Finite Group Databasebeta

← #10595243

Subgroups — #10595243

Order 576.

786 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3799C211yesmaximal, characteristic, complemented
3C3×#938C211yesmaximal, characteristic, complemented
4C3×#936C211yesmaximal, characteristic, complemented
5#63981C212yesmaximal, complemented
6#63975C212yesmaximal, complemented
7#63974C212yesmaximal, complemented
8#63958C212yesmaximal, complemented
9C4×#626C211yesmaximal, characteristic, complemented
10#63894C211yesmaximal, characteristic, complemented
11#63871C211yesmaximal, characteristic, complemented
12#5676—31nomaximal
13#5612—31nomaximal
14C23×C3⋊D3C2212yes—
15C2×C32×D4C2211yescharacteristic
16C2×#638C2212yes—
17D4×C3⋊D3—24no—
18C2×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×C3×#67C2212yescomplemented
21C2×C3×C4×D3C2211yescharacteristic, complemented
22C2×C3×Dic6C2211yescharacteristic, complemented
23C2×#626C2211yescharacteristic
24C2×#626C2212yescomplemented
25C2×#624C2212yescomplemented
26C2×#624C2212yescomplemented
27#3799—24no—
28C3×#217C2212yescomplemented
29C3×#217C2212yescomplemented
30C3×#217C2212yescomplemented
31C3×C4⋊Dic3C2211yescharacteristic, complemented
32C3×C4×Dic3C2211yescharacteristic, complemented
33Dic3⋊Dic3C2212yescomplemented
34Dic3⋊Dic3C2211yescharacteristic, complemented
35#3722C2212yes—
36#3722C2211yescharacteristic
37#3722C2212yescomplemented
38(C2×D3)⋊Dic3C2211yescharacteristic, complemented
39C2×#239—31no—
40C2×D3×D4—31no—
41C2×D3×D4—31no—
42C2×D3×D4—61no—
43C2×#235—31no—
44C3×#124—31no—
45C3×#121—31no—
46#946—31no—
47#938—32no—
48#938D311yescharacteristic, complemented
49#937—32no—
50#936D311yescharacteristic, complemented
51C4×#67—31no—
52#897—31no—
53#893—31no—
54#884—32no—
55#883—32no—
56#883—32no—
57#882—32no—
58D3×C22⋊C4—32no—
59D3×C22⋊C4—32no—
60C4×D12—31no—
61C4×D12—31no—
62C23×C32C2312yes—
63C22×C3⋊D3—24no—
64C22×C3⋊D3—24no—
65C22×C3⋊D3—42no—
66C22×C3⋊D3C2312yes—
67C22×C3⋊D3C2311yescharacteristic
68C22×C3×D3C2311yescharacteristic
69C32×D4—22no—
70C2×C32×C4C2311yescharacteristic
71#638—24no—
72C2×C3⋊Dic3C2311yescharacteristic
73C4⋊(C3⋊D3)D414yescomplemented
74C4×C3⋊D3—22no—
75C3×#67—42no—
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2312yes—
78C2×C3×Dic3C2311yescharacteristic
79C2×C3×Dic3C2312yes—
80C2×C3×Dic3C2311yescharacteristic
81C3×C4×D3—22no—
82C3×Dic6—22no—
83#626—24no—
84#626—22no—
85#624—24no—
86#624—42no—
87#546—91noSylow
88C23×D3—32no—
89C23×D3—32no—
90C23×D3—62no—
91C2×C3×Q8—31no—
92C2×C3×D4—21no—
93C2×C3×D4—32no—
94C2×C3×D4C2×D311yescharacteristic
95C2×C3×D4C2×D311yescharacteristic
96C22×C3×C4—31no—
97C2×#67—32no—
98C2×#67—32no—
99C2×#67—31no—
100C2×#67—32no—
101C2×#67—62no—
102C2×#67C2×D312yescomplemented
103#239—64no—
104D3×D4—64no—
105D3×D4—64no—
106D3×D4—68no—
107#235—64no—
108C2×D12—31no—
109C2×D12—31no—
110C2×D12—31no—
111C2×D12—32no—
112C2×D12—61no—
113C2×C4×D3—31no—
114C2×C4×D3—31no—
115C2×C4×D3—32no—
116C2×C4×D3—32no—
117C2×C4×D3—32no—
118C2×C4×D3—32no—
119C2×C4×D3—61no—
120C2×C4×D3C2×D311yescharacteristic, complemented
121C2×Dic6C2×D311yescharacteristic, complemented
