Finite Group Databasebeta

← #11781942

Subgroups — #11781942

Order 1152.

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

normal subgroups only

This group has 364 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×#4838C211yesmaximal, characteristic
2#10590365C211yesmaximal, characteristic
3#10590363C211yesmaximal, characteristic
4#10590311C211yesmaximal, characteristic, complemented
5#10589352C211yesmaximal, characteristic, complemented
6C3×#4486C211yesmaximal, characteristic, complemented
7#10589097C211yesmaximal, characteristic, complemented
8#70873—31nomaximal
9#68451—31nomaximal
10#64078C2211yescharacteristic
11C2×C3×ΓL(1,𝔽25)—21no—
12C3×C4×D12C2211yescharacteristic
13C3×#869C2211yescharacteristic
14C2×#3716—21no—
15C2×#3715—21no—
16C32×C4⋊C8C2211yescharacteristic
17C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×(C2×D3)⋊C8—21no—
19C3×C4⋊(C3⋊C8)C2211yescharacteristic
20C3×C4×C3⋊C8C2211yescharacteristic
21#63575—21no—
22#63575—21no—
23#63567—21no—
24#63565—21no—
25C3×#436—31no—
26#5083—31no—
27#5078—31no—
28#5019—31no—
29#5019—31no—
30#4841—31no—
31#4838D311yescharacteristic
32#4682—31no—
33C3×#332—31no—
34#4486—31no—
35#4486—31no—
36#4486—61no—
37#4486D311yescharacteristic, complemented
38#4480—31no—
39#4457—31no—
40C2×#634—21no—
41C2×C3×D12D411yescharacteristic
42C2×C3×C4×D3—21no—
43C2×C3×Dic6—21no—
44C2×C32×C8—21no—
45C32×C42C2311yescharacteristic
46C4⋊(C3⋊Dic3)—21no—
47C4⋊(C3⋊Dic3)—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
56C3×ΓL(1,𝔽25)—42no—
57#3716—42no—
58#3715—42no—
59#1693—91noSylow
60C2×C3×M16—61no—
61C3×C4⋊Q8—31no—
62C3×C4×D4—31no—
63C2×#215—61no—
64C2×#214—61no—
65C2×#214—61no—
66C2×#204—61no—
67C2×ΓL(1,𝔽25)—21no—
68C4×D12C2×D311yescharacteristic
69#869—31no—
70#869—31no—
71#869—61no—
72#869C2×D311yescharacteristic
73C3×C4⋊C8—21no—
74C3×C4⋊C8—31no—
75C3×C4⋊C8C2×D311yescharacteristic
76C3×C4⋊C8C2×D311yescharacteristic
77C3×C22⋊C8—61no—
78C3×C4×C8—31no—
79#835—61no—
80(C2×D3)⋊C8—21no—
81#818—61no—
82#818—61no—
83#816—61no—
84#810—61no—
85#810—61no—
86#807—61no—
87#807—61no—
88C4⋊(C3⋊C8)—31no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—61no—
91C4⋊(C3⋊C8)C2×D311yescharacteristic
92C4×C3⋊C8C2×D311yescharacteristic
93C22×C3×D3—21no—
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4#4111yescharacteristic
97C2×C3⋊Dic3—41no—
98#634—41no—
99#634—41no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—41no—
102C3×D12—21no—
103C3×D12—21no—
104C3×C4×D3—41no—
105C3×C4×D3—41no—
106C3×Dic6—41no—
107C3×Dic6—41no—
108C32×C8—41no—
109C3⋊(C3⋊C8)—41no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112C3×C3⋊C8SD1612yescomplemented
113#500—91no—
114#461—91no—
115#461—91no—
116#436—31no—
117#334—91no—
118#332—31no—
119#332—91no—
120C2×C3×Q8—61no—
121C2×C3×D4—31no—
122C22×C3×C4—61no—
123C2×D12D1211yescharacteristic
124C2×C4×D3—21no—
125C2×Dic6—21no—
126C2×Dic6—61no—
127C2×Dic6—61no—
128C2×Dic6—121no—
129C3×M16—122no—
130C2×C3×C8—21no—
131C2×C3×C8—21no—
132C2×C3×C8—31no—
133C2×C3×C8—31no—
134C2×C3×C8—41no—
135C2×C3×C8—61no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—61no—
139C3×C22⋊C4—61no—
140C3×C42—21no—
141C3×C42C22×D311yescharacteristic
142C3×C42C22×D311yescharacteristic
143#215—122no—
144#214—122no—
145#214—122no—
146(C2×D3)⋊C4—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—62no—
150C4⋊Dic3—61no—
151C4⋊Dic3—61no—
152C4⋊Dic3—61no—
153C4⋊Dic3—61no—
154C4⋊Dic3—121no—
155C4⋊Dic3D1211yescharacteristic
156C2×C3⋊C8—21no—
157C2×C3⋊C8—61no—
158C2×C3⋊C8—61no—
159C2×C3⋊C8—121no—
160C2×C3⋊C8#6711yescharacteristic
161C2×C3⋊C8#6711yescharacteristic
162#204—122no—
163ΓL(1,𝔽25)—42no—
164C22×C32#12111yescharacteristic
165C2×C3×D3—41no—
166C2×C3×D3—41no—
167C2×C3×D3—41no—
168C32×C4—21no—
169C32×C4C2×SD1611yescharacteristic
170C32×C4AGL(1,ℤ/8)11yescharacteristic
171C32×C4AGL(1,ℤ/8)11yescharacteristic
172C32×C4#13711yescharacteristic
173C3⋊Dic3—81no—
174C3×Dic3—41no—
175C3×Dic3—81no—
