Finite Group Databasebeta

← #10594079

Subgroups — #10594079

Order 576.

725 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 725 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C22×#624C211yesmaximal, characteristic, complemented
2C2×#3804C211yesmaximal, characteristic, complemented
3#64091C211yesmaximal, characteristic, complemented
4C3×#886C211yesmaximal, characteristic, complemented
5C3×#877C211yesmaximal, characteristic, complemented
6#63981C211yesmaximal, characteristic, complemented
7#63966C211yesmaximal, characteristic, complemented
8#63910C212yesmaximal, complemented
9#63895C212yesmaximal, complemented
10#63894C212yesmaximal, complemented
11#63880C212yesmaximal, complemented
12#5656—31nomaximal
13#5524—31nomaximal
14C23×C3⋊D3C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C2×C4⋊(C3⋊D3)C2212yes—
17C2×C3×#67C2211yescharacteristic
18C22×C3×Dic3C2211yescharacteristic
19C22×C3×Dic3C2211yescharacteristic
20C2×#627C2211yescharacteristic, complemented
21C2×#626C2212yes—
22C2×#626C2211yescharacteristic, complemented
23C2×#624—24no—
24C2×#624C2212yes—
25C2×#624C2212yes—
26C2×#624C2211yescharacteristic, complemented
27#3804—24no—
28C2×D3×Dic3C2211yescharacteristic, complemented
29C32×C22⋊C4C2211yescharacteristic
30#3752C2212yes—
31C3×#217C2211yescharacteristic
32C3×(C2×D3)⋊C4C2212yescomplemented
33C3×C4⋊Dic3C2212yescomplemented
34C3×Dic3⋊C4C2212yescomplemented
35C3×C4×Dic3C2212yescomplemented
36Dic3⋊Dic3C2212yescomplemented
37#3722C2212yes—
38#3722C2212yescomplemented
39C2×#237—31no—
40C2×#235—31no—
41C22×C4×D3—31no—
42C22×C4×D3—31no—
43C3×#123—31no—
44C3×#117—31no—
45#937—31no—
46#929—31no—
47#897—32no—
48#896—32no—
49#895—32no—
50#893—32no—
51#886—31no—
52#886D311yescharacteristic, complemented
53#884—32no—
54#882—31no—
55#882—31no—
56#882—31no—
57#882—61no—
58#67⋊C4—32no—
59#877D311yescharacteristic, complemented
60C4×D12—32no—
61C4×D12—32no—
62C23×C32C2311yescharacteristic
63C22×C3⋊D3—22no—
64C22×C3⋊D3—24no—
65C22×C3⋊D3C2311yescharacteristic
66C22×C3⋊D3C2312yes—
67C22×C3×D3C2311yescharacteristic
68C2×C32×C4C2312yes—
69#638—22no—
70C2×C3⋊Dic3C2311yescharacteristic
71C4⋊(C3⋊D3)—24no—
72C3×#67—22no—
73C2×C3×Dic3—22no—
74C2×C3×Dic3D414yescomplemented
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2312yes—
77C2×C3×Dic3C2312yes—
78C2×C3×Dic3C2311yescharacteristic
79C2×C3×Dic3C2311yescharacteristic
80#627—22no—
81#626—24no—
82#626—22no—
83#624—22no—
84#624—24no—
85#624—24no—
86D3×Dic3—22no—
87#525—91noSylow
88C23×D3—31no—
89C23×D3—31no—
90C23×D3—61no—
91C2×C3×D4—31no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C2×#67—31no—
95C2×#67—31no—
96C2×#67—32no—
97C2×#67—31no—
98C2×#67—61no—
99C2×#67C2×D311yescharacteristic
100C22×Dic3—31no—
101C22×Dic3C2×D311yescharacteristic
102C22×Dic3C2×D311yescharacteristic
103#237—64no—
104#235—64no—
105C2×D12—32no—
106C2×D12—31no—
107C2×D12—32no—
108C2×D12—32no—
109C2×D12—62no—
110C2×C4×D3—31no—
111C2×C4×D3—32no—
112C2×C4×D3—31no—
