Finite Group Databasebeta

← #10595308

Subgroups — #10595308

Order 576.

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

normal subgroups only

This group has 810 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
1C2×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#3811C211yesmaximal, characteristic, complemented
3C3×#939C212yesmaximal, complemented
4#63983C211yesmaximal, characteristic, complemented
5#63981C212yesmaximal, complemented
6#63978C212yesmaximal, complemented
7#63975C211yesmaximal, characteristic, complemented
8C2×#3722C211yesmaximal, characteristic, complemented
9#63923C211yesmaximal, characteristic, complemented
10#63897C211yesmaximal, characteristic, complemented
11#63853C212yesmaximal, complemented
12#5605—32nomaximal
13C23×C3⋊D3C2211yescharacteristic
14C23×C3⋊D3C2211yescharacteristic, complemented
15C22×D32C2211yescharacteristic, complemented
16C2×C32×D4C2211yescharacteristic, complemented
17C2×#638C2211yescharacteristic
18C2×#638C2211yescharacteristic, complemented
19D4×C3⋊D3—24no—
20C2×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C4×C3⋊D3C2211yescharacteristic, complemented
22C2×C3×#67C2212yescomplemented
23C2×C3×#67C2212yescomplemented
24C22×C3×Dic3C2212yescomplemented
25#3811—24no—
26C2×#626C2212yes—
27C2×#626C2212yescomplemented
28C2×#624C2211yescharacteristic, complemented
29C3×#217C2212yescomplemented
30C3×(C2×D3)⋊C4C2212yescomplemented
31C3×Dic3⋊C4C2212yescomplemented
32Dic3⋊Dic3C2212yescomplemented
33#3722C2212yescomplemented
34#3722C2211yescharacteristic, complemented
35#3722C2212yescomplemented
36#3722C2211yescharacteristic, complemented
37(C2×D3)⋊Dic3C2212yescomplemented
38C2×D3×D4—32no—
39C2×D3×D4—32no—
40C2×D3×D4—61no—
41C3×#121—32no—
42#939—32no—
43#939D312yescomplemented
44#937—32no—
45#937—32no—
46#930—32no—
47C2×(C2×D3)⋊C4—32no—
48#894—32no—
49#894—32no—
50#883—32no—
51#883—32no—
52#882—32no—
53#882—32no—
54C23×C32C2311yescharacteristic
55C23×C32C2311yescharacteristic
56C22×C3⋊D3—22no—
57C22×C3⋊D3—42no—
58C22×C3⋊D3—42no—
59C22×C3⋊D3D414yescomplemented
60C22×C3⋊D3C2311yescharacteristic
61C22×C3⋊D3C2311yescharacteristic
62C22×C3⋊D3C2311yescharacteristic
63C22×C3×D3C2312yes—
64C2×D32—22no—
65C2×D32—42no—
66C32×D4—22no—
67C2×C32×C4C2311yescharacteristic
68#638—22no—
69#638—22no—
70C2×C3⋊Dic3C2311yescharacteristic
71C4⋊(C3⋊D3)—22no—
72C4×C3⋊D3—22no—
73C3×#67—24no—
74C3×#67—42no—
75C2×C3×Dic3—42no—
76C2×C3×Dic3C2312yes—
77C2×C3×Dic3C2312yes—
78C2×C3×Dic3C2312yes—
79#626—24no—
80#626—42no—
81#624—22no—
82#534—91noSylow
83C23×D3—32no—
84C23×D3—32no—
85C23×D3—32no—
86C23×D3—61no—
87C23×D3—61no—
88C2×C3×D4—21no—
89C2×C3×D4—32no—
90C2×C3×D4—32no—
91C2×C3×D4C2×D312yescomplemented
92C22×C3×C4—32no—
93C2×#67—32no—
94C2×#67—32no—
95C2×#67—32no—
96C2×#67—32no—
97C2×#67—61no—
98C2×#67—61no—
99C2×#67C2×D312yescomplemented
100C2×#67C2×D312yescomplemented
101C22×Dic3C2×D312yescomplemented
102D3×D4—68no—
