Finite Group Databasebeta

← #10591691

Subgroups — #10591691

Order 576.

438 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3757C211yesmaximal, characteristic
2C2×C3×#217C211yesmaximal, characteristic
3C2×C3×#217C211yesmaximal, characteristic
4C2×#3724C211yesmaximal, characteristic, complemented
5C2×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
6C2×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
7#63579C211yesmaximal, characteristic, complemented
8#4948—31nomaximal
9#4948—31nomaximal
10C24×C32C2211yescharacteristic
11C22×C3⋊Dic3C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C22×C3×Dic3C2211yescharacteristic
14C22×C3×Dic3C2211yescharacteristic
15C22×C3×Dic3C2211yescharacteristic
16C22×C3×Dic3C2211yescharacteristic
17#3757—22no—
18C3×#217—22no—
19C3×#217—22no—
20#3724—22no—
21Dic3⋊Dic3—22no—
22Dic3⋊Dic3—22no—
23C2×C3×C22⋊C4—31no—
24C2×C3×C22⋊C4—31no—
25C2×#217—31no—
26C2×#217—31no—
27C2×#217—61no—
28C2×#217D311yescharacteristic
29C2×#217D311yescharacteristic
30C2×C4⋊Dic3—31no—
31C2×C4⋊Dic3—31no—
32C2×Dic3⋊C4—31no—
33C2×Dic3⋊C4—31no—
34C2×Dic3⋊C4—31no—
35C2×Dic3⋊C4—31no—
36C3⋊#95—31no—
37C3⋊#95—31no—
38C23×C32—21no—
39C23×C32—21no—
40C23×C32—21no—
41C23×C32—21no—
42C23×C32—21no—
43C23×C32—21no—
44C23×C32Q812yes—
45C23×C32C2311yescharacteristic
46C2×C3⋊Dic3—21no—
47C2×C3⋊Dic3—21no—
48C2×C3⋊Dic3—21no—
49C2×C3⋊Dic3—21no—
50C2×C3⋊Dic3—21no—
51C2×C3⋊Dic3D412yes—
52C2×C3×Dic3—21no—
53C2×C3×Dic3—21no—
54C2×C3×Dic3—21no—
55C2×C3×Dic3—21no—
56C2×C3×Dic3—21no—
57C2×C3×Dic3—21no—
58C2×C3×Dic3—21no—
59C2×C3×Dic3—21no—
60C2×C3×Dic3—21no—
61C2×C3×Dic3—21no—
62C2×C3×Dic3D412yes—
63C2×C3×Dic3D412yes—
64#396—91noSylow
65C24×C3—21no—
66C24×C3C2×D311yescharacteristic
67C24×C3C2×D311yescharacteristic
68C22×C3×C4—31no—
69C22×C3×C4—31no—
70C22×C3×C4—31no—
71C22×C3×C4—31no—
72C22×Dic3—31no—
73C22×Dic3—31no—
74C22×Dic3—31no—
75C22×Dic3—31no—
76C22×Dic3—61no—
77C22×Dic3—61no—
78C22×Dic3C2×D311yescharacteristic
79C22×Dic3C2×D311yescharacteristic
80C22×Dic3C2×D311yescharacteristic
81C22×Dic3C2×D311yescharacteristic
82C3×C22⋊C4—62no—
83C3×C22⋊C4—62no—
84#217—22no—
85#217—22no—
86#217—62no—
87#217—62no—
88#217—64no—
89C4⋊Dic3—62no—
90C4⋊Dic3—62no—
91Dic3⋊C4—62no—
92Dic3⋊C4—62no—
93Dic3⋊C4—62no—
94Dic3⋊C4—62no—
95C22×C32—22no—
96C22×C32—22no—
97C22×C32—22no—
98C22×C32—41no—
99C22×C32—41no—
100C22×C32—41no—
101C22×C32—41no—
102C22×C32C2×D411yescharacteristic
103C22×C32C2×D411yescharacteristic
104C22×C32C2×D411yescharacteristic
105C22×C32C2×Q811yescharacteristic
106C22×C32#4111yescharacteristic
107C22×C32#4111yescharacteristic
108C22×C32#4111yescharacteristic
109C3⋊Dic3—22no—
110C3⋊Dic3—41no—
111C3×Dic3—22no—
112C3×Dic3—22no—
113C3×Dic3—41no—
114C3×Dic3—41no—
115C2×C4⋊C4—91no—
116C2×C4⋊C4—91no—
117C2×C4⋊C4—91no—
118C2×C22⋊C4—31no—
119C2×C22⋊C4—31no—
120C2×C22⋊C4—91no—
121#95—91no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—22no—
137C23×C3—21no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3—41no—
141C23×C3—41no—
142C23×C3Dic612yes—
143C23×C3Dic612yes—
144C23×C3C22×D311yescharacteristic
145C23×C3C22×D311yescharacteristic
146C2×C3×C4—32no—
147C2×C3×C4—32no—
148C2×C3×C4—61no—
149C2×C3×C4—61no—
150C2×C3×C4—61no—
151C2×C3×C4—61no—
152C2×C3×C4—61no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155C2×C3×C4—61no—
156C2×C3×C4—61no—
157C2×C3×C4—61no—
158C2×Dic3—21no—
159C2×Dic3—21no—
160C2×Dic3—21no—
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3—21no—
165C2×Dic3—21no—
166C2×Dic3—21no—
167C2×Dic3—21no—
168C2×Dic3—32no—
169C2×Dic3—32no—
170C2×Dic3—62no—
171C2×Dic3—61no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—62no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—61no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—121no—
