Finite Group Databasebeta

← #10591717

Subgroups — #10591717

Order 576.

406 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 100 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×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
5#63579C211yesmaximal, characteristic, complemented
6#63579C211yesmaximal, characteristic, complemented
7#63579C211yesmaximal, characteristic, complemented
8#4953—31nomaximal
9#4952—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—
20Dic3⋊Dic3—22no—
21C2×C3×C22⋊C4—31no—
22C2×C3×C22⋊C4—31no—
23C2×#217—31no—
24C2×#217—31no—
25C2×#217—61no—
26C2×#217D311yescharacteristic
27C2×#217D311yescharacteristic
28C2×C4⋊Dic3—31no—
29C2×Dic3⋊C4—31no—
30C3⋊#95—31no—
31C3⋊#95—31no—
32C3⋊#95—31no—
33C3⋊#95—31no—
34C3⋊#95—31no—
35C3⋊#95—31no—
36C23×C32—21no—
37C23×C32—21no—
38C23×C32—21no—
39C23×C32—21no—
40C23×C32—21no—
41C23×C32—21no—
42C23×C32D412yes—
43C23×C32C2311yescharacteristic
44C2×C3⋊Dic3—21no—
45C2×C3⋊Dic3—21no—
46C2×C3⋊Dic3—21no—
47C2×C3⋊Dic3—21no—
48C2×C3⋊Dic3—21no—
49C2×C3⋊Dic3—21no—
50C2×C3×Dic3—21no—
51C2×C3×Dic3—21no—
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×Dic3D412yes—
62#397—91noSylow
63C24×C3—21no—
64C24×C3C2×D311yescharacteristic
65C24×C3C2×D311yescharacteristic
66C22×C3×C4—31no—
67C22×C3×C4—31no—
68C22×C3×C4—31no—
69C22×C3×C4—31no—
70C22×Dic3—31no—
71C22×Dic3—31no—
72C22×Dic3—31no—
73C22×Dic3—31no—
74C22×Dic3—61no—
75C22×Dic3—61no—
76C22×Dic3C2×D311yescharacteristic
77C22×Dic3C2×D311yescharacteristic
78C22×Dic3C2×D311yescharacteristic
79C22×Dic3C2×D311yescharacteristic
80C3×C22⋊C4—62no—
81C3×C22⋊C4—62no—
82#217—22no—
83#217—22no—
84#217—64no—
85#217—62no—
86#217—62no—
87C4⋊Dic3—62no—
88Dic3⋊C4—62no—
89C22×C32—22no—
90C22×C32—22no—
91C22×C32—22no—
92C22×C32—41no—
93C22×C32—41no—
94C22×C32—41no—
95C22×C32—41no—
96C22×C32C2×D411yescharacteristic
97C22×C32C2×D411yescharacteristic
98C22×C32#4111yescharacteristic
99C22×C32#4111yescharacteristic
100C22×C32#4111yescharacteristic
101C22×C32#4111yescharacteristic
102C22×C32#4111yescharacteristic
103C3⋊Dic3—41no—
104C3⋊Dic3—41no—
105C3×Dic3—22no—
106C3×Dic3—41no—
107C3×Dic3—41no—
108C3×Dic3—41no—
109C2×C4⋊C4—91no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—91no—
113#95—91no—
114#95—91no—
115#95—91no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—22no—
131C23×C3—21no—
132C23×C3—41no—
133C23×C3—41no—
134C23×C3—41no—
135C23×C3—41no—
136C23×C3D1212yes—
137C23×C3#6712yes—
138C23×C3C22×D311yescharacteristic
139C23×C3C22×D311yescharacteristic
140C2×C3×C4—32no—
141C2×C3×C4—61no—
142C2×C3×C4—61no—
143C2×C3×C4—61no—
144C2×C3×C4—61no—
145C2×C3×C4—61no—
146C2×C3×C4—61no—
147C2×C3×C4—61no—
148C2×C3×C4—61no—
149C2×C3×C4—61no—
150C2×C3×C4—61no—
151C2×C3×C4—61no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—21no—
160C2×Dic3—21no—
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—61no—
166C2×Dic3—61no—
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—121no—
178C2×Dic3—121no—
179C2×Dic3—121no—
180C2×Dic3—121no—
181C2×Dic3#6712yes—
182C2×C32—42no—
