Finite Group Databasebeta

← #11777904

Subgroups — #11777904

Order 1152.

351 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 96 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
2C4×#3715C211yesmaximal, characteristic
3#10590258C211yesmaximal, characteristic, complemented
4#10589345C211yesmaximal, characteristic, complemented
5C3×#4484C211yesmaximal, characteristic, complemented
6#10589081C211yesmaximal, characteristic
7#10589079C211yesmaximal, characteristic, complemented
8#70860—31nomaximal
9#68226—31nomaximal
10C2×C3×ΓL(1,𝔽25)—21no—
11C3×C4×D12C2211yescharacteristic
12C3×C4×Dic6C2211yescharacteristic
13C2×#3715C411yescharacteristic
14C32×C4⋊C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C4×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×(C2×D3)⋊C8—21no—
18C3×C4×C3⋊C8C2211yescharacteristic
19C3×C4×C3⋊C8C2211yescharacteristic
20#63574C411yescharacteristic
21Dic3⋊(C3⋊C8)—21no—
22#63565C411yescharacteristic
23#63562C411yescharacteristic
24#63554—21no—
25C3×#436—31no—
26C4×#215—31no—
27#5024—31no—
28C4×#214—31no—
29#4838D311yescharacteristic
30#4679—31no—
31C3×Q8⋊C8—31no—
32D4⋊(C3⋊C8)—31no—
33#4484D311yescharacteristic, complemented
34#4479—31no—
35#4479—31no—
36#4479—61no—
37#4459—31no—
38#4454—31no—
39#4454—31no—
40C2×C3×D12C2×C411yescharacteristic
41C2×C3×C4×D3—21no—
42C2×C3×Dic6C2×C411yescharacteristic
43C2×C32×C8—21no—
44C32×C42C2311yescharacteristic
45C2×C3⋊(C3⋊C8)—21no—
46C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
47C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
48C3×(C2×D3)⋊C4—21no—
49C3×C4⋊Dic3C2×C411yescharacteristic
50C3×C4⋊Dic3C2×C411yescharacteristic
51C3×Dic3⋊C4—21no—
52C3×C4×Dic3—21no—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8D411yescharacteristic
55C2×C3×C3⋊C8D411yescharacteristic
56C3×ΓL(1,𝔽25)—42no—
57#3715—22no—
58#1592—91noSylow
59C2×C3×M16—61no—
60C3×C4×Q8—31no—
61C3×C4×D4—31no—
62C2×#215—31no—
63C2×#214—31no—
64C2×ΓL(1,𝔽25)—21no—
65C4×D12C2×D311yescharacteristic
66C4×Dic6C2×D311yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×D311yescharacteristic
69C3×C4⋊C8C2×D311yescharacteristic
70C3×C22⋊C8—61no—
71C3×C4×C8—31no—
72C3×C4×C8—31no—
73#835—31no—
74#832—31no—
75(C2×D3)⋊C8—21no—
76#815—61no—
77Dic3⋊C8—61no—
78#810—31no—
79#809—31no—
80#808—31no—
81#808—31no—
82C4⋊(C3⋊C8)—31no—
83C4⋊(C3⋊C8)—31no—
84C4⋊(C3⋊C8)—61no—
85C4⋊(C3⋊C8)—61no—
86#803—61no—
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—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C3×D12—21no—
100C3×D12—21no—
101C3×C4×D3—41no—
102C3×C4×D3—41no—
103C3×Dic6—21no—
104C3×Dic6—21no—
105C32×C8—41no—
106C3⋊(C3⋊C8)—21no—
107C3⋊(C3⋊C8)—41no—
108C3⋊(C3⋊C8)M1612yescomplemented
109C3×C3⋊C8—21no—
110C3×C3⋊C8—21no—
111C3×C3⋊C8—41no—
112#446—91no—
113C4×SD16—91no—
114#436—31no—
115#335—91no—
116Q8⋊C8—31no—
117D4⋊C8—91no—
118#322—91no—
119C2×C3×Q8—31no—
120C2×C3×D4—31no—
121C22×C3×C4—61no—
122C2×D12C4×D311yescharacteristic
123C2×C4×D3—21no—
124C2×Dic6C4×D311yescharacteristic
125C3×M16—122no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—31no—
129C2×C3×C8—31no—
130C2×C3×C8—41no—
131C2×C3×C8—61no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—61no—
135C3×C22⋊C4—61no—
136C3×C42—21no—
137C3×C42—61no—
138C3×C42C22×D311yescharacteristic
139C3×C42C22×D311yescharacteristic
140#215—62no—
141#214—62no—
142(C2×D3)⋊C4—21no—
143C4⋊Dic3C4×D311yescharacteristic
144C4⋊Dic3C4×D311yescharacteristic
145Dic3⋊C4—21no—
146C4×Dic3—21no—
147C2×C3⋊C8—21no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—31no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—31no—
152C2×C3⋊C8—61no—
153C2×C3⋊C8—61no—
154C2×C3⋊C8—61no—
155C2×C3⋊C8—61no—
156C2×C3⋊C8—121no—
157C2×C3⋊C8#6711yescharacteristic
158C2×C3⋊C8#6711yescharacteristic
159ΓL(1,𝔽25)—42no—
160C22×C32C4×D411yescharacteristic
161C2×C3×D3—41no—
162C2×C3×D3—41no—
163C2×C3×D3—41no—
164C32×C4—21no—
165C32×C4C2×M1611yescharacteristic
