
344C2—721no—
345C2#1059045711yescharacteristic
346C2#1059072711yescharacteristic
347C2#1059099511yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32C4×D4
prime radicalC2×C4⋊(C3⋊D3)C2×C4
cosocleC32×C42C23
hypercenterC4⋊C8D32
hypocenterC32#1588
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4⋊C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#4830C2semidirect
5. C3×#4479C2semidirect
6. #10589083C2semidirect
7. #10589079C2semidirect
28. #4830D3semidirect
37. #4479D3semidirect
104. C3×Dic6M16semidirect
223. Dic6ΓL(1,𝔽25)semidirect
296. C32#1588semidirect
342. C3#68562semidirect
343. C3#70936semidirect

Maximal subgroups

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

subgroupindex
C3×#48302
C4×#37172
#105903382
#105893492
C3×#44792
#105890832
#105890792
#709363
#685623