Finite Group Databasebeta

← #11775616

Subgroups — #11775616

Order 1152.

566 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 73 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×#63982C211yesmaximal, characteristic
2C2×C22⋊#622C211yesmaximal, characteristic
3#10591720C211yesmaximal, characteristic
4#10590072C214yesmaximal, complemented
5C23×C3⋊Dic3C2211yescharacteristic
6C22×#627C411yescharacteristic, complemented
7C22×#622—21no—
8C2×C3×#217—21no—
9C2×C3×#217—21no—
10#63982—21no—
11#63982—21no—
12#63982C412yescomplemented
13#63982C2212yes—
14C2×#3724—21no—
15C2×#3724C411yescharacteristic, complemented
16C2×Dic3⋊Dic3—21no—
17C22⋊#622—21no—
18C22⋊#622C2212yes—
19#63579C2212yes—
20C2×#878—61no—
21#4942—61no—
22C24×C32D411yescharacteristic
23C22×C3⋊Dic3—42no—
24C22×C3⋊Dic3C2×C411yescharacteristic
25C22×C3⋊Dic3C2×C411yescharacteristic
26C22×C3⋊Dic3D411yescharacteristic
27C22×C3⋊Dic3D411yescharacteristic
28C22×C3⋊Dic3D411yescharacteristic
29C22×C3⋊Dic3C2311yescharacteristic
30C22×C3×Dic3—21no—
31C22×C3×Dic3—21no—
32C22×C3×Dic3—22no—
33C2×#627—21no—
34C2×#627—42no—
35C2×#627C2×C412yes—
36C2×#622—22no—
37C2×#622—42no—
38C3×#217—22no—
39C3×#217—22no—
40C3×#217—42no—
41#3724—21no—
42#3724—42no—
43#3724C2×C412yes—
44Dic3⋊Dic3—22no—
45Dic3⋊Dic3—22no—
46#1520—91nomaximal, Sylow
47C23×Dic3—61no—
48C23×Dic3—61no—
49C22×Dic6—61no—
50C2×C3×C22⋊C4—61no—
51C2×C3×C22⋊C4—61no—
52C2×#217—21no—
53C2×#217—21no—
54C2×C4⋊Dic3—61no—
55C2×Dic3⋊C4—61no—
56C2×Dic3⋊C4—61no—
57C2×Dic3⋊C4—62no—
58#878—62no—
59#878—62no—
60#878—62no—
61#878—62no—
62C3⋊#95—62no—
63C23×C32—41no—
64C23×C32—41no—
65C23×C32C22⋊C411yescharacteristic
66C23×C32C22⋊C411yescharacteristic
67C23×C32SD1612yes—
68C23×C32Q1612yes—
69C23×C32C2×D411yescharacteristic
70C2×C3⋊Dic3—22no—
71C2×C3⋊Dic3—22no—
72C2×C3⋊Dic3—22no—
73C2×C3⋊Dic3—41no—
74C2×C3⋊Dic3—41no—
75C2×C3⋊Dic3—41no—
76C2×C3⋊Dic3C22⋊C411yescharacteristic
77C2×C3⋊Dic3C22⋊C411yescharacteristic
78C2×C3⋊Dic3C22×C411yescharacteristic
79C2×C3⋊Dic3C2×D411yescharacteristic
80C2×C3×Dic3—22no—
81C2×C3×Dic3—22no—
82C2×C3×Dic3—42no—
83C2×C3×Dic3—42no—
84C2×C3×Dic3—42no—
85C2×C3×Dic3—42no—
86C2×C3×Dic3—42no—
87#627—42no—
88#627—42no—
89#622—42no—
90C2×C22⋊Q8—91no—
91C2×C22⋊C8—91no—
92#380—91no—
93#328—94no—
94C24×C3—21no—
95C24×C3—21no—
96C22×C3×C4—61no—
97C22×C3×C4—61no—
98C22×C3×C4—62no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—22no—
102C22×Dic3—61no—
103C22×Dic3—61no—
104C22×Dic3—61no—
105C22×Dic3—61no—
106C22×Dic3—61no—
107C22×Dic3—61no—
108C22×Dic3—61no—
109C22×Dic3—61no—
110C22×Dic3—61no—
111C22×Dic3—61no—
112C22×Dic3—61no—
113C22×Dic3—61no—
114C22×Dic3—124no—
115C22×Dic3—124no—
116C2×Dic6—62no—
117C2×Dic6—62no—
118C2×Dic6—124no—
119C3×C22⋊C4—62no—
120C3×C22⋊C4—62no—
121C3×C22⋊C4—122no—
122#217—22no—
123#217—22no—
124#217—42no—
125C4⋊Dic3—62no—
126C4⋊Dic3—62no—
127Dic3⋊C4—62no—
128Dic3⋊C4—62no—
129Dic3⋊C4—62no—
130Dic3⋊C4—62no—
131Dic3⋊C4—124no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—42no—
139C22×C32—42no—
140C22×C32#10312yes—
141C22×C32#10312yes—
142C22×C32C2×C22⋊C411yescharacteristic
143C22×C32C2×SD1611yescharacteristic
144C22×C32C2×Q1611yescharacteristic
145C3⋊Dic3—21no—
146C3⋊Dic3—21no—
147C3⋊Dic3—21no—
148C3⋊Dic3#9912yes—
149C3×Dic3—42no—
150C3×Dic3—42no—
151C3×Dic3—82no—
152C22×Q8—91no—
153C23×C4—91no—
154C22×C8—181no—
155C22⋊Q8—92no—
156C22⋊Q8—92no—
157C22⋊Q8—181no—
