Finite Group Databasebeta

← #11796036

Subgroups — #11796036

Order 1152.

621 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C2×M16×C3⋊D3C211yesmaximal, characteristic
2#10590625C212yesmaximal, complemented
3C3×#4553C212yesmaximal, complemented
4#10589110C212yesmaximal, complemented
5D3×#349—32nomaximal
6C22×C4×C3⋊D3C411yescharacteristic
7C2×C32×M16C2211yescharacteristic
8C2×#3748C411yescharacteristic
9M16×C3⋊D3—22no—
10M16×C3⋊D3—22no—
11C2×#3743C411yescharacteristic
12C2×#3743C2211yescharacteristic
13C2×C8×C3⋊D3C411yescharacteristic
14C2×C8×C3⋊D3C2211yescharacteristic
15C3×#812C2214yescomplemented
16#63544—22no—
17#63542—22no—
18C2×D3×M16—32no—
19C2×D3×M16—61no—
20D3×M32—34no—
21C3×#349—32no—
22#4553D312yescomplemented
23#4514—34no—
24C23×C3⋊D3C811yescharacteristic
25C22×C32×C4C2×C411yescharacteristic
26C22×C3⋊Dic3C811yescharacteristic
27C2×C4×C3⋊D3—21no—
28C2×C4×C3⋊D3—21no—
29C2×C4×C3⋊D3—21no—
30C2×C4×C3⋊D3—21no—
31C2×C4×C3⋊D3C811yescharacteristic
32C2×C4×C3⋊D3C811yescharacteristic
33C2×C4×C3⋊D3C2×C411yescharacteristic
34C2×C4×C3⋊D3C2×C411yescharacteristic
35C32×M16—22no—
36C2×C32×C8C2×C411yescharacteristic
37C2×C32×C8C2311yescharacteristic
38#3748—22no—
39C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
40C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
41#3743—22no—
42#3743—22no—
43C8×C3⋊D3—21no—
44C8×C3⋊D3—21no—
45C8×C3⋊D3D412yes—
46C8×C3⋊D3D412yes—
47C3×C3⋊C16—24no—
48C2×#349—91noSylow
49C22×C4×D3—32no—
50C22×C4×D3—61no—
51C2×C3×M16—21no—
52C2×C3×M16C2×D312yes—
53C2×#208—32no—
54C2×#208—61no—
55D3×M16—64no—
56D3×M16—64no—
57D3×M16—64no—
58D3×M16—64no—
59C2×ΓL(1,𝔽25)—32no—
60C2×ΓL(1,𝔽25)—32no—
61C2×ΓL(1,𝔽25)—61no—
62C2×ΓL(1,𝔽25)—61no—
63C2×D3×C8—32no—
64C2×D3×C8—32no—
65C2×D3×C8—61no—
66C2×D3×C8—61no—
67C3×M32—34no—
68#812C2×D314yescomplemented
69#798—64no—
70D3×C16—64no—
71C23×C32C2×C811yescharacteristic
72C22×C3⋊D3—21no—
73C22×C3⋊D3—21no—
74C22×C3⋊D3—42no—
75C22×C3⋊D3C2×C811yescharacteristic
76C22×C3⋊D3C2×C811yescharacteristic
77C2×C32×C4—21no—
78C2×C32×C4—21no—
79C2×C32×C4C2×C811yescharacteristic
80C2×C32×C4C22×C411yescharacteristic
81C2×C3⋊Dic3—21no—
82C2×C3⋊Dic3—21no—
83C2×C3⋊Dic3C2×C811yescharacteristic
84C2×C3⋊Dic3C2×C811yescharacteristic
85C4×C3⋊D3—21no—
86C4×C3⋊D3—21no—
87C4×C3⋊D3—21no—
88C4×C3⋊D3—21no—
89C4×C3⋊D3—21no—
90C4×C3⋊D3—21no—
91C4×C3⋊D3C22⋊C411yescharacteristic
92C4×C3⋊D3C22⋊C411yescharacteristic
93C4×C3⋊D3M1611yescharacteristic
94C4×C3⋊D3M1611yescharacteristic
95C32×C8—21no—
96C32×C8C2×D412yes—
97C3⋊(C3⋊C8)—21no—
98C3⋊(C3⋊C8)C22⋊C412yes—
