Finite Group Databasebeta

← #10595131

Subgroups — #10595131

Order 576.

674 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3804C212yesmaximal, complemented
2#64149C211yesmaximal, characteristic, complemented
3C3×#938C211yesmaximal, characteristic, complemented
4C3×D4×Dic3C211yesmaximal, characteristic, complemented
5#63971C212yesmaximal, complemented
6#63953C212yesmaximal, complemented
7#63952C212yesmaximal, complemented
8#63919C211yesmaximal, characteristic, complemented
9#63909C211yesmaximal, characteristic, complemented
10#63894C211yesmaximal, characteristic, complemented
11C4×D3×Dic3C211yesmaximal, characteristic, complemented
12#5596—31nomaximal
13#5548—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C2×#638C2212yes—
17C2×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×#67C2212yes—
19C22×C3×Dic3C2212yes—
20C2×C3×C4×D3C2211yescharacteristic
21C2×#627C2212yes—
22C2×#626C2212yes—
23C2×#624C2212yes—
24#3804—28no—
25C2×D3×Dic3C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C4×C3⋊Dic3C2211yescharacteristic
28C3×#217C2212yes—
29C3×#217C2212yes—
30C3×C4⋊Dic3C2211yescharacteristic
31C3×C4⋊Dic3C2211yescharacteristic
32C3×C4×Dic3C2211yescharacteristic
33#3724C2212yes—
34#3722C2212yes—
35#3722C2212yes—
36Dic32C2211yescharacteristic
37Dic32C2211yescharacteristic
38C2×#237—32no—
39C2×#235—32no—
40C3×#121—31no—
41C3×C4×D4—31no—
42#940—31no—
43#940—31no—
44#940—61no—
45#939—32no—
46#938D311yescharacteristic, complemented
47D4×Dic3D311yescharacteristic, complemented
48#930—31no—
49C4×#67—32no—
50C2×C4×Dic3—31no—
51#895—31no—
52#893—31no—
53#888—31no—
54#888—31no—
55#885—32no—
56#885—32no—
57#884—32no—
58#67⋊C4—32no—
59C4×D12—31no—
60C42×D3—31no—
61C23×C32C2312yes—
62C22×C3⋊D3C2312yes—
63C22×C3×D3C2311yescharacteristic
64C32×D4—22no—
65C2×C32×C4C2311yescharacteristic
66#638—24no—
67#638—24no—
68C2×C3⋊Dic3C2311yescharacteristic
69C2×C3⋊Dic3C2311yescharacteristic
70C4⋊(C3⋊D3)—22no—
71C3×#67—24no—
72C2×C3×Dic3—24no—
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2312yes—
77C2×C3×Dic3C2311yescharacteristic
78C3×C4×D3—22no—
79#627—24no—
80#626—24no—
81#624—24no—
82D3×Dic3—22no—
83D3×Dic3D414yescomplemented
84#532—91noSylow
85C2×C3×D4—21no—
86C2×C3×D4—32no—
87C2×C3×D4C2×D311yescharacteristic
88C2×C3×D4C2×D311yescharacteristic
89C22×C3×C4—32no—
90C22×C3×C4—31no—
91C2×#67—32no—
92C2×#67—32no—
93C2×#67—32no—
94C2×#67—32no—
95C2×#67—32no—
96C2×#67—62no—
97C2×#67—62no—
98C2×#67C2×D312yes—
99C22×Dic3—31no—
100C22×Dic3—31no—
101C22×Dic3C2×D312yes—
102#237—68no—
103#235—68no—
104C2×D12—32no—
105C2×D12—31no—
106C2×D12—31no—
107C2×D12—61no—
108C2×C4×D3—31no—
109C2×C4×D3—31no—
110C2×C4×D3—32no—
111C2×C4×D3—32no—
112C2×C4×D3C2×D311yescharacteristic
113C2×Dic6—32no—
114C2×Dic6—32no—
115C3×C4⋊C4—31no—
116C3×C4⋊C4—31no—
117C3×C22⋊C4—32no—
118C3×C22⋊C4—32no—
119C3×C42—31no—
120#217C2×D312yes—
121#217C2×D312yes—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—32no—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—32no—
126C4⋊Dic3C2×D311yescharacteristic
127C4⋊Dic3C2×D311yescharacteristic
128Dic3⋊C4—32no—
129Dic3⋊C4—32no—
130C4×Dic3—31no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C4×Dic3—31no—
134C4×Dic3—31no—
135C4×Dic3—31no—
136C4×Dic3—61no—
137C4×Dic3C2×D311yescharacteristic
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32C2411yescharacteristic
142C2×C3⋊D3—22no—