183C2×C32#12111yescharacteristic
184C2×C32#12311yescharacteristic
185C2×C32#12311yescharacteristic
186C2×C32#12311yescharacteristic
187C2×C32#12411yescharacteristic
188C2×C32#12611yescharacteristic
189C2×C32#12611yescharacteristic
190C24D3211yescharacteristic
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—91no—
196C22×C4—91no—
197C4⋊C4—182no—
198C22⋊C4—62no—
199C22⋊C4—62no—
200C22⋊C4—182no—
201C22×C3—24no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—42no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3C2×D1211yescharacteristic
230C22×C3#23511yescharacteristic
231C22×C3#23511yescharacteristic
232C22×C3#23511yescharacteristic
233C22×C3D3×D411yescharacteristic
234C22×C3#23711yescharacteristic
235C22×C3#23711yescharacteristic
236C22×C3#23711yescharacteristic
237C22×C3#23711yescharacteristic
238C22×C3#23711yescharacteristic
239C22×C3#23711yescharacteristic
240C22×C3#23711yescharacteristic
241C22×C3C2×#6711yescharacteristic
242C22×C3C2×#6711yescharacteristic
243C3×C4—62no—
244C3×C4—121no—
245C3×C4—121no—
246C3×C4—121no—
247Dic3—22no—
248Dic3—41no—
249Dic3—41no—
250Dic3—41no—
251Dic3—122no—
252Dic3—122no—
253Dic3—121no—
254Dic3—121no—
255Dic3—121no—
256Dic3—121no—
257C32#39711yescharacteristic, complemented, Sylow
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23#62612yes—
265C23C2×D3211yescharacteristic
266C2×C4—32no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—44no—
292C2×C3—42no—
293C2×C3—42no—
294C2×C3#87911yescharacteristic
295C2×C3#87911yescharacteristic
296C2×C3#87911yescharacteristic
297C2×C3#87911yescharacteristic
298C2×C3#88311yescharacteristic
299C2×C3#88511yescharacteristic
300C2×C3#88611yescharacteristic
301C2×C3#88611yescharacteristic
302C2×C3#92911yescharacteristic
303C2×C3#93011yescharacteristic
304C2×C3#93511yescharacteristic
305C2×C3#93511yescharacteristic
306C2×C3#93611yescharacteristic
307C2×C3#93911yescharacteristic
308C22—22no—
309C22—22no—
310C22—22no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22#380411yescharacteristic
316C22#380411yescharacteristic
317C22#380411yescharacteristic
318C22#380511yescharacteristic
319C22#380511yescharacteristic
320C22C2×#62611yescharacteristic
321C22D3×#6711yescharacteristic
322C4—62no—
323C4—121no—
324C4—121no—
325C4—121no—
326C4—361no—
327C4—361no—
328C3—21no—
329C3#495211yescharacteristic, complemented
330C3#495311yescharacteristic, complemented
331C2—42no—
332C2#6395611yescharacteristic
333C2#6395611yescharacteristic
334C2#6395911yescharacteristic
335C2#6396211yescharacteristic
336C2#6396611yescharacteristic
337C2#6396911yescharacteristic
338C2#6397811yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
Fitting, prime radicalC24×C32C22
hypercenterC24D32
hypocenterC32#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊Dic3C2semidirect
5. #63579C2semidirect
6. #63579C2semidirect
7. #63579C2semidirect
257. C32#397semidirect
329. C3#4952semidirect
330. C3#4953semidirect

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×Dic3⋊Dic32
#635792
#49533
#49523