158C22⋊Q8—181no—
159C2×C4⋊C4—91no—
160C2×C4⋊C4—181no—
161C2×C4⋊C4—181no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C22⋊C8—92no—
165C22⋊C8—181no—
166#95—92no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—22no—
174C23×C3—22no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—42no—
178C23×C3—42no—
179C23×C3—42no—
180C23×C3—42no—
181C2×C3×C4—62no—
182C2×C3×C4—62no—
183C2×C3×C4—122no—
184C2×C3×C4—122no—
185C2×C3×C4—122no—
186C2×C3×C4—122no—
187C2×C3×C4—122no—
188C2×Dic3—22no—
189C2×Dic3—22no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3—61no—
196C2×Dic3—61no—
197C2×Dic3—61no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202C2×Dic3—61no—
203C2×Dic3—64no—
204C2×Dic3—64no—
205C2×Dic3—122no—
206C2×Dic3—122no—
207C2×Dic3—122no—
208C2×Dic3—122no—
209C2×Dic3—122no—
210C2×Dic3—122no—
211C2×Dic3—122no—
212C2×Dic3—122no—
213C2×Dic3—122no—
214C2×Dic3—122no—
215Dic6—124no—
216Dic6—124no—
217C2×C32—21no—
218C2×C32—22no—
219C2×C32—21no—
220C2×C32—22no—
221C2×C32C2×#9911yescharacteristic
222C2×C32C2×#10311yescharacteristic
223C2×C32#42111yescharacteristic
224C24GO+(4,𝔽2)11yescharacteristic
225C2×Q8—92no—
226C2×Q8—181no—
227C2×Q8—362no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—62no—
231C22×C4—91no—
232C22×C4—91no—
233C22×C4—91no—
234C22×C4—91no—
235C22×C4—91no—
236C22×C4—91no—
237C22×C4—362no—
238C2×C8—182no—
239C2×C8—362no—
240C4⋊C4—92no—
241C4⋊C4—181no—
242C4⋊C4—182no—
243C4⋊C4—182no—
244C4⋊C4—362no—
245C22⋊C4—62no—
246C22⋊C4—62no—
247C22⋊C4—122no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—44no—
271C22×C3—44no—
272C22×C3—44no—
273C22×C3—44no—
274C3×C4—122no—
275C3×C4—122no—
276C3×C4—242no—
277Dic3—42no—
278Dic3—42no—
279Dic3—62no—
280Dic3—62no—
281Dic3—62no—
282Dic3—62no—
283Dic3—82no—
284Dic3—121no—
285Dic3—121no—
286Dic3—121no—
287Dic3—121no—
288C32#152011yescharacteristic, complemented, Sylow
289C23—41no—
290C23—41no—
291C23D32⋊C411yescharacteristic
292C23D32⋊C411yescharacteristic
293C23#377512yes—
294C23#377612yes—
295C23C2×GO+(4,𝔽2)11yescharacteristic
296Q8—362no—
297Q8—362no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4—182no—
310C2×C4—182no—
311C2×C4—182no—
312C2×C4—361no—
313C2×C4—361no—
314C2×C4—361no—
315C8—362no—
316C2×C3—22no—
317C2×C3—22no—
318C2×C3—24no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—42no—
328C2×C3—42no—
329C2×C3—42no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—42no—
337C22—42no—
338C22#6374012yes—
339C22#6374012yes—
340C22C2×D32⋊C411yescharacteristic
341C22C2×#377511yescharacteristic
342C22C2×#377611yescharacteristic
343C4—92no—
344C4—122no—
345C4—122no—
346C4—181no—
347C4—181no—
348C4—181no—
349C4—242no—
350C3—21no—
351C3—21no—
352C2—21no—
353C2—21no—
354C2—22no—
355C2—22no—
356C2#1059349011yescharacteristic
357C2C2×#6374011yescharacteristic
358C2#1059350611yescharacteristic
subgroupG/H
centerC22C2×D32⋊C4
commutatorC2×C3⋊Dic3C22×C4
FrattiniC23C2×GO+(4,𝔽2)
Fitting, prime radicalC24×C32D4
socleC22×C32C2×C22⋊C4
cosocleC22×C3⋊Dic3C23
hypercenterC24GO+(4,𝔽2)
hypocenterC32#1520
LCSG ▷ C2×C3⋊Dic3 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
4. #10590072C2semidirect
6. C22×#627C4semidirect
12. #63982C4semidirect
15. C2×#3724C4semidirect
288. C32#1520semidirect

Maximal subgroups

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

subgroupindex
C2×#639822
C2×C22⋊#6222
#105917202
#105900722
#15209