Finite Group Databasebeta

← #10595124

Subgroups — #10595124

Order 576.

470 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 128 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
1#64145C211yesmaximal, characteristic, complemented
2C3×#938C211yesmaximal, characteristic
3C3×D4×Dic3C211yesmaximal, characteristic, complemented
4#63964C212yesmaximal, complemented
5#63961C212yesmaximal, complemented
6#63959C212yesmaximal, complemented
7#63956C212yesmaximal, complemented
8#63919C211yesmaximal, characteristic, complemented
9#63857C211yesmaximal, characteristic, complemented
10#63855C211yesmaximal, characteristic, complemented
11#63844C211yesmaximal, characteristic, complemented
12#5603—31nomaximal
13#5561—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×#67C2212yes—
17C22×C3×Dic3C2212yes—
18C2×C3×C4×D3C2211yescharacteristic
19C2×#627C2212yes—
20#3757C2212yes—
21#3757C2212yes—
22C4×C3⋊Dic3C2211yescharacteristic
23C3×#217C2212yes—
24C3×#217C2212yes—
25C3×C4⋊Dic3C2211yescharacteristic
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4×Dic3C2211yescharacteristic
28#3724C2212yes—
29#3724C2212yes—
30Dic3⋊Dic3C2211yescharacteristic
31Dic3⋊Dic3C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2211yescharacteristic
34(C2×D3)⋊Dic3C2211yescharacteristic
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×#121—31no—
38C3×C4×D4—31no—
39#938D311yescharacteristic
40#936—31no—
41#936—31no—
42#936—32no—
43#936—61no—
44#935—32no—
45D4×Dic3D311yescharacteristic, complemented
46#929—32no—
47#926—31no—
48#924—32no—
49#924—31no—
50#897—31no—
51#888—31no—
52#888—31no—
53#887—31no—
54#885—32no—
55#879—32no—
56#879—32no—
57#878—32no—
58#872—31no—
59C4×Dic6—31no—
60C23×C32C2312yes—
61C22×C3×D3C2311yescharacteristic
62C32×D4—22no—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2312yes—
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67#634—22no—
68C3×#67—42no—
69C2×C3×Dic3—42no—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2312yes—
74C2×C3×Dic3C2311yescharacteristic
75C3×C4×D3—22no—
76#627—42no—
77#542—91noSylow
78C2×C3×D4—21no—
79C2×C3×D4—32no—
80C2×C3×D4C2×D311yescharacteristic
81C2×C3×D4C2×D311yescharacteristic
82C22×C3×C4—32no—
83C22×C3×C4—31no—
84C2×#67C2×D312yes—
85C22×Dic3C2×D312yes—
86C2×C4×D3C2×D311yescharacteristic
87C2×Dic6—31no—
88C2×Dic6—31no—
89C2×Dic6—32no—
90C2×Dic6—32no—
91C2×Dic6—61no—
92C3×C4⋊C4—31no—
93C3×C4⋊C4—31no—
94C3×C22⋊C4—32no—
95C3×C22⋊C4—32no—
96C3×C42—31no—
97#217—32no—
98#217—32no—
99#217—32no—
100#217—32no—
101#217—32no—
102#217—31no—
103#217—31no—
104#217—62no—
105#217—62no—
106#217C2×D312yes—
107#217C2×D312yes—
108(C2×D3)⋊C4—31no—
109(C2×D3)⋊C4—31no—
110(C2×D3)⋊C4—32no—
111C4⋊Dic3—31no—
112C4⋊Dic3—31no—
113C4⋊Dic3—32no—
114C4⋊Dic3C2×D311yescharacteristic
115C4⋊Dic3C2×D311yescharacteristic
116Dic3⋊C4—32no—
117Dic3⋊C4—31no—
118Dic3⋊C4—31no—
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—32no—
123C4×Dic3—31no—
124C4×Dic3—31no—
125C4×Dic3—32no—
126C4×Dic3—32no—
127C4×Dic3—61no—
128C4×Dic3C2×D311yescharacteristic
129C22×C32—22no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32C2411yescharacteristic
133C2×C3×D3—21no—
134C2×C3×D3—21no—
135C2×C3×D3—21no—
136C32×C4#4112yes—
137C3⋊Dic3—22no—
138C3⋊Dic3—21no—
139C3⋊Dic3—21no—
140C3×Dic3—21no—
141C3×Dic3—22no—
142C3×Dic3—22no—
143C3×Dic3—21no—
144C3×Dic3—21no—
145C4⋊Q8—91no—
146#126—92no—
147#124—92no—
148#124—91no—
149#123—92no—
150C22⋊Q8—92no—
151C22⋊Q8—91no—
152#121—31no—
153C4×Q8—91no—
154C4×D4—31no—
155#117—91no—
156C23×C3—22no—
157C23×C3—31no—
158C23×C3C22×D312yes—
159C23×C3C22×D312yes—
160C22×D3C22×D311yescharacteristic
