Finite Group Databasebeta

← #11774556

Subgroups — #11774556

Order 1152.

687 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 687 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C4×#3802C211yesmaximal, characteristic
2#10592536C211yesmaximal, characteristic, complemented
3#10590297C211yesmaximal, characteristic, complemented
4C3×#4483C212yesmaximal, complemented
5#10589081C212yesmaximal, complemented
6#68172—32nomaximal
7C2×#3802C411yescharacteristic
8C2×C4×D32—21no—
9C2×#3743—21no—
10C42×C3⋊D3C2211yescharacteristic
11C3×C4×D12C2212yes—
12#63921—21no—
13#63902—21no—
14C4×#625—21no—
15#63900C411yescharacteristic
16#63898C412yes—
17C32×C4⋊C8C2211yescharacteristic, complemented
18#63641—21no—
19C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
20C3×C4×C3⋊C8C2212yescomplemented
21#63572—21no—
22#63557—21no—
23C4×D3×D4—32no—
24#4837—32no—
25#4837—61no—
26#4685—32no—
27C3×D4⋊C8—32no—
28D4⋊(C3⋊C8)—32no—
29#4483D312yescomplemented
30#4459—32no—
31C22×D32—21no—
32C2×C4×C3⋊D3C2×C411yescharacteristic
33C2×C4×C3⋊D3D411yescharacteristic
34C2×C4×C3⋊D3D411yescharacteristic
35C2×C3×D12C2×C412yes—
36C2×C3×C4×D3—22no—
37C2×#625—21no—
38C2×#624—21no—
39C2×D3×Dic3—22no—
40#3802—24no—
41C4×D32—42no—
42C4×D32—42no—
43C2×C32×C8—21no—
44C32×C42C2311yescharacteristic
45C4×C3⋊Dic3C2×C411yescharacteristic
46C4×C3⋊Dic3D411yescharacteristic
47C4×C3⋊Dic3D411yescharacteristic
48C2×C3⋊(C3⋊C8)—21no—
49#3743—42no—
50C3×(C2×D3)⋊C4—22no—
51C3×C4⋊Dic3C2×C412yes—
52C2×C3×C3⋊C8—22no—
53#3724—21no—
54(C2×D3)⋊Dic3—22no—
55#1486—91noSylow
56C2×D3×D4—32no—
57C22×C4×D3—62no—
58C3×C4×D4—32no—
59D4×Dic3—32no—
60C4×#67—62no—
61C2×ΓL(1,𝔽25)—62no—
62C2×ΓL(1,𝔽25)—62no—
63#893—32no—
64D3×C4⋊C4—32no—
65#67⋊C4—62no—
66D3×C22⋊C4—62no—
67C4×D12C2×D312yes—
68C42×D3—32no—
69C42×D3—61no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8C2×D312yescomplemented
72C3×C4×C8—32no—
73(C2×D3)⋊C8—62no—
74Dic3⋊C8—62no—
75Dic3⋊C8—62no—
76Dic3⋊C8—62no—
77C4⋊(C3⋊C8)—32no—
78C4⋊(C3⋊C8)—61no—
79C4×C3⋊C8C2×D312yescomplemented
80C22×C3⋊D3C22⋊C411yescharacteristic
81C22×C3×D3—22no—
82C2×D32—24no—
83C2×D32—42no—
84C2×D32—42no—
85C2×C32×C4C22×C411yescharacteristic
86C2×C32×C4C2×D411yescharacteristic
87C2×C32×C4C2×D411yescharacteristic
88C2×C3⋊Dic3C22⋊C411yescharacteristic
89C2×C3⋊Dic3C22⋊C411yescharacteristic
90C2×C3⋊Dic3C22⋊C411yescharacteristic
91C4×C3⋊D3—22no—
92C4×C3⋊D3—41no—
93C4×C3⋊D3D812yes—
94C4×C3⋊D3SD1612yes—
95C2×C3×Dic3—22no—
96C3×D12—22no—
97C3×D12M1614yescomplemented
98C3×C4×D3—42no—
99C3×C4×D3—42no—
100#625—24no—
101#624—42no—
102D3×Dic3—44no—
103C32×C8—41no—
104C3⋊(C3⋊C8)—41no—
105C3×C3⋊C8—42no—
