Finite Group Databasebeta

← #10591298

Subgroups — #10591298

Order 576.

386 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3750C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic
4C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic
5C3×C3⋊#95C211yesmaximal, characteristic, complemented
6#63579C211yesmaximal, characteristic, complemented
7#63579C211yesmaximal, characteristic, complemented
8#4952—31nomaximal
9#4675—31nomaximal
10C23×C3×D3C2211yescharacteristic
11C22×C32×C4C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C22×C3⋊Dic3C2211yescharacteristic
14C22×C3×Dic3C2211yescharacteristic
15C22×C3×Dic3C2211yescharacteristic
16C22×C3×Dic3C2211yescharacteristic
17#3750—22no—
18C3×(C2×D3)⋊C4—22no—
19(C2×D3)⋊Dic3—22no—
20(C2×D3)⋊Dic3—22no—
21C2×C3×C22⋊C4—31no—
22C2×#217—31no—
23C2×#217—31no—
24C2×(C2×D3)⋊C4—31no—
25C2×(C2×D3)⋊C4—31no—
26C2×(C2×D3)⋊C4D311yescharacteristic
27C2×Dic3⋊C4—31no—
28C2×Dic3⋊C4—31no—
29C2×Dic3⋊C4—61no—
30C3×#95—31no—
31C3⋊#95—31no—
32C3⋊#95—31no—
33C3⋊#95—31no—
34C3⋊#95—31no—
35C3⋊#95D311yescharacteristic, complemented
36C23×C32C2311yescharacteristic
37C22×C3×D3—21no—
38C22×C3×D3—21no—
39C22×C3×D3—21no—
40C22×C3×D3—21no—
41C22×C3×D3—21no—
42C22×C3×D3—21no—
43C22×C3×D3D412yes—
44C2×C32×C4—21no—
45C2×C32×C4—21no—
46C2×C32×C4—21no—
47C2×C3⋊Dic3—21no—
48C2×C3⋊Dic3—21no—
49C2×C3⋊Dic3—21no—
50C2×C3⋊Dic3—21no—
51C2×C3⋊Dic3—21no—
52C2×C3⋊Dic3D412yes—
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—
62#397—91noSylow
63C24×C3—31no—
64C23×D3C2×D311yescharacteristic
65C22×C3×C4—21no—
66C22×C3×C4—31no—
67C22×C3×C4—31no—
68C22×C3×C4—31no—
69C22×C3×C4C2×D311yescharacteristic
70C22×C3×C4C2×D311yescharacteristic
71C22×Dic3—31no—
72C22×Dic3—31no—
73C22×Dic3—31no—
74C22×Dic3—31no—
75C22×Dic3—61no—
76C22×Dic3—61no—
77C22×Dic3C2×D311yescharacteristic
78C22×Dic3C2×D311yescharacteristic
79C22×Dic3C2×D311yescharacteristic
80C3×C22⋊C4—62no—
81#217—62no—
82#217—62no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—62no—
85(C2×D3)⋊C4—62no—
86Dic3⋊C4—62no—
87Dic3⋊C4—62no—
88Dic3⋊C4—64no—
89C22×C32C2×D411yescharacteristic
90C22×C32C2×D411yescharacteristic
91C22×C32#4111yescharacteristic
92C22×C32#4111yescharacteristic
93C22×C32#4111yescharacteristic
94C22×C32#4111yescharacteristic
95C22×C32#4111yescharacteristic
96C2×C3×D3—22no—
97C2×C3×D3—22no—
98C2×C3×D3—22no—
99C2×C3×D3—41no—
100C2×C3×D3—41no—
101C2×C3×D3—41no—
102C2×C3×D3—41no—
103C32×C4—41no—
104C3⋊Dic3—22no—
105C3⋊Dic3—41no—
106C3×Dic3—41no—
107C3×Dic3—41no—
108C3×Dic3—41no—
109C2×C4⋊C4—91no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—91no—
112C2×C22⋊C4—91no—
113#95—31no—
114#95—91no—
115#95—91no—
116C23×C3—21no—
117C23×C3—32no—
118C23×C3—61no—
119C23×C3—61no—
120C23×C3—61no—
121C23×C3—61no—
122C23×C3—61no—
123C23×C3—61no—
124C23×C3C22×D311yescharacteristic
125C23×C3C22×D311yescharacteristic
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3—21no—
131C22×D3—21no—
132C22×D3#6712yes—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—41no—
141C2×C3×C4—41no—
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×Dic3—21no—
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—32no—
161C2×Dic3—32no—
162C2×Dic3—62no—
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—61no—
166C2×Dic3—61no—
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—62no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—121no—
177C2×Dic3—121no—
178C2×C32#12111yescharacteristic