161C3×D4—22no—
162C3×D4—22no—
163C3×D4—42no—
164C3×D4—122no—
165C2×C3×C4—21no—
166C2×C3×C4—31no—
167C2×C3×C4—32no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—32no—
171C2×C3×C4—62no—
172C2×C3×C4—122no—
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175#67—42no—
176C2×Dic3—31no—
177C2×Dic3—31no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—31no—
181C2×Dic3—31no—
182C2×Dic3—42no—
183C2×Dic3—61no—
184C2×Dic3—61no—
185C2×Dic3—62no—
186C2×Dic3C22×D311yescharacteristic
187C2×Dic3C22×D312yes—
188C2×Dic3C22×D311yescharacteristic
189C2×Dic3C22×D311yescharacteristic
190C2×Dic3C22×D312yes—
191C4×D3—22no—
192Dic6—62no—
193Dic6—62no—
194Dic6—122no—
195Dic6—122no—
196Dic6—122no—
197C2×C32—42no—
198C2×C32C2×#4111yescharacteristic
199C2×C32H(5,𝔽2)11yescharacteristic
200C2×C322−1+411yescharacteristic
201C3×D3—41no—
202C2×Q8—92no—
203C2×Q8—91no—
204C2×D4—32no—
205C2×D4D3211yescharacteristic
206C22×C4—32no—
207C22×C4—31no—
208C4⋊C4—31no—
209C4⋊C4—31no—
210C4⋊C4—91no—
211C4⋊C4—91no—
212C4⋊C4—92no—
213C4⋊C4—92no—
214C4⋊C4—92no—
215C22⋊C4—32no—
216C22⋊C4—32no—
217C22⋊C4—91no—
218C22⋊C4—91no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C22⋊C4—92no—
222C42—31no—
223C42—92no—
224C42—91no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—21no—
232C22×C3—42no—
233C22×C3—42no—
234C22×C3—42no—
235C22×C3—61no—
236C22×C3—61no—
237C22×C3—61no—
238C22×C3C23×D311yescharacteristic
239C22×C3C23×D311yescharacteristic
240C2×D3—21no—
241C2×D3—21no—
242C2×D3—21no—
243C3×C4—22no—
244C3×C4—61no—
245C3×C4—62no—
246C3×C4—62no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4#23512yes—
250C3×C4#23712yes—
251Dic3—21no—
252Dic3—22no—
253Dic3—22no—
254Dic3—21no—
255Dic3—21no—
256Dic3—62no—
257Dic3—62no—
258Dic3—61no—
259Dic3—61no—
260Dic3—61no—
261Dic3—61no—
262Dic3—122no—
263Dic3—121no—
264Dic3—121no—
265C32#54211yescharacteristic, complemented, Sylow
266C23—31no—
267C23C2×D3212yes—
268Q8—182no—
269Q8—362no—
270D4—22no—
271D4—122no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—32no—
275C2×C4—31no—
276C2×C4—32no—
277C2×C4—62no—
278C2×C4—91no—
279C2×C4—92no—
280C2×C4—91no—
281C2×C4—122no—
282C2×C4C2×D3211yescharacteristic
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—42no—
287C2×C3—42no—
288C2×C3—82no—
289C2×C3—121no—
290C2×C3C2×#23511yescharacteristic
291C2×C3C2×#23711yescharacteristic
292C2×C3#100411yescharacteristic
293C2×C3#100411yescharacteristic
294C2×C3#101011yescharacteristic
295C2×C3#101011yescharacteristic
296D3—41no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—61no—
301C22—61no—
302C22—61no—
303C22C22×D3211yescharacteristic
304C4—61no—
305C4—61no—
306C4—62no—
307C4—62no—
308C4—61no—
309C4—181no—
310C4—181no—
311C4—182no—
312C4#380412yes—
313C3—21no—
314C3#556111yescharacteristic, complemented
315C3#560311yescharacteristic, complemented
316C2—42no—
317C2—121no—
318C2#6430911yescharacteristic
319C2#6431311yescharacteristic
320C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
prime radicalC3×#938C2
hypercenterC2×D4D32
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64145C2semidirect
3. C3×D4×Dic3C2semidirect
4. #63964C2semidirect
5. #63961C2semidirect
6. #63959C2semidirect
7. #63956C2semidirect
8. #63919C2semidirect
9. #63857C2semidirect
10. #63855C2semidirect
11. #63844C2semidirect
45. D4×Dic3D3semidirect
265. C32#542semidirect
314. C3#5561semidirect
315. C3#5603semidirect

Maximal subgroups

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

subgroupindex
#641452
C3×#9382
C3×D4×Dic32
#639642
#639612
#639592
#639562
#639192
#638572
#638552
#638442
#56033
#55613