122C3×C4⋊C4—31no—
123C3×C22⋊C4—32no—
124C3×C22⋊C4—32no—
125C3×C22⋊C4—32no—
126C3×C42—31no—
127#217—31no—
128#217C2×D312yescomplemented
129#217C2×D312yescomplemented
130#217C2×D312yescomplemented
131(C2×D3)⋊C4—31no—
132(C2×D3)⋊C4—31no—
133(C2×D3)⋊C4—32no—
134(C2×D3)⋊C4—31no—
135(C2×D3)⋊C4—32no—
136(C2×D3)⋊C4—32no—
137(C2×D3)⋊C4—32no—
138C4⋊Dic3—31no—
139C4⋊Dic3—32no—
140C4⋊Dic3C2×D311yescharacteristic, complemented
141Dic3⋊C4—31no—
142Dic3⋊C4—32no—
143C4×Dic3C2×D311yescharacteristic, complemented
144C22×C32—22no—
145C22×C32—22no—
146C22×C32—22no—
147C22×C32C2411yescharacteristic
148C2×C3⋊D3—21no—
149C2×C3⋊D3—21no—
150C2×C3⋊D3—22no—
151C2×C3⋊D3—22no—
152C2×C3⋊D3—48no—
153C2×C3⋊D3C2×D414yes—
154C2×C3⋊D3#4112yescomplemented
155C2×C3×D3—21no—
156C2×C3×D3—21no—
157C2×C3×D3—21no—
158C32×C4C2×D412yes—
159C3⋊Dic3—21no—
160C3×Dic3—21no—
161C3×Dic3—22no—
162C3×Dic3—21no—
163C3×Dic3—21no—
164C3×Dic3—22no—
165C2×#41—91no—
166C22×D4—91no—
167#124—31no—
168#123—92no—
169C22⋊Q8—91no—
170#121—31no—
171#121—92no—
172#120—92no—
173C4×D4—91no—
174C4×D4—91no—
175C2×C22⋊C4—92no—
176C23×C3—22no—
177C23×C3—31no—
178C23×C3C22×D312yes—
179C23×C3C22×D312yes—
180C22×D3—31no—
181C22×D3—31no—
182C22×D3—32no—
183C22×D3—32no—
184C22×D3—61no—
185C22×D3—62no—
186C22×D3—64no—
187C22×D3—64no—
188C22×D3—64no—
189C22×D3—64no—
190C22×D3—68no—
191C22×D3—124no—
192C22×D3—122no—
193C22×D3—122no—
194C22×D3—124no—
195C22×D3C22×D311yescharacteristic
196C3×Q8—62no—
197C3×D4—24no—
198C3×D4—22no—
199C3×D4—22no—
200C3×D4—122no—
201C2×C3×C4—21no—
202C2×C3×C4—31no—
203C2×C3×C4—32no—
204C2×C3×C4—31no—
205C2×C3×C4—32no—
206C2×C3×C4—31no—
207C2×C3×C4—62no—
208C2×C3×C4C22×D311yescharacteristic
209C2×C3×C4C22×D311yescharacteristic
210#67—42no—
211#67—64no—
212#67—62no—
213#67—64no—
214#67—64no—
215#67—68no—
216C2×Dic3—31no—
217C2×Dic3—31no—
218C2×Dic3—61no—
219C2×Dic3C22×D311yescharacteristic
220C2×Dic3C22×D311yescharacteristic
221C2×Dic3C22×D312yes—
222C2×Dic3C22×D312yes—
223C2×Dic3C22×D311yescharacteristic
224D12—34no—
225D12—34no—
226D12—64no—
227D12—62no—
228D12—64no—
229C4×D3—22no—
230C4×D3—64no—
231C4×D3—64no—
232C4×D3—64no—
233C4×D3—62no—
234C4×D3—62no—
235C4×D3—122no—
236C4×D3—122no—
237Dic6—22no—
238C2×C32—42no—
239C2×C32C22×D411yescharacteristic
240C2×C32C2×#4111yescharacteristic
241C2×C32H(5,𝔽2)11yescharacteristic
242C3⋊D3—22no—
243C3⋊D3—24no—
244C3×D3—41no—
245C24—92no—
246#41—184no—
247C2×Q8—31no—
248C2×D4—32no—
249C2×D4—91no—
250C2×D4—92no—
251C2×D4—92no—
252C2×D4—91no—
253C2×D4—184no—
254C2×D4D3211yescharacteristic
255C22×C4—31no—
256C22×C4—92no—
257C22×C4—92no—
258C22×C4—91no—
259C4⋊C4—31no—
260C4⋊C4—91no—
261C4⋊C4—92no—
262C22⋊C4—32no—
263C22⋊C4—32no—
264C22⋊C4—32no—
265C22⋊C4—91no—
266C22⋊C4—92no—
267C22⋊C4—92no—
268C22⋊C4—91no—
269C42—31no—
270C22×C3—24no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—21no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—61no—
281C22×C3—61no—
282C22×C3—61no—
283C22×C3C23×D311yescharacteristic
284C22×C3C23×D311yescharacteristic
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—21no—
288C2×D3—34no—