179C2×C32#12311yescharacteristic
180C2×C32#12311yescharacteristic
181C2×C32#12311yescharacteristic
182C2×C32#12411yescharacteristic
183C2×C32#12611yescharacteristic
184C2×C32#12611yescharacteristic
185C3×D3—42no—
186C24—31no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—91no—
191C22×C4—91no—
192C22×C4D3211yescharacteristic
193C4⋊C4—182no—
194C22⋊C4—62no—
195C22⋊C4—182no—
196C22⋊C4—182no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—62no—
205C22×C3—62no—
206C22×C3—62no—
207C22×C3—121no—
208C22×C3—121no—
209C22×C3—121no—
210C22×C3—121no—
211C22×C3#23511yescharacteristic
212C22×C3#23511yescharacteristic
213C22×C3#23511yescharacteristic
214C22×C3#23511yescharacteristic
215C22×C3#23511yescharacteristic
216C22×C3D3×D411yescharacteristic
217C22×C3D3×D411yescharacteristic
218C22×C3D3×D411yescharacteristic
219C22×C3#23711yescharacteristic
220C22×C3#23711yescharacteristic
221C22×C3#23711yescharacteristic
222C22×C3#23711yescharacteristic
223C22×C3#23911yescharacteristic
224C22×C3C2×#6711yescharacteristic
225C2×D3—22no—
226C2×D3—22no—
227C2×D3—22no—
228C2×D3—41no—
229C2×D3—41no—
230C2×D3—41no—
231C2×D3—41no—
232C3×C4—42no—
233C3×C4—41no—
234C3×C4—41no—
235C3×C4—121no—
236C3×C4—121no—
237C3×C4—121no—
238Dic3—41no—
239Dic3—41no—
240Dic3—41no—
241Dic3—64no—
242Dic3—62no—
243Dic3—62no—
244Dic3—122no—
245Dic3—121no—
246Dic3—121no—
247C32#39711yescharacteristic, complemented, Sylow
248C23—32no—
249C23—61no—
250C23—61no—
251C23—61no—
252C23—61no—
253C23—61no—
254C23—61no—
255C23C2×D3211yescharacteristic
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—92no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—181no—
273C2×C4—181no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—122no—
282C2×C3#87611yescharacteristic
283C2×C3#87911yescharacteristic
284C2×C3#87911yescharacteristic
285C2×C3#88311yescharacteristic
286C2×C3#88311yescharacteristic
287C2×C3#88411yescharacteristic
288C2×C3#88511yescharacteristic
289C2×C3#88511yescharacteristic
290C2×C3#89411yescharacteristic
291C2×C3#89411yescharacteristic
292C2×C3#89811yescharacteristic
293C2×C3#92911yescharacteristic
294C2×C3#93511yescharacteristic
295C2×C3#93911yescharacteristic
296D3—42no—
297C22—62no—
298C22—62no—
299C22—62no—
300C22—121no—
301C22—121no—
302C22—121no—
303C22—121no—
304C22#379511yescharacteristic
305C22#379511yescharacteristic
306C22#379611yescharacteristic
307C22#379811yescharacteristic
308C22#380411yescharacteristic
309C22D3×#6711yescharacteristic
310C22#381111yescharacteristic
311C4—41no—
312C4—121no—
313C4—121no—
314C4—121no—
315C4—182no—
316C4—361no—
317C3—21no—
318C3#467511yescharacteristic, complemented
319C3#495211yescharacteristic, complemented
320C2—122no—
321C2#6385311yescharacteristic
322C2#6385811yescharacteristic
323C2#6385911yescharacteristic
324C2#6388411yescharacteristic
325C2#6390511yescharacteristic
326C2#6391311yescharacteristic
327C2#6397111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC23×C3×D3C22
hypercenterC22×C4D32
hypocenterC32#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3750C2semidirect
5. C3×C3⋊#95C2semidirect
6. #63579C2semidirect
7. #63579C2semidirect
35. C3⋊#95D3semidirect
247. C32#397semidirect
318. C3#4675semidirect
319. C3#4952semidirect

Maximal subgroups

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

subgroupindex
C2×#37502
C2×C3×(C2×D3)⋊C42
C2×(C2×D3)⋊Dic32
C3×C3⋊#952
#635792
#49523
#46753