103D3×D4—68no—
104D3×D4—68no—
105C2×D12—32no—
106C2×D12—32no—
107C2×D12—32no—
108C2×D12—61no—
109C2×C4×D3—32no—
110C2×C4×D3—32no—
111C2×C4×D3—61no—
112C3×C4⋊C4—32no—
113C3×C22⋊C4—32no—
114C3×C22⋊C4—32no—
115#217—32no—
116#217C2×D312yescomplemented
117(C2×D3)⋊C4—32no—
118(C2×D3)⋊C4—32no—
119(C2×D3)⋊C4—32no—
120(C2×D3)⋊C4—32no—
121(C2×D3)⋊C4—32no—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—32no—
124(C2×D3)⋊C4C2×D312yescomplemented
125C4⋊Dic3—32no—
126Dic3⋊C4—32no—
127Dic3⋊C4C2×D312yescomplemented
128C22×C32—21no—
129C22×C32—22no—
130C22×C32—22no—
131C22×C32C2×D412yes—
132C22×C32C2411yescharacteristic
133C2×C3⋊D3—22no—
134C2×C3⋊D3—21no—
135C2×C3⋊D3—22no—
136C2×C3⋊D3—22no—
137C2×C3⋊D3—44no—
138C2×C3⋊D3—44no—
139C2×C3⋊D3C2×D412yes—
140C2×C3⋊D3C2×D412yes—
141C2×C3×D3—22no—
142C2×C3×D3—22no—
143C2×C3×D3—22no—
144D32—44no—
145C32×C4—21no—
146C3⋊Dic3—21no—
147C3×Dic3—22no—
148C3×Dic3—22no—
149C3×Dic3—22no—
150C22×D4—91no—
151C22×D4—91no—
152#123—91no—
153#123—92no—
154#123—91no—
155#121—32no—
156#121—92no—
157#120—91no—
158#120—92no—
159#120—91no—
160C2×C22⋊C4—91no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—32no—
164C23×C3C22×D312yes—
165C23×C3C22×D312yes—
166C22×D3—32no—
167C22×D3—32no—
168C22×D3—38no—
169C22×D3—32no—
170C22×D3—61no—
171C22×D3—64no—
172C22×D3—61no—
173C22×D3—61no—
174C22×D3—64no—
175C22×D3—64no—
176C22×D3—64no—
177C22×D3—122no—
178C22×D3—122no—
179C22×D3—122no—
180C22×D3—122no—
181C22×D3—122no—
182C22×D3—122no—
183C22×D3—124no—
184C22×D3—124no—
185C22×D3C22×D312yes—
186C3×D4—24no—
187C3×D4—24no—
188C3×D4—64no—
189C3×D4—122no—
190C2×C3×C4—21no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—122no—
195C2×C3×C4C22×D312yes—
196#67—24no—
197#67—42no—
198#67—64no—
199#67—64no—
200#67—64no—
201#67—64no—
202#67—64no—
203#67—122no—
204C2×Dic3—32no—
205C2×Dic3—42no—
206C2×Dic3—61no—
207C2×Dic3C22×D312yes—
208C2×Dic3C22×D312yes—
209C2×Dic3C22×D312yes—
210D12—64no—
211D12—64no—
212D12—64no—
213D12—122no—
214C4×D3—64no—
215C4×D3—64no—
216C4×D3—64no—
217C2×C32—22no—
218C2×C32—41no—
219C2×C32C22×D411yescharacteristic
220C2×C32H(5,𝔽2)12yes—
221C3⋊D3—22no—
222C3⋊D3—22no—
223C3⋊D3—41no—
224C3×D3—42no—
225C24—91no—
226C24—91no—
227C24—91no—
228C2×D4—32no—
229C2×D4—32no—
230C2×D4—92no—
231C2×D4—92no—
232C2×D4—91no—
233C2×D4—91no—
234C2×D4—91no—
235C2×D4—184no—
236C2×D4—184no—
237C2×D4D3211yescharacteristic, complemented
238C22×C4—32no—
239C22×C4—91no—
240C22×C4—91no—
241C4⋊C4—32no—
242C4⋊C4—92no—
243C22⋊C4—32no—
244C22⋊C4—32no—
245C22⋊C4—91no—
246C22⋊C4—92no—
247C22⋊C4—91no—
248C22⋊C4—92no—
249C22⋊C4—92no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—22no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—22no—
258C22×C3—42no—
259C22×C3—42no—
260C22×C3—62no—
261C22×C3—62no—
262C22×C3—62no—
263C22×C3D3×D414yes—
264C22×C3C23×D312yes—
265C2×D3—22no—
266C2×D3—22no—
267C2×D3—22no—
268C2×D3—34no—
269C2×D3—34no—
