Finite Group Databasebeta

← #10594329

Subgroups — #10594329

Order 576.

627 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3799C211yesmaximal, characteristic, complemented
2C2×#3797C211yesmaximal, characteristic, complemented
3#64104C211yesmaximal, characteristic, complemented
4C3×#897C211yesmaximal, characteristic, complemented
5C3×#887C211yesmaximal, characteristic, complemented
6#63910C212yesmaximal, complemented
7C4×#626C211yesmaximal, characteristic, complemented
8#63895C212yesmaximal, complemented
9#63894C212yesmaximal, complemented
10#63864C211yesmaximal, characteristic, complemented
11#63852C212yesmaximal, complemented
12#5633—31nomaximal
13#5576—31nomaximal
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4⋊(C3⋊D3)C2212yes—
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×Dic6C2211yescharacteristic
19C2×C3×Dic6C2211yescharacteristic
20C2×#627C2212yes—
21C2×#626C2212yes—
22C2×#626C2211yescharacteristic
23C2×#624C2212yes—
24C2×#624C2212yes—
25#3799—24no—
26#3797—24no—
27C32×C4⋊C4C2211yescharacteristic
28#3752C2212yes—
29C3×(C2×D3)⋊C4C2212yes—
30C3×C4⋊Dic3C2212yes—
31C3×Dic3⋊C4C2212yes—
32C3×C4×Dic3C2212yes—
33C3×C4×Dic3C2211yescharacteristic
34Dic3⋊Dic3C2211yescharacteristic
35#3722C2212yes—
36#3722C2212yes—
37#3722C2211yescharacteristic
38(C2×D3)⋊Dic3C2211yescharacteristic
39C2×#239—31no—
40C2×D3×Q8—31no—
41C2×D3×Q8—31no—
42C2×#235—31no—
43C3×C22⋊Q8—31no—
44C3×C4×Q8—31no—
45#947—31no—
46C4×#67—31no—
47#897—32no—
48#897D311yescharacteristic, complemented
49#896—32no—
50#895—31no—
51#895—31no—
52#895—32no—
53#895—61no—
54#893—32no—
55#887D311yescharacteristic, complemented
56#885—32no—
57#884—32no—
58#875—31no—
59#875—32no—
60C4×D12—31no—
61C4×D12—32no—
62C22×C3⋊D3C2312yes—
63C22×C3⋊D3C2311yescharacteristic
64C22×C3×D3C2311yescharacteristic
65C2×C32×C4C2312yes—
66C2×C32×C4C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C4⋊(C3⋊D3)—24no—
69C4⋊(C3⋊D3)—22no—
70C4×C3⋊D3—22no—
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2312yes—
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2312yes—
76C3×C4×D3—22no—
77C3×Dic6—22no—
78C3×Dic6D414yescomplemented
79#627—24no—
80#626—22no—
81#626—24no—
82#624—24no—
83#624—24no—
84#548—91noSylow
85C2×C3×Q8—31no—
86C2×C3×Q8—31no—
87C22×C3×C4—31no—
88C2×#67—31no—
89C2×#67—32no—
90#239—64no—
91D3×Q8—64no—
92D3×Q8—64no—
93#235—64no—
94C2×D12—31no—
95C2×D12—32no—
96C2×D12—32no—
97C2×D12—31no—
98C2×D12—32no—
99C2×D12—31no—
100C2×D12—62no—
101C2×D12—61no—
102C2×C4×D3—31no—
103C2×C4×D3—31no—
104C2×C4×D3—32no—
105C2×C4×D3—32no—
106C2×C4×D3—32no—
107C2×C4×D3—32no—
108C2×C4×D3—61no—
109C2×C4×D3C2×D311yescharacteristic
110C2×Dic6—32no—
111C2×Dic6—32no—