166C32×C4#13111yescharacteristic
167C32×C4AGL(1,ℤ/8)11yescharacteristic
168C32×C4#13711yescharacteristic
169C3×Dic3—41no—
170C3×Dic3—41no—
171C3×Dic3—41no—
172C2×SD16—91no—
173C2×M16—61no—
174C4×Q8—31no—
175C4×D4—31no—
176#106—91no—
177C4⋊C8—91no—
178C4⋊C8—181no—
179C4⋊C8D3211yescharacteristic
180#103—91no—
181#102—91no—
182C22⋊C8—61no—
183#97—181no—
184C4×C8—31no—
185C4×C8—31no—
186C4×C8—91no—
187C23×C3—61no—
188C22×D3—21no—
189C3×Q8—61no—
190C3×Q8—61no—
191C3×D4—61no—
192C3×D4—61no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—121no—
200C2×C3×C4—121no—
201C2×C3×C4C2×C4×D311yescharacteristic
202C2×C3×C4C2×C4×D311yescharacteristic
203C2×C3×C4#23511yescharacteristic
204C2×C3×C4D3×D411yescharacteristic
205C2×C3×C4#23711yescharacteristic
206C2×C3×C4C2×#6711yescharacteristic
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210D12—21no—
211D12—21no—
212C4×D3—41no—
213C4×D3—41no—
214Dic6—21no—
215Dic6—21no—
216C3×C8—41no—
217C3×C8—41no—
218C3×C8—61no—
219C3×C8—61no—
220C3×C8—81no—
221C3×C8—121no—
222C3⋊C8—21no—
223C3⋊C8—21no—
224C3⋊C8—32no—
225C3⋊C8—32no—
226C3⋊C8—41no—
227C3⋊C8—62no—
228C3⋊C8—61no—
229C3⋊C8—61no—
230C3⋊C8—121no—
231C3⋊C8—121no—
232C3⋊C8—121no—
233C3⋊C8—241no—
234C2×C32#43611yescharacteristic
235C2×C32#44011yescharacteristic
236C2×C32#44411yescharacteristic
237C3×D3—81no—
238C2×Q8—31no—
239C2×D4—31no—
240C22×C4—61no—
241SD16—182no—
242M16—122no—
243C2×C8—21no—
244C2×C8—31no—
245C2×C8—31no—
246C2×C8—61no—
247C2×C8—91no—
248C2×C8—91no—
249C2×C8—181no—
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—61no—
253C22⋊C4—61no—
254C42—61no—
255C42C2×D3211yescharacteristic
256C22×C3—21no—
257C22×C3—121no—
258C22×C3—121no—
259C22×C3—121no—
260C22×C3#67⋊C411yescharacteristic
261C22×C3C4×#6711yescharacteristic
262C2×D3—41no—
263C2×D3—41no—
264C2×D3—41no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—121no—
273C3×C4—121no—
274C3×C4—121no—
275C3×C4#90111yescharacteristic
276C3×C4D3×M1611yescharacteristic
277C3×C4D3×M1611yescharacteristic
278C3×C4#90711yescharacteristic
279C3×C4#91411yescharacteristic
280C3×C4#91811yescharacteristic
281C3×C4#93211yescharacteristic
282C3×C4#95111yescharacteristic
283Dic3—41no—
284Dic3—41no—
285Dic3—41no—
286C32#159211yescharacteristic, complemented, Sylow
287C23—61no—
288Q8—61no—
289Q8—61no—
290D4—61no—
291D4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—121no—
296C2×C4—121no—
297C2×C4#379511yescharacteristic
298C2×C4C4×D3211yescharacteristic
299C2×C4D3×#6711yescharacteristic
300C8—41no—
301C8—61no—
302C8—61no—
303C8—92no—
304C8—121no—
305C8—181no—
306C8—361no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—241no—
311C2×C3#472911yescharacteristic
312C2×C3#481711yescharacteristic
313C2×C3#482311yescharacteristic
314C2×C3#501811yescharacteristic
315C2×C3#512811yescharacteristic
316C2×C3#514511yescharacteristic
317D3—81no—
318C22—121no—
319C22—121no—
320C22—121no—
321C22#6387911yescharacteristic
322C4—21no—
323C4—121no—
324C4—121no—
325C4—121no—
326C4#6379111yescharacteristic
327C4#6380411yescharacteristic
328C4#6393411yescharacteristic
329C4#6394311yescharacteristic
330C3—21no—
331C3#6822611yescharacteristic, complemented
332C3#7086011yescharacteristic, complemented
333C2—241no—
334C2#1059047811yescharacteristic
335C2#1059073511yescharacteristic
336C2#1059097611yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32C4×D4
prime radicalC2×C3×D12C2×C4
cosocleC32×C42C23
hypercenterC4⋊C8D32
hypocenterC32#1592
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
3. #10590258C2semidirect
4. #10589345C2semidirect
5. C3×#4484C2semidirect
7. #10589079C2semidirect
33. #4484D3semidirect
108. C3⋊(C3⋊C8)M16semidirect
286. C32#1592semidirect
331. C3#68226semidirect
332. C3#70860semidirect

Maximal subgroups

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

subgroupindex
C3×#48382
C4×#37152
#105902582
#105893452
C3×#44842
#105890812
#105890792
#708603
#682263