Finite Group Databasebeta

← #10594325

Subgroups — #10594325

Order 576.

458 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C4⋊C4×C3⋊D3C211yesmaximal, characteristic
2C3×#890C211yesmaximal, characteristic, complemented
3C3×#888C211yesmaximal, characteristic, complemented
4#63918C211yesmaximal, characteristic, complemented
5#63874C211yesmaximal, characteristic
6#63871C212yesmaximal
7#63869C211yesmaximal, characteristic, complemented
8#63865C212yesmaximal
9#63849C211yesmaximal, characteristic
10#63847C212yesmaximal, complemented
11#63839C212yesmaximal, complemented
12#5729—31nomaximal
13#5689—31nomaximal
14C2×C4×C3⋊D3C2212yes—
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C3×Dic6C2212yes—
17C2×#627C2212yes—
18C32×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊Dic3)C2211yescharacteristic
20#3750C2212yes—
21C3×C4⋊Dic3C2211yescharacteristic
22C3×C4⋊Dic3C2212yes—
23C3×C4⋊Dic3C2211yescharacteristic
24C3×Dic3⋊C4C2212yes—
25C3×Dic3⋊C4C2212yes—
26C3×C4×Dic3C2211yescharacteristic
27C3×C4×Dic3C2211yescharacteristic
28#3724C2211yescharacteristic
29#3724C2211yescharacteristic
30#3724C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34#3722C2211yescharacteristic
35#3722C2211yescharacteristic
36Dic32C2212yes—
37C3×C4⋊Q8—31no—
38C3×#125—31no—
39#946—32no—
40#944—32no—
41#897—32no—
42#896—32no—
43#896—32no—
44#892—31no—
45#892—31no—
46D3×C4⋊C4—31no—
47D3×C4⋊C4—31no—
48D3×C4⋊C4—61no—
49#890—31no—
50#890D311yescharacteristic, complemented
51#889—32no—
52#889—32no—
53#888—32no—
54#888—31no—
55#888D311yescharacteristic, complemented
56#872—31no—
57#872—31no—
58#870—31no—
59#869—31no—
60C22×C3⋊D3C2311yescharacteristic
61C2×C32×C4C2312yes—
62C2×C32×C4C2311yescharacteristic
63C2×C3⋊Dic3C2312yes—
64C2×C3⋊Dic3C2311yescharacteristic
65C4×C3⋊D3—42no—
66C4×C3⋊D3Q814yescomplemented
67C2×C3×Dic3C2312yes—
68C2×C3×Dic3C2312yes—
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C3×Dic6—42no—
74#627—42no—
75#544—91noSylow
76C2×C3×Q8—32no—
77C2×C4×D3—31no—
78C2×C4×D3—31no—
79C2×C4×D3—32no—
80C2×C4×D3—32no—
81C2×C4×D3—61no—
82C2×C4×D3—62no—
83C2×Dic6—32no—
84C2×Dic6—32no—
85C2×Dic6C2×D312yes—
86C3×C4⋊C4—21no—
87C3×C4⋊C4—31no—
88C3×C4⋊C4—32no—
89C3×C4⋊C4—32no—
90C3×C4⋊C4—32no—
91C3×C4⋊C4—31no—
92C3×C4⋊C4C2×D311yescharacteristic
93C3×C4⋊C4C2×D311yescharacteristic
94C3×C42—31no—
95C3×C42—31no—
96(C2×D3)⋊C4—31no—
97(C2×D3)⋊C4—31no—
98(C2×D3)⋊C4—31no—
99(C2×D3)⋊C4—31no—
100(C2×D3)⋊C4—32no—
101(C2×D3)⋊C4—32no—
102C4⋊Dic3—31no—
103C4⋊Dic3—31no—
104C4⋊Dic3—32no—
105C4⋊Dic3—32no—
106C4⋊Dic3—61no—
107C4⋊Dic3C2×D311yescharacteristic
108C4⋊Dic3C2×D311yescharacteristic
109C4⋊Dic3C2×D312yes—
110Dic3⋊C4—32no—
111Dic3⋊C4—32no—
112Dic3⋊C4—32no—
113Dic3⋊C4—32no—
114Dic3⋊C4—31no—
115Dic3⋊C4—31no—
116Dic3⋊C4—31no—
117Dic3⋊C4—31no—
118Dic3⋊C4—32no—
119Dic3⋊C4—32no—
120Dic3⋊C4—62no—
121Dic3⋊C4C2×D312yes—
122Dic3⋊C4C2×D312yes—
123C4×Dic3—32no—
124C4×Dic3—32no—
125C4×Dic3C2×D311yescharacteristic
126C4×Dic3C2×D311yescharacteristic
127C22×C32C2411yescharacteristic
128C2×C3⋊D3—21no—
129C2×C3⋊D3—21no—
130C2×C3⋊D3C2×Q812yes—
131C32×C4—22no—
132C32×C4C2×Q812yes—
133C3⋊Dic3—22no—
134C3⋊Dic3C2×Q812yes—
135C3×Dic3—21no—
136C3×Dic3—22no—
137C3×Dic3—21no—
138C3×Dic3—21no—
139C3×Dic3—22no—
140C3×Dic3—21no—
141C4⋊Q8—31no—
142C4⋊Q8—91no—
143C4⋊Q8—92no—
144#125—31no—
145#125—91no—
146#125—92no—
147C22⋊Q8—92no—