106C2×C4×D4—91no—
107#431—91no—
108#423—91no—
109D4⋊C8—32no—
110D4⋊C8—92no—
111C23×D3—62no—
112C2×C3×D4—32no—
113C22×C3×C4—62no—
114C2×#67—62no—
115C22×Dic3—62no—
116D3×D4—64no—
117D3×D4—64no—
118C2×D12C4×D312yes—
119C2×C4×D3—22no—
120C2×C4×D3—32no—
121C2×C4×D3—32no—
122C2×C4×D3—32no—
123C2×C4×D3—61no—
124C2×C4×D3—61no—
125C2×C4×D3—61no—
126C2×C4×D3—62no—
127C2×C4×D3—62no—
128C2×C4×D3—124no—
129C2×C4×D3—124no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—62no—
133C3×C4⋊C4—32no—
134C3×C22⋊C4—62no—
135C3×C42—21no—
136C3×C42C22×D312yes—
137#217—62no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—62no—
140C4⋊Dic3C4×D312yes—
141Dic3⋊C4—62no—
142C4×Dic3—32no—
143C4×Dic3—32no—
144C4×Dic3—32no—
145C4×Dic3—61no—
146C4×Dic3—61no—
147C4×Dic3—61no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—62no—
150C2×C3⋊C8—62no—
151ΓL(1,𝔽25)—124no—
152ΓL(1,𝔽25)—124no—
153C22×C32C2×C22⋊C411yescharacteristic
154C2×C3⋊D3—21no—
155C2×C3⋊D3—21no—
156C2×C3⋊D3#10212yes—
157C2×C3×D3—24no—
158C2×C3×D3—42no—
159C2×C3×D3—42no—
160D32—48no—
161C32×C4—21no—
162C32×C4C2×M1612yes—
163C32×C4C2×D811yescharacteristic
164C32×C4C2×SD1611yescharacteristic
165C3⋊Dic3—21no—
166C3⋊Dic3—21no—
167C3⋊Dic3#10212yes—
168C3×Dic3—42no—
169C22×D4—91no—
170C23×C4—181no—
171C2×M16—181no—
172C4×D4—32no—
173C4×D4—92no—
174C4×D4—181no—
175C4×D4—181no—
176C2×C4⋊C4—91no—
177C2×C22⋊C4—181no—
178C2×C42—91no—
179C4⋊C8—91no—
180C4⋊C8—181no—
181C4⋊C8—181no—
182C4⋊C8D3211yescharacteristic, complemented
183C22⋊C8—181no—
184C4×C8—32no—
185C23×C3—62no—
186C22×D3—22no—
187C22×D3—32no—
188C22×D3—61no—
189C22×D3—68no—
190C22×D3—124no—
191C22×D3—124no—
192C3×D4—34no—
193C3×D4—62no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—62no—
198C2×C3×C4—122no—
199C2×C3×C4—122no—
200C2×C3×C4C2×C4×D312yes—
201C2×C3×C4D3×D412yes—
202C2×C3×C4D3×D412yes—
203#67—68no—
204C2×Dic3—22no—
205C2×Dic3—32no—
206C2×Dic3—32no—
207C2×Dic3—32no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—124no—
212D12—22no—
213D12ΓL(1,𝔽25)14yescomplemented
214C4×D3—34no—
215C4×D3—34no—
216C4×D3—42no—
217C4×D3—42no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—62no—
221C4×D3—62no—
222C4×D3—122no—
223C4×D3—124no—
224C4×D3—124no—
225C4×D3—122no—
226C4×D3—122no—
227C3×C8—42no—
228C3×C8—42no—
229C3×C8—122no—
230C3⋊C8—42no—
231C3⋊C8—122no—
232C3⋊C8—122no—
233C2×C32#40711yescharacteristic
234C2×C32C2×#10211yescharacteristic
235C2×C32#42111yescharacteristic
236C3⋊D3—22no—
237C3×D3—44no—
238C24—181no—
239C2×D4—32no—
240C2×D4—181no—
241C2×D4—184no—
242C22×C4—62no—
243C22×C4—91no—
244C22×C4—91no—
245C22×C4—91no—
246C22×C4—181no—