183C2×Dic3—121no—
184C2×Dic3—121no—
185C2×Dic3—121no—
186C2×Dic3#6712yes—
187C2×Dic3#6712yes—
188C2×C32—42no—
189C2×C32#12111yescharacteristic
190C2×C32#12111yescharacteristic
191C2×C32#12111yescharacteristic
192C2×C32C22⋊Q811yescharacteristic
193C2×C32C22⋊Q811yescharacteristic
194C2×C32C22⋊Q811yescharacteristic
195C2×C32#12611yescharacteristic
196C24D3211yescharacteristic
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—91no—
202C22×C4—91no—
203C4⋊C4—182no—
204C4⋊C4—182no—
205C4⋊C4—182no—
206C22⋊C4—62no—
207C22⋊C4—62no—
208C22⋊C4—182no—
209C22×C3—24no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—22no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—42no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—42no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3C2×Dic611yescharacteristic
238C22×C3C2×Dic611yescharacteristic
239C22×C3#23511yescharacteristic
240C22×C3#23511yescharacteristic
241C22×C3D3×D411yescharacteristic
242C22×C3D3×D411yescharacteristic
243C22×C3D3×D411yescharacteristic
244C22×C3D3×D411yescharacteristic
245C22×C3#23711yescharacteristic
246C22×C3#23711yescharacteristic
247C22×C3#23711yescharacteristic
248C22×C3#23711yescharacteristic
249C22×C3C2×#6711yescharacteristic
250C22×C3C2×#6711yescharacteristic
251C3×C4—62no—
252C3×C4—62no—
253C3×C4—121no—
254C3×C4—121no—
255Dic3—22no—
256Dic3—22no—
257Dic3—41no—
258Dic3—41no—
259Dic3—62no—
260Dic3—62no—
261Dic3—122no—
262Dic3—122no—
263Dic3—121no—
264Dic3—121no—
265C32#39611yescharacteristic, complemented, Sylow
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23#62712yes—
273C23C2×D3211yescharacteristic
274C2×C4—32no—
275C2×C4—32no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—61no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—92no—
287C2×C4—181no—
288C2×C4—181no—
289C2×C4—181no—
290C2×C4—181no—
291C2×C4—181no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—44no—
300C2×C3—42no—
301C2×C3—42no—
302C2×C3#87811yescharacteristic
303C2×C3#87811yescharacteristic
304C2×C3#87811yescharacteristic
305C2×C3#87811yescharacteristic
306C2×C3#87911yescharacteristic
307C2×C3#87911yescharacteristic
308C2×C3#88411yescharacteristic
309C2×C3#88411yescharacteristic
310C2×C3#92411yescharacteristic
311C2×C3#92411yescharacteristic
312C2×C3#93811yescharacteristic
313C2×C3#93811yescharacteristic
314C2×C3#93911yescharacteristic
315C2×C3#93911yescharacteristic
316C22—22no—
317C22—22no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22#380411yescharacteristic
324C22#380411yescharacteristic
325C22#380511yescharacteristic
326C22C2×#62711yescharacteristic
327C22D3×#6711yescharacteristic
328C22D3×#6711yescharacteristic
329C22#381111yescharacteristic
330C4—62no—
331C4—62no—
332C4—121no—
333C4—121no—
334C4—182no—
335C4—361no—
336C3—21no—
337C3#494811yescharacteristic, complemented
338C3#494811yescharacteristic, complemented
339C2—42no—
340C2#6395611yescharacteristic
341C2#6395811yescharacteristic
342C2#6396411yescharacteristic
343C2#6396411yescharacteristic
344C2#6397011yescharacteristic
345C2#6397111yescharacteristic
346C2#6398211yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
Fitting, prime radicalC24×C32C22
hypercenterC24D32
hypocenterC32#396
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#3724C2semidirect
5. C2×Dic3⋊Dic3C2semidirect
6. C2×Dic3⋊Dic3C2semidirect
7. #63579C2semidirect
265. C32#396semidirect
337. C3#4948semidirect
338. C3#4948semidirect

Maximal subgroups

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

subgroupindex
C2×#37572
C2×C3×#2172
C2×#37242
C2×Dic3⋊Dic32
#635792
#49483