148C22⋊Q8—92no—
149#117—91no—
150#117—91no—
151C2×C4⋊C4—91no—
152C22×D3—31no—
153C22×D3—31no—
154C22×D3—61no—
155C3×Q8—122no—
156C2×C3×C4—21no—
157C2×C3×C4—22no—
158C2×C3×C4—32no—
159C2×C3×C4—32no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4C22×D312yes—
165C2×C3×C4C22×D312yes—
166C2×C3×C4C22×D311yescharacteristic
167C2×C3×C4C22×D311yescharacteristic
168C2×Dic3—31no—
169C2×Dic3—31no—
170C2×Dic3—32no—
171C2×Dic3—32no—
172C2×Dic3—61no—
173C2×Dic3—62no—
174C2×Dic3C22×D312yes—
175C2×Dic3C22×D312yes—
176C2×Dic3C22×D311yescharacteristic
177C2×Dic3C22×D311yescharacteristic
178C2×Dic3C22×D311yescharacteristic
179C2×Dic3C22×D311yescharacteristic
180C4×D3—34no—
181C4×D3—34no—
182C4×D3—64no—
183C4×D3—122no—
184C4×D3—122no—
185C4×D3—124no—
186Dic6—42no—
187Dic6—122no—
188Dic6—122no—
189C2×C32C22×Q811yescharacteristic
190C2×C32H(5,𝔽2)11yescharacteristic
191C2×C322−1+411yescharacteristic
192C3⋊D3—22no—
193C2×Q8—32no—
194C2×Q8—92no—
195C22×C4—91no—
196C22×C4—92no—
197C4⋊C4—31no—
198C4⋊C4—31no—
199C4⋊C4—32no—
200C4⋊C4—32no—
201C4⋊C4—32no—
202C4⋊C4—91no—
203C4⋊C4—91no—
204C4⋊C4—92no—
205C4⋊C4—92no—
206C4⋊C4—92no—
207C4⋊C4—92no—
208C4⋊C4—91no—
209C4⋊C4D3211yescharacteristic
210C22⋊C4—91no—
211C22⋊C4—91no—
212C22⋊C4—92no—
213C42—31no—
214C42—31no—
215C42—92no—
216C22×C3—21no—
217C22×C3C23×D311yescharacteristic
218C22×C3C23×D311yescharacteristic
219C2×D3—32no—
220C2×D3—32no—
221C2×D3—62no—
222C2×D3—62no—
223C2×D3—62no—
224C2×D3—61no—
225C2×D3—61no—
226C2×D3—61no—
227C2×D3—61no—
228C3×C4—22no—
229C3×C4—22no—
230C3×C4—22no—
231C3×C4—42no—
232C3×C4—61no—
233C3×C4—62no—
234C3×C4—61no—
235C3×C4—62no—
236C3×C4—61no—
237C3×C4—61no—
238C3×C4D3×Q812yes—
239C3×C4D3×Q812yes—
240Dic3—21no—
241Dic3—22no—
242Dic3—21no—
243Dic3—21no—
244Dic3—22no—
245Dic3—21no—
246Dic3—32no—
247Dic3—32no—
248Dic3—62no—
249Dic3—62no—
250Dic3—62no—
251Dic3—122no—
252C32#54411yescharacteristic, complemented, Sylow
253C23—91no—
254Q8—122no—
255Q8—362no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—91no—
263C2×C4—94no—
264C2×C4—92no—
265C2×C4—362no—
266C2×C4C2×D3212yes—
267C2×C4C2×D3211yescharacteristic
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3#100411yescharacteristic
272C2×C3C2×D3×Q811yescharacteristic
273C2×C3C2×D3×Q811yescharacteristic
274C2×C3#100711yescharacteristic
275C2×C3#100911yescharacteristic
276C2×C3#101011yescharacteristic
277D3—64no—
278D3—62no—
279D3—62no—
280C22—92no—
281C22—181no—
282C22—181no—
283C22C22×D3211yescharacteristic
284C4—22no—
285C4—61no—
286C4—62no—
287C4—61no—
288C4—61no—
289C4—61no—
290C4—62no—
291C4—92no—
292C4—182no—
293C4#379712yes—
294C3—21no—
295C3#568911yescharacteristic, complemented
296C3#572911yescharacteristic, complemented
297C2—182no—
298C2C2×#379711yescharacteristic
299C2#6431411yescharacteristic
300C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC22×C3⋊D3C23
hypercenterC4⋊C4D32
hypocenterC32#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#890C2semidirect
3. C3×#888C2semidirect
4. #63918C2semidirect
7. #63869C2semidirect
10. #63847C2semidirect
11. #63839C2semidirect
50. #890D3semidirect
55. #888D3semidirect
66. C4×C3⋊D3Q8semidirect
252. C32#544semidirect
295. C3#5689semidirect
296. C3#5729semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊D32
C3×#8902
C3×#8882
#639182
#638742
#638712
#638692
#638652
#638492
#638472
#638392
#57293
#56893