99C22×M16—91no—
100C2×M32—92no—
101#349—32no—
102#349—92no—
103C23×D3—32no—
104C23×D3—61no—
105C22×C3×C4—21no—
106C22×C3×C4C4×D312yes—
107C22×Dic3—32no—
108C22×Dic3—61no—
109C2×C4×D3—32no—
110C2×C4×D3—32no—
111C2×C4×D3—32no—
112C2×C4×D3—32no—
113C2×C4×D3—61no—
114C2×C4×D3—61no—
115C2×C4×D3—61no—
116C2×C4×D3—61no—
117C2×C4×D3—62no—
118C2×C4×D3—62no—
119C2×C4×D3—62no—
120C2×C4×D3—62no—
121C2×C4×D3—62no—
122C2×C4×D3—62no—
123C2×C4×D3—62no—
124C2×C4×D3—62no—
125C3×M16—24no—
126C3×M16—24no—
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C2×C3×C8C4×D312yes—
130C2×C3×C8C22×D312yes—
131#208—64no—
132#208—64no—
133C2×C3⋊C8—32no—
134C2×C3⋊C8—32no—
135C2×C3⋊C8—61no—
136C2×C3⋊C8—61no—
137ΓL(1,𝔽25)—64no—
138ΓL(1,𝔽25)—64no—
139ΓL(1,𝔽25)—64no—
140ΓL(1,𝔽25)—64no—
141D3×C8—34no—
142D3×C8—34no—
143D3×C8—62no—
144D3×C8—62no—
145D3×C8—62no—
146D3×C8—62no—
147D3×C8—62no—
148D3×C8—62no—
149C3×C16—64no—
150C3⋊C16—24no—
151C22×C32—21no—
152C22×C32—41no—
153C22×C32C22×C811yescharacteristic
154C2×C3⋊D3—22no—
155C2×C3⋊D3—21no—
156C2×C3⋊D3—42no—
157C2×C3⋊D3—42no—
158C2×C3⋊D3—41no—
159C2×C3⋊D3C22⋊C811yescharacteristic
160C2×C3⋊D3C22⋊C811yescharacteristic
161C32×C4—21no—
162C32×C4C2×C22⋊C411yescharacteristic
163C32×C4C2×M1611yescharacteristic
164C3⋊Dic3—21no—
165C3⋊Dic3C22⋊C811yescharacteristic
166C3⋊Dic3C22⋊C811yescharacteristic
167C23×C4—91no—
168C2×M16—91no—
169C2×M16—91no—
170C2×M16—91no—
171C2×M16—182no—
172C2×M16—182no—
173C2×M16D3211yescharacteristic
174C22×C8—91no—
175C22×C8—91no—
176M32—34no—
177M32—182no—
178C2×C16—182no—
179C23×C3—21no—
180C23×C3D3×C812yes—
181C22×D3—32no—
182C22×D3—32no—
183C22×D3—61no—
184C22×D3—61no—
185C22×D3—62no—
186C22×D3—62no—
187C22×D3—62no—
188C22×D3—62no—
189C22×D3—124no—
190C22×D3—124no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4D3×C812yes—
198C2×C3×C4C2×C4×D312yes—
199C2×Dic3—32no—
200C2×Dic3—32no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C4×D3—32no—
208C4×D3—32no—
209C4×D3—32no—
210C4×D3—32no—
211C4×D3—61no—
212C4×D3—61no—
213C4×D3—61no—
214C4×D3—61no—
215C4×D3—62no—
216C4×D3—62no—
217C4×D3—62no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—62no—
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—62no—
226C4×D3—62no—
227C3×C8—22no—
228C3×C8—22no—
229C3×C8—22no—
230C3×C8D3×D414yes—
231C3⋊C8—34no—
232C3⋊C8—62no—
233C3⋊C8—62no—
234C3⋊C8—62no—
235C2×C32—21no—
236C2×C32—41no—
237C2×C32C2×C22⋊C811yescharacteristic
238C3⋊D3—21no—
239C3⋊D3—41no—
240C3⋊D3#34912yescomplemented
241C24—91no—
242C22×C4—91no—
243C22×C4—91no—
244C22×C4—91no—