112C2×Dic6C2×D311yescharacteristic
113C2×Dic6C2×D311yescharacteristic
114C3×C4⋊C4—21no—
115C3×C4⋊C4—32no—
116C3×C4⋊C4—32no—
117C3×C4⋊C4C2×D311yescharacteristic
118C3×C4⋊C4C2×D311yescharacteristic
119C3×C22⋊C4—32no—
120C3×C42—32no—
121C3×C42—31no—
122#217—31no—
123(C2×D3)⋊C4—31no—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—32no—
126(C2×D3)⋊C4—31no—
127(C2×D3)⋊C4—32no—
128(C2×D3)⋊C4—32no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4—31no—
132(C2×D3)⋊C4—62no—
133(C2×D3)⋊C4C2×D312yes—
134C4⋊Dic3—31no—
135C4⋊Dic3C2×D312yes—
136Dic3⋊C4—31no—
137Dic3⋊C4C2×D312yes—
138C4×Dic3C2×D312yes—
139C4×Dic3C2×D311yescharacteristic
140C22×C32C2411yescharacteristic
141C2×C3⋊D3—21no—
142C2×C3⋊D3—21no—
143C2×C3⋊D3—22no—
144C2×C3⋊D3—22no—
145C2×C3⋊D3—22no—
146C2×C3⋊D3#4112yescomplemented
147C2×C3×D3—21no—
148C2×C3×D3—21no—
149C2×C3×D3—21no—
150C32×C4—22no—
151C32×C4C2×D412yes—
152C3⋊Dic3—21no—
153C3×Dic3—21no—
154C3×Dic3—22no—
155C3×Dic3—21no—
156C3×Dic3—21no—
157C3×Dic3C2×D414yes—
158C2×#41—91no—
159C22×Q8—91no—
160#124—92no—
161#124—91no—
162C22⋊Q8—31no—
163C22⋊Q8—92no—
164#121—92no—
165#121—91no—
166C4×Q8—31no—
167C4×D4—92no—
168C4×D4—91no—
169C23×C3—31no—
170C22×D3—31no—
171C22×D3—31no—
172C22×D3—32no—
173C22×D3—32no—
174C22×D3—61no—
175C22×D3—62no—
176C22×D3C22×D311yescharacteristic
177C3×Q8—34no—
178C3×Q8—62no—
179C2×C3×C4—21no—
180C2×C3×C4—22no—
181C2×C3×C4—32no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—62no—
187C2×C3×C4C22×D312yes—
188C2×C3×C4C22×D312yes—
189C2×C3×C4C22×D311yescharacteristic
190C2×C3×C4C22×D311yescharacteristic
191#67—62no—
192#67—64no—
193C2×Dic3—31no—
194C2×Dic3—31no—
195C2×Dic3—61no—
196C2×Dic3C22×D311yescharacteristic
197C2×Dic3C22×D311yescharacteristic
198C2×Dic3C22×D311yescharacteristic
199C2×Dic3C22×D312yes—
200C2×Dic3C22×D312yes—
201D12—64no—
202D12—64no—
203D12—64no—
204D12—62no—
205D12—62no—
206D12—62no—
207D12—122no—
208D12—124no—
209C4×D3—22no—
210C4×D3—64no—
211C4×D3—64no—
212C4×D3—62no—
213C4×D3—62no—
214C4×D3—64no—
215C4×D3—64no—
216C4×D3—122no—
217Dic6—22no—
218Dic6—64no—
219Dic6—64no—
220Dic6D1214yescomplemented
221C2×C32C22×D411yescharacteristic
222C2×C32C2×#4111yescharacteristic
223C2×C322−1+411yescharacteristic
224C3⋊D3—22no—
225C3⋊D3—42no—
226C3×D3—41no—
227#41—184no—
228C2×Q8—31no—
229C2×Q8—31no—
230C2×Q8—92no—
231C2×Q8—184no—
232C2×D4—91no—
233C2×D4—92no—
234C2×D4—92no—
235C2×D4—91no—
236C22×C4—31no—
237C22×C4—92no—
238C22×C4—92no—
239C22×C4—91no—
240C4⋊C4—32no—
241C4⋊C4—32no—
242C4⋊C4—91no—
243C4⋊C4D3211yescharacteristic
244C22⋊C4—32no—
245C22⋊C4—91no—
246C22⋊C4—92no—
247C22⋊C4—92no—
248C22⋊C4—91no—
249C22⋊C4—92no—
250C42—32no—
251C42—31no—