176C2×SD16—181no—
177C2×SD16—181no—
178C2×M16—61no—
179C4⋊Q8—31no—
180C4⋊Q8—91no—
181C4×D4—31no—
182#107—181no—
183#107—181no—
184C4⋊C8—31no—
185C4⋊C8—91no—
186C4⋊C8D3211yescharacteristic
187#103—181no—
188#103—181no—
189C22⋊C8—61no—
190C4×C8—31no—
191C23×C3—61no—
192C22×D3—21no—
193C3×Q8—121no—
194C3×Q8—121no—
195C3×D4—61no—
196C3×D4—61no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—61no—
201C2×C3×C4—121no—
202C2×C3×C4—121no—
203C2×C3×C4—121no—
204C2×C3×C4C2×D1211yescharacteristic
205C2×C3×C4#23511yescharacteristic
206C2×C3×C4#23511yescharacteristic
207C2×C3×C4D3×D411yescharacteristic
208C2×C3×C4D3×D411yescharacteristic
209C2×C3×C4C2×#6711yescharacteristic
210C2×Dic3—21no—
211C2×Dic3—41no—
212C2×Dic3—122no—
213C2×Dic3—121no—
214C2×Dic3—121no—
215D12—21no—
216D12—21no—
217C4×D3—41no—
218C4×D3—41no—
219Dic6—41no—
220Dic6—41no—
221Dic6—121no—
222Dic6—121no—
223Dic6—121no—
224Dic6—121no—
225Dic6—241no—
226Dic6—241no—
227C3×C8—32no—
228C3×C8—41no—
229C3×C8—41no—
230C3×C8—61no—
231C3×C8—81no—
232C3×C8—121no—
233C3⋊C8—21no—
234C3⋊C8—41no—
235C3⋊C8—121no—
236C3⋊C8—121no—
237C3⋊C8—241no—
238C3⋊C8#21412yescomplemented
239C2×C32#46111yescharacteristic
240C2×C32#46911yescharacteristic
241C2×C32#47311yescharacteristic
242C3×D3—81no—
243C2×Q8—61no—
244C2×Q8—181no—
245C2×D4—31no—
246C22×C4—61no—
247SD16—362no—
248SD16—362no—
249M16—122no—
250C2×C8—21no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—61no—
254C2×C8—181no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—181no—
259C4⋊C4—181no—
260C22⋊C4—61no—
261C42C2×D3211yescharacteristic
262C22×C3—21no—
263C22×C3—121no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3#88411yescharacteristic
267C22×C3#93011yescharacteristic
268C2×D3—41no—
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—121no—
279C3×C4—241no—
280C3×C4#90811yescharacteristic
281C3×C4#90811yescharacteristic
282C3×C4#91111yescharacteristic
283C3×C4D3×SD1611yescharacteristic
284C3×C4#91411yescharacteristic
285C3×C4#91811yescharacteristic
286C3×C4C2×#21411yescharacteristic
287C3×C4#95111yescharacteristic
288Dic3—41no—
289Dic3—81no—
290Dic3—242no—
291Dic3—241no—
292Dic3—241no—
293C32#169311yescharacteristic, complemented, Sylow
294C23—61no—
295Q8—121no—
296Q8—121no—
297Q8—361no—
298Q8—361no—
299D4—61no—
300D4—61no—
301C2×C4—61no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—121no—
305C2×C4—361no—
306C2×C4#379811yescharacteristic
307C2×C4D3×D1211yescharacteristic
308C2×C4D3×#6711yescharacteristic
309C8—32no—
310C8—41no—
311C8—61no—
312C8—121no—
313C8—361no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—241no—
318C2×C3#475511yescharacteristic
319C2×C3#478211yescharacteristic
320C2×C3#481811yescharacteristic
321C2×C3#501911yescharacteristic
322C2×C3#513511yescharacteristic
323C2×C3#514311yescharacteristic
324D3—81no—
325C22—121no—
326C22—121no—
327C22—121no—
328C22#6390611yescharacteristic
329C4—21no—
330C4—121no—
331C4—241no—
332C4—721no—
333C4#6379411yescharacteristic
334C4#6379511yescharacteristic
335C4D3×#21411yescharacteristic
336C4#6394311yescharacteristic
337C3—21no—
338C3#6845111yescharacteristic, complemented
339C3#7087311yescharacteristic, complemented
340C2—241no—
341C2#1059055911yescharacteristic
342C2#1059082711yescharacteristic
343C2#1059099911yescharacteristic
subgroupG/H
centerC22#63906
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#121
prime radicalC2×C3×D12D4
hypercenterC4⋊C8D32
hypocenterC32#1693
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
4. #10590311C2semidirect
5. #10589352C2semidirect
6. C3×#4486C2semidirect
7. #10589097C2semidirect
37. #4486D3semidirect
112. C3×C3⋊C8SD16semidirect
238. C3⋊C8#214semidirect
293. C32#1693semidirect
338. C3#68451semidirect
339. C3#70873semidirect

Maximal subgroups

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

subgroupindex
C3×#48382
#105903652
#105903632
#105903112
#105893522
C3×#44862
#105890972
#708733
#684513