113C2×C4×D3—32no—
114C2×C4×D3—32no—
115C2×C4×D3—31no—
116C2×C4×D3—32no—
117C2×C4×D3—64no—
118C2×C4×D3—64no—
119C2×Dic6—31no—
120C2×Dic6—31no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—32no—
123C3×C22⋊C4—21no—
124C3×C22⋊C4—31no—
125C3×C22⋊C4—32no—
126C3×C22⋊C4C2×D311yescharacteristic
127C3×C22⋊C4C2×D311yescharacteristic
128C3×C42—32no—
129#217C2×D311yescharacteristic
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4—32no—
132(C2×D3)⋊C4—32no—
133(C2×D3)⋊C4—32no—
134(C2×D3)⋊C4—32no—
135(C2×D3)⋊C4—32no—
136(C2×D3)⋊C4—62no—
137(C2×D3)⋊C4C2×D312yescomplemented
138C4⋊Dic3—32no—
139C4⋊Dic3C2×D312yescomplemented
140Dic3⋊C4—32no—
141Dic3⋊C4C2×D312yescomplemented
142C4×Dic3C2×D312yescomplemented
143C22×C32—22no—
144C22×C32C2×D412yes—
145C22×C32C2411yescharacteristic
146C2×C3⋊D3—28no—
147C2×C3⋊D3—22no—
148C2×C3⋊D3—22no—
149C2×C3⋊D3—21no—
150C2×C3⋊D3—22no—
151C2×C3⋊D3#4114yescomplemented
152C2×C3×D3—21no—
153C2×C3×D3—22no—
154C32×C4—22no—
155C3⋊Dic3—21no—
156C3×Dic3—22no—
157C3×Dic3—21no—
158C3×Dic3—22no—
159C3×Dic3C2×D412yes—
160C3×Dic3C2×D412yes—
161C2×#41—91no—
162C23×C4—91no—
163#123—31no—
164#123—91no—
165C22⋊Q8—92no—
166#121—92no—
167#120—91no—
168#120—91no—
169C4×D4—92no—
170C4×D4—92no—
171#117—31no—
172C23×C3—21no—
173C23×C3—31no—
174C23×C3C22×D311yescharacteristic
175C23×C3C22×D311yescharacteristic
176C22×D3—32no—
177C22×D3—32no—
178C22×D3—31no—
179C22×D3—31no—
180C22×D3—62no—
181C22×D3—61no—
182C22×D3—64no—
183C22×D3—64no—
184C22×D3—62no—
185C22×D3—62no—
186C22×D3—124no—
187C22×D3—122no—
188C22×D3C22×D311yescharacteristic
189C3×D4—62no—
190C2×C3×C4—22no—
191C2×C3×C4—34no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—31no—
197C2×C3×C4—62no—
198C2×C3×C4C22×D312yes—
199C2×C3×C4C22×D312yes—
200#67—22no—
201#67—64no—
202#67—62no—
203#67—62no—
204#67—62no—
205#67—122no—
206C2×Dic3—22no—
207C2×Dic3—31no—
208C2×Dic3—31no—
209C2×Dic3—61no—
210C2×Dic3—62no—
211C2×Dic3D1214yescomplemented
212C2×Dic3C22×D312yes—
213C2×Dic3C22×D312yes—
214C2×Dic3C22×D311yescharacteristic
215C2×Dic3C22×D311yescharacteristic
216C2×Dic3C22×D311yescharacteristic
217D12—64no—
218D12—64no—
219D12—62no—
220D12—64no—
221D12—124no—
222C4×D3—64no—
223C4×D3—64no—
224C4×D3—62no—
225C4×D3—62no—
226C4×D3—64no—
227C4×D3—64no—
228C4×D3—62no—
229Dic6—62no—
230Dic6—62no—
231C2×C32—22no—
232C2×C32C22×D411yescharacteristic
233C2×C32C2×#4112yes—
234C3⋊D3—24no—
235C3⋊D3—41no—
236C3×D3—41no—
237C24—91no—
238#41—184no—
239C2×Q8—91no—
240C2×D4—31no—
241C2×D4—92no—
242C2×D4—91no—
243C2×D4—91no—
244C2×D4—92no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—92no—
248C22×C4—91no—
249C22×C4—92no—
250C22×C4—91no—
251C22×C4—184no—
252C4⋊C4—32no—
253C4⋊C4—32no—
254C4⋊C4—92no—
255C22⋊C4—31no—
256C22⋊C4—32no—
257C22⋊C4—92no—
258C22⋊C4—92no—
259C22⋊C4—92no—
260C22⋊C4D3211yescharacteristic
261C42—32no—
262C22×C3—22no—
263C22×C3—22no—
264C22×C3—21no—
265C22×C3—22no—
266C22×C3—42no—
267C22×C3—61no—
268C22×C3—62no—
269C22×C3C2×D1212yes—