252C22×C3—21no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3C23×D311yescharacteristic
257C22×C3C23×D311yescharacteristic
258C2×D3—21no—
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—32no—
262C2×D3—32no—
263C2×D3—62no—
264C2×D3—61no—
265C2×D3—61no—
266C2×D3—61no—
267C2×D3—61no—
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—62no—
272C2×D3—62no—
273C2×D3—62no—
274C2×D3—121no—
275C2×D3—121no—
276C2×D3—122no—
277C2×D3—122no—
278C2×D3—122no—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—34no—
283C3×C4—42no—
284C3×C4—62no—
285C3×C4—61no—
286C3×C4—61no—
287C3×C4—61no—
288C3×C4C2×D1212yes—
289C3×C4D3×D412yes—
290Dic3—22no—
291Dic3—21no—
292Dic3—21no—
293Dic3—21no—
294Dic3—61no—
295Dic3—61no—
296Dic3—121no—
297Dic3C2×D1214yes—
298C32#54811yescharacteristic, complemented, Sylow
299C23—31no—
300C23—91no—
301C23—92no—
302Q8—34no—
303Q8—62no—
304Q8—184no—
305D4—184no—
306D4—184no—
307D4—182no—
308D4—182no—
309C2×C4—32no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—32no—
313C2×C4—31no—
314C2×C4—62no—
315C2×C4—91no—
316C2×C4—184no—
317C2×C4—184no—
318C2×C4—182no—
319C2×C4C2×D3212yes—
320C2×C4C2×D3211yescharacteristic
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—121no—
325C2×C3C22×D1211yescharacteristic
326C2×C3C2×D3×D411yescharacteristic
327C2×C3#100711yescharacteristic
328C2×C3#100711yescharacteristic
329C2×C3D3×#4111yescharacteristic
330C2×C3D3×#4111yescharacteristic
331D3—41no—
332D3—62no—
333D3—62no—
334D3—122no—
335D3—122no—
336D3—122no—
337D3—242no—
338C22—61no—
339C22—61no—
340C22—61no—
341C22—92no—
342C22—181no—
343C22—181no—
344C22—182no—
345C22—182no—
346C22—182no—
347C22C22×D3211yescharacteristic
348C4—22no—
349C4—34no—
350C4—61no—
351C4—62no—
352C4—61no—
353C4—61no—
354C4—181no—
355C4D3×D1212yes—
356C3—21no—
357C3#557611yescharacteristic, complemented
358C3#563311yescharacteristic, complemented
359C2—121no—
360C2—182no—
361C2—362no—
362C2C2×D3×D1211yescharacteristic
363C2#6431211yescharacteristic
364C2#6431611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3799C2semidirect
2. C2×#3797C2semidirect
3. #64104C2semidirect
4. C3×#897C2semidirect
5. C3×#887C2semidirect
6. #63910C2semidirect
7. C4×#626C2semidirect
8. #63895C2semidirect
9. #63894C2semidirect
10. #63864C2semidirect
11. #63852C2semidirect
48. #897D3semidirect
55. #887D3semidirect
78. C3×Dic6D4semidirect
146. C2×C3⋊D3#41semidirect
220. Dic6D12semidirect
298. C32#548semidirect
357. C3#5576semidirect
358. C3#5633semidirect

Maximal subgroups

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

subgroupindex
C2×#37992
C2×#37972
#641042
C3×#8972
C3×#8872
#639102
C4×#6262
#638952
#638942
#638642
#638522
#56333
#55763