247C22×C4—182no—
248C22×C4—362no—
249C22×C4—362no—
250M16—362no—
251C2×C8—21no—
252C2×C8—62no—
253C2×C8—181no—
254C4⋊C4—32no—
255C4⋊C4—181no—
256C22⋊C4—62no—
257C22⋊C4—182no—
258C42—91no—
259C42—91no—
260C42—91no—
261C42C2×D3211yescharacteristic
262C22×C3—21no—
263C22×C3—64no—
264C22×C3—122no—
265C22×C3—122no—
266C22×C3D3×C22⋊C412yes—
267C2×D3—24no—
268C2×D3—34no—
269C2×D3—42no—
270C2×D3—42no—
271C2×D3—62no—
272C2×D3—62no—
273C2×D3—62no—
274C2×D3—121no—
275C2×D3—121no—
276C2×D3—1216no—
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—122no—
283C3×C4C2×ΓL(1,𝔽25)12yes—
284C3×C4D3×M1612yes—
285C3×C4D3×D812yes—
286C3×C4D3×SD1612yes—
287Dic3—34no—
288Dic3—42no—
289Dic3—62no—
290Dic3—62no—
291Dic3—62no—
292Dic3—62no—
293Dic3—121no—
294C32#148611yescharacteristic, complemented, Sylow
295C23—62no—
296C23—91no—
297C23—184no—
298C23—362no—
299C23—362no—
300D4—34no—
301D4—62no—
302D4—184no—
303C2×C4—62no—
304C2×C4—91no—
305C2×C4—92no—
306C2×C4—92no—
307C2×C4—91no—
308C2×C4—91no—
309C2×C4—122no—
310C2×C4—122no—
311C2×C4—182no—
312C2×C4—361no—
313C2×C4—364no—
314C2×C4—362no—
315C2×C4C4×D3211yescharacteristic
316C2×C4#380211yescharacteristic
317C2×C4#381111yescharacteristic
318C8—41no—
319C8—122no—
320C8—361no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—124no—
325C2×C3#472612yes—
326C2×C3D3×#10212yes—
327C2×C3#482112yes—
328D3—44no—
329D3—64no—
330D3—122no—
331C22—64no—
332C22—92no—
333C22—122no—
334C22—122no—
335C22—181no—
336C22—181no—
337C22—368no—
338C22#6392111yescharacteristic
339C4—21no—
340C4—92no—
341C4—122no—
342C4—181no—
343C4—181no—
344C4D3×ΓL(1,𝔽25)12yes—
345C4#6392611yescharacteristic
346C4#6394011yescharacteristic
347C3—21no—
348C3#6817212yescomplemented
349C2—124no—
350C2—182no—
351C2#1059059311yescharacteristic
352C2#1059093711yescharacteristic
353C2#1059100911yescharacteristic
subgroupG/H
centerC2×C4#3811
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32C2×C22⋊C4
prime radicalC2×#3802C4
cosocleC32×C42C23
hypercenterC4⋊C8D32
hypocenterC32#1486
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
2. #10592536C2semidirect
3. #10590297C2semidirect
4. C3×#4483C2semidirect
5. #10589081C2semidirect
17. C32×C4⋊C8C22semidirect
19. C4⋊(C3⋊(C3⋊C8))C22semidirect
20. C3×C4×C3⋊C8C22semidirect
29. #4483D3semidirect
71. C3×C4⋊C8C2×D3semidirect
79. C4×C3⋊C8C2×D3semidirect
97. C3×D12M16semidirect
182. C4⋊C8D32semidirect
213. D12ΓL(1,𝔽25)semidirect
294. C32#1486semidirect
348. C3#68172semidirect

Maximal subgroups

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

subgroupindex
C4×#38022
#105925362
#105902972
C3×#44832
#105890812
#681723