270C22×C3D3×D412yes—
271C22×C3C23×D311yescharacteristic
272C22×C3C23×D311yescharacteristic
273C2×D3—21no—
274C2×D3—22no—
275C2×D3—34no—
276C2×D3—34no—
277C2×D3—64no—
278C2×D3—68no—
279C2×D3—68no—
280C2×D3—62no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—62no—
284C2×D3—61no—
285C2×D3—61no—
286C2×D3—62no—
287C2×D3—62no—
288C2×D3—122no—
289C2×D3—122no—
290C2×D3—128no—
291C2×D3—122no—
292C2×D3—121no—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—32no—
296C3×C4—32no—
297C3×C4—42no—
298C3×C4—61no—
299C3×C4—62no—
300C3×C4—62no—
301Dic3—21no—
302Dic3—22no—
303Dic3—22no—
304Dic3—61no—
305Dic3—61no—
306Dic3—121no—
307Dic3C2×D1212yes—
308Dic3C2×D1212yes—
309C32#52511yescharacteristic, complemented, Sylow
310C23—31no—
311C23—92no—
312C23—91no—
313C23—184no—
314C23—182no—
315C23C2×D3211yescharacteristic
316Q8—182no—
317D4—62no—
318D4—184no—
319D4—182no—
320D4—184no—
321D4—182no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—34no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—31no—
328C2×C4—62no—
329C2×C4—91no—
330C2×C4—184no—
331C2×C4—184no—
332C2×C4—182no—
333C2×C4—182no—
334C2×C4C2×D3212yes—
335C2×C3—22no—
336C2×C3—22no—
337C2×C3—21no—
338C2×C3—22no—
339C2×C3—42no—
340C2×C3—121no—
341C2×C3C22×D1211yescharacteristic
342C2×C3C2×D3×D411yescharacteristic
343C2×C3D3×#4112yes—
344C2×C3D3×#4112yes—
345D3—41no—
346D3—64no—
347D3—64no—
348D3—124no—
349D3—121no—
350D3—121no—
351D3—241no—
352C22—22no—
353C22—61no—
354C22—62no—
355C22—94no—
356C22—188no—
357C22—182no—
358C22—182no—
359C22—181no—
360C22—182no—
361C22D3×D1212yes—
362C22C22×D3211yescharacteristic
363C4—22no—
364C4—32no—
365C4—32no—
366C4—62no—
367C4—61no—
368C4—62no—
369C4—181no—
370C3—21no—
371C3#552411yescharacteristic, complemented
372C3#565611yescharacteristic, complemented
373C2—22no—
374C2—121no—
375C2—184no—
376C2—361no—
377C2C2×D3×D1211yescharacteristic
378C2#6431212yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#624C2semidirect
2. C2×#3804C2semidirect
3. #64091C2semidirect
4. C3×#886C2semidirect
5. C3×#877C2semidirect
6. #63981C2semidirect
7. #63966C2semidirect
8. #63910C2semidirect
9. #63895C2semidirect
10. #63894C2semidirect
11. #63880C2semidirect
20. C2×#627C22semidirect
22. C2×#626C22semidirect
26. C2×#624C22semidirect
28. C2×D3×Dic3C22semidirect
32. C3×(C2×D3)⋊C4C22semidirect
33. C3×C4⋊Dic3C22semidirect
34. C3×Dic3⋊C4C22semidirect
35. C3×C4×Dic3C22semidirect
36. Dic3⋊Dic3C22semidirect
38. #3722C22semidirect
52. #886D3semidirect
59. #877D3semidirect
74. C2×C3×Dic3D4semidirect
137. (C2×D3)⋊C4C2×D3semidirect
139. C4⋊Dic3C2×D3semidirect
141. Dic3⋊C4C2×D3semidirect
142. C4×Dic3C2×D3semidirect
151. C2×C3⋊D3#41semidirect
211. C2×Dic3D12semidirect
309. C32#525semidirect
371. C3#5524semidirect
372. C3#5656semidirect

Maximal subgroups

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

subgroupindex
C22×#6242
C2×#38042
#640912
C3×#8862
C3×#8772
#639812
#639662
#639102
#638952
#638942
#638802
#56563
#55243