289C2×D3—34no—
290C2×D3—32no—
291C2×D3—32no—
292C2×D3—62no—
293C2×D3—64no—
294C2×D3—62no—
295C2×D3—62no—
296C2×D3—61no—
297C2×D3—61no—
298C2×D3—62no—
299C2×D3—61no—
300C2×D3—62no—
301C2×D3—62no—
302C2×D3—62no—
303C2×D3—61no—
304C2×D3—122no—
305C2×D3—122no—
306C2×D3—128no—
307C2×D3—128no—
308C2×D3—1216no—
309C3×C4—22no—
310C3×C4—61no—
311C3×C4—62no—
312C3×C4—61no—
313C3×C4—61no—
314C3×C4—62no—
315C3×C4D3×D412yes—
316C3×C4D3×D412yes—
317Dic3—22no—
318Dic3—21no—
319Dic3—22no—
320Dic3—21no—
321Dic3—21no—
322Dic3—62no—
323Dic3—61no—
324Dic3—61no—
325C32#54611yescharacteristic, complemented, Sylow
326C23—31no—
327C23—91no—
328C23—92no—
329C23—184no—
330C23—184no—
331C23—362no—
332C23C2×D3212yes—
333Q8—62no—
334D4—22no—
335D4—94no—
336D4—122no—
337D4—184no—
338D4—182no—
339D4—184no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—62no—
346C2×C4—91no—
347C2×C4—184no—
348C2×C4—182no—
349C2×C4—362no—
350C2×C4C2×D3211yescharacteristic
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—44no—
355C2×C3—42no—
356C2×C3—42no—
357C2×C3—121no—
358C2×C3C2×D3×D411yescharacteristic
359C2×C3C2×D3×D411yescharacteristic
360C2×C3#100411yescharacteristic
361C2×C3D3×#4111yescharacteristic
362C2×C3D3×#4111yescharacteristic
363C2×C3#100911yescharacteristic
364D3—41no—
365D3—64no—
366D3—62no—
367D3—64no—
368D3—62no—
369D3—64no—
370D3—124no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—92no—
378C22—94no—
379C22—182no—
380C22—182no—
381C22—181no—
382C22—181no—
383C22—368no—
384C22C22×D3211yescharacteristic
385C4—61no—
386C4—61no—
387C4—62no—
388C4—61no—
389C4—62no—
390C4—181no—
391C4#381112yes—
392C3—21no—
393C3#561211yescharacteristic, complemented
394C3#567611yescharacteristic, complemented
395C2—42no—
396C2—121no—
397C2—182no—
398C2—184no—
399C2#6431211yescharacteristic
400C2#6431411yescharacteristic
401C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#546
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×#3799C2semidirect
3. C3×#938C2semidirect
4. C3×#936C2semidirect
5. #63981C2semidirect
6. #63975C2semidirect
7. #63974C2semidirect
8. #63958C2semidirect
9. C4×#626C2semidirect
10. #63894C2semidirect
11. #63871C2semidirect
20. C2×C3×#67C22semidirect
21. C2×C3×C4×D3C22semidirect
22. C2×C3×Dic6C22semidirect
24. C2×#626C22semidirect
25. C2×#624C22semidirect
26. C2×#624C22semidirect
28. C3×#217C22semidirect
29. C3×#217C22semidirect
30. C3×#217C22semidirect
31. C3×C4⋊Dic3C22semidirect
32. C3×C4×Dic3C22semidirect
33. Dic3⋊Dic3C22semidirect
34. Dic3⋊Dic3C22semidirect
37. #3722C22semidirect
38. (C2×D3)⋊Dic3C22semidirect
48. #938D3semidirect
50. #936D3semidirect
73. C4⋊(C3⋊D3)D4semidirect
102. C2×#67C2×D3semidirect
120. C2×C4×D3C2×D3semidirect
121. C2×Dic6C2×D3semidirect
128. #217C2×D3semidirect
129. #217C2×D3semidirect
130. #217C2×D3semidirect
140. C4⋊Dic3C2×D3semidirect
143. C4×Dic3C2×D3semidirect
154. C2×C3⋊D3#41semidirect
325. C32#546semidirect
393. C3#5612semidirect
394. C3#5676semidirect

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×#37992
C3×#9382
C3×#9362
#639812
#639752
#639742
#639582
C4×#6262
#638942
#638712
#56763
#56123