245C22×C4—91no—
246C22×C4—91no—
247C22×C4—181no—
248C22×C4—181no—
249C22×C4—181no—
250C22×C4—181no—
251C22×C4#62411yescharacteristic
252M16—22no—
253M16—182no—
254M16—182no—
255M16—182no—
256C2×C8—91no—
257C2×C8—92no—
258C2×C8—92no—
259C2×C8—91no—
260C2×C8—181no—
261C2×C8—181no—
262C2×C8#62411yescharacteristic
263C2×C8C2×D3211yescharacteristic
264C16—64no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—42no—
269C22×C3—42no—
270C22×C3C2×D3×C812yes—
271C2×D3—32no—
272C2×D3—32no—
273C2×D3—61no—
274C2×D3—61no—
275C2×D3—64no—
276C2×D3—64no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—124no—
280C2×D3—124no—
281C2×D3—124no—
282C2×D3—124no—
283C2×D3—122no—
284C2×D3—122no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4D3×C22⋊C412yes—
290C3×C4D3×M1612yes—
291Dic3—32no—
292Dic3—32no—
293Dic3—61no—
294Dic3—61no—
295Dic3—62no—
296Dic3—62no—
297C32C2×#34911yescharacteristic, complemented, Sylow
298C23—91no—
299C23—91no—
300C23—181no—
301C23—181no—
302C23—362no—
303C23#371011yescharacteristic
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—91no—
309C2×C4—91no—
310C2×C4—91no—
311C2×C4—91no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4—181no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4#371011yescharacteristic
321C2×C4C2×#62411yescharacteristic
322C8—21no—
323C8—92no—
324C8—181no—
325C8#381112yes—
326C2×C3—22no—
327C2×C3—22no—
328C2×C3—21no—
329C2×C3—42no—
330C2×C3—42no—
331C2×C3D3×C22⋊C812yes—
332D3—34no—
333D3—62no—
334D3—62no—
335D3—62no—
336D3—122no—
337D3—122no—
338C22—21no—
339C22—41no—
340C22—91no—
341C22—91no—
342C22—182no—
343C22—181no—
344C22—361no—
345C22—362no—
346C22—362no—
347C22C2×#371011yescharacteristic
348C4—21no—
349C4—91no—
350C4—91no—
351C4—181no—
352C4#6381811yescharacteristic
353C4#6397411yescharacteristic
354C3—21no—
355C3D3×#34912yescomplemented
356C2—21no—
357C2—41no—
358C2—92no—
359C2—181no—
360C2—361no—
361C2#1059115211yescharacteristic
subgroupG/H
centerC4#63974
commutatorC22×C32C22×C8
FrattiniC2×C8C2×D32
FittingC2×C32×M16C22
socleC2×C32C2×C22⋊C8
prime radicalC23×C3⋊D3C8
cosocleC2×C32×C8C23
hypercenterC2×M16D32
hypocenterC32C2×#349
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10590625C2semidirect
3. C3×#4553C2semidirect
4. #10589110C2semidirect
15. C3×#812C22semidirect
22. #4553D3semidirect
68. #812C2×D3semidirect
240. C3⋊D3#349semidirect
297. C32C2×#349semidirect
355. C3D3×#349semidirect

Maximal subgroups

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

subgroupindex
C2×M16×C3⋊D32
#105906252
C3×#45532
#105891102
D3×#3493