270C2×D3—62no—
271C2×D3—62no—
272C2×D3—64no—
273C2×D3—62no—
274C2×D3—62no—
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—62no—
281C2×D3—122no—
282C2×D3—122no—
283C2×D3—128no—
284C2×D3—128no—
285C2×D3—128no—
286C2×D3—128no—
287C2×D3—128no—
288C3×C4—22no—
289C3×C4—22no—
290C3×C4—62no—
291C3×C4—62no—
292C3×C4—62no—
293Dic3—22no—
294Dic3—22no—
295Dic3—22no—
296Dic3—62no—
297Dic3—62no—
298C32#53411yescharacteristic, complemented, Sylow
299C23—32no—
300C23—91no—
301C23—91no—
302C23—94no—
303C23—91no—
304C23—182no—
305C23—182no—
306C23—362no—
307C23—362no—
308C23—362no—
309C23C2×D3211yescharacteristic
310C23C2×D3211yescharacteristic
311D4—22no—
312D4—64no—
313D4—122no—
314D4—182no—
315D4—184no—
316D4—182no—
317D4—182no—
318D4—362no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—91no—
323C2×C4—122no—
324C2×C4—182no—
325C2×C4—182no—
326C2×C4C2×D3211yescharacteristic
327C2×C3—24no—
328C2×C3—22no—
329C2×C3—21no—
330C2×C3—42no—
331C2×C3—42no—
332C2×C3—42no—
333C2×C3—122no—
334C2×C3C2×D3×D412yes—
335C2×C3#100412yes—
336C2×C3#100912yes—
337D3—42no—
338D3—64no—
339D3—64no—
340D3—64no—
341D3—122no—
342D3—122no—
343D3—122no—
344C22—22no—
345C22—22no—
346C22—21no—
347C22—62no—
348C22—62no—
349C22—62no—
350C22—92no—
351C22—92no—
352C22—182no—
353C22—182no—
354C22—182no—
355C22—181no—
356C22—364no—
357C22—364no—
358C22—364no—
359C22#381112yes—
360C22C22×D3211yescharacteristic
361C4—21no—
362C4—62no—
363C4—62no—
364C4—62no—
365C4—181no—
366C3—21no—
367C3#560512yescomplemented
368C2—22no—
369C2—41no—
370C2—122no—
371C2—182no—
372C2—182no—
373C2—361no—
374C2#6431412yes—
375C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#534
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×C3⋊D3C2semidirect
2. C2×#3811C2semidirect
3. C3×#939C2semidirect
4. #63983C2semidirect
5. #63981C2semidirect
6. #63978C2semidirect
7. #63975C2semidirect
8. C2×#3722C2semidirect
9. #63923C2semidirect
10. #63897C2semidirect
11. #63853C2semidirect
14. C23×C3⋊D3C22semidirect
15. C22×D32C22semidirect
16. C2×C32×D4C22semidirect
18. C2×#638C22semidirect
21. C2×C4×C3⋊D3C22semidirect
22. C2×C3×#67C22semidirect
23. C2×C3×#67C22semidirect
24. C22×C3×Dic3C22semidirect
27. C2×#626C22semidirect
28. C2×#624C22semidirect
29. C3×#217C22semidirect
30. C3×(C2×D3)⋊C4C22semidirect
31. C3×Dic3⋊C4C22semidirect
32. Dic3⋊Dic3C22semidirect
33. #3722C22semidirect
34. #3722C22semidirect
35. #3722C22semidirect
36. #3722C22semidirect
37. (C2×D3)⋊Dic3C22semidirect
43. #939D3semidirect
59. C22×C3⋊D3D4semidirect
91. C2×C3×D4C2×D3semidirect
99. C2×#67C2×D3semidirect
100. C2×#67C2×D3semidirect
101. C22×Dic3C2×D3semidirect
116. #217C2×D3semidirect
124. (C2×D3)⋊C4C2×D3semidirect
127. Dic3⋊C4C2×D3semidirect
237. C2×D4D32semidirect
298. C32#534semidirect
367. C3#5605semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D32
C2×#38112
C3×#9392
#639832
#639812
#639782
#639752
C2×#37222
#639232
#638972
#638532
#56053