143C2×C3⋊D3—22no—
144C2×C3⋊D3—22no—
145C2×C3×D3—21no—
146C2×C3×D3—21no—
147C2×C3×D3C2×D412yes—
148C32×C4#4112yescomplemented
149C3⋊Dic3C2×D412yes—
150C3⋊Dic3#4112yescomplemented
151C3×Dic3—22no—
152C3×Dic3—22no—
153C3×Dic3—21no—
154C3×Dic3—21no—
155C3×Dic3C2×D412yes—
156C2×#41—92no—
157C4⋊Q8—91no—
158#127—91no—
159#124—92no—
160#121—31no—
161#121—92no—
162#121—91no—
163C4×D4—31no—
164C4×D4—91no—
165C4×D4—92no—
166C2×C42—91no—
167C23×C3—22no—
168C23×C3—31no—
169C23×C3C22×D312yes—
170C23×C3C22×D312yes—
171C22×D3—32no—
172C22×D3—32no—
173C22×D3—62no—
174C22×D3C22×D311yescharacteristic
175C3×D4—22no—
176C3×D4—22no—
177C3×D4—42no—
178C3×D4—64no—
179C2×C3×C4—21no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—32no—
184C2×C3×C4—32no—
185C2×C3×C4—64no—
186C2×C3×C4—62no—
187C2×C3×C4C22×D311yescharacteristic
188C2×C3×C4C22×D311yescharacteristic
189#67—24no—
190#67—64no—
191#67—64no—
192#67—64no—
193#67—64no—
194#67—64no—
195#67—124no—
196#67—124no—
197C2×Dic3—24no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—34no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—62no—
206C2×Dic3C22×D311yescharacteristic
207C2×Dic3C22×D311yescharacteristic
208C2×Dic3C22×D312yes—
209C2×Dic3C22×D312yes—
210C2×Dic3C22×D311yescharacteristic
211D12—64no—
212D12—62no—
213D12—62no—
214D12—122no—
215C4×D3—22no—
216C4×D3—34no—
217C4×D3—64no—
218C4×D3—62no—
219C4×D3—64no—
220Dic6—64no—
221Dic6—64no—
222C2×C32—42no—
223C2×C32C22×D411yescharacteristic
224C2×C32C2×#4111yescharacteristic
225C2×C32C2×#4111yescharacteristic
226C3⋊D3—42no—
227C3×D3—22no—
228#41—188no—
229C2×Q8—92no—
230C2×D4—32no—
231C2×D4—92no—
232C2×D4—92no—
233C2×D4—92no—
234C2×D4—91no—
235C2×D4D3211yescharacteristic
236C22×C4—32no—
237C22×C4—31no—
238C22×C4—92no—
239C22×C4—91no—
240C22×C4—91no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—92no—
244C22⋊C4—32no—
245C22⋊C4—32no—
246C22⋊C4—92no—
247C22⋊C4—92no—
248C42—31no—
249C42—91no—
250C42—91no—
251C42—91no—
252C22×C3—22no—
253C22×C3—22no—
254C22×C3—22no—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—32no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3C23×D311yescharacteristic
266C22×C3C23×D311yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—62no—
272C2×D3—62no—
273C2×D3—62no—
274C2×D3—62no—
275C2×D3—122no—
276C2×D3—122no—
277C2×D3—122no—
278C2×D3D3×D412yes—
279C3×C4—22no—
280C3×C4—32no—
281C3×C4—61no—
282C3×C4—62no—
283C3×C4—62no—
284C3×C4—61no—
285C3×C4#23512yescomplemented
286C3×C4#23712yescomplemented
287Dic3—22no—
288Dic3—22no—
289Dic3—21no—
290Dic3—21no—
291Dic3—32no—
292Dic3—32no—
293Dic3—32no—
294Dic3—32no—
295Dic3—62no—
296Dic3—62no—
297Dic3D3×D412yes—
298C32#53211yescharacteristic, complemented, Sylow
299C23—31no—
300C23—92no—
301C23C2×D3212yes—
302Q8—184no—
303D4—22no—
304D4—64no—
305D4—184no—
306D4—184no—
307D4—184no—
308D4—182no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—32no—
313C2×C4—32no—
314C2×C4—64no—
315C2×C4—62no—
316C2×C4—94no—
317C2×C4—91no—
318C2×C4—91no—
319C2×C4—184no—
320C2×C4—182no—
321C2×C4C2×D3211yescharacteristic
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—42no—
326C2×C3—42no—
327C2×C3—62no—
328C2×C3—82no—
329C2×C3C2×#23511yescharacteristic
330C2×C3C2×D3×D411yescharacteristic
331C2×C3C2×D3×D411yescharacteristic
332C2×C3C2×#23711yescharacteristic
333C2×C3D3×#4111yescharacteristic
334C2×C3D3×#4111yescharacteristic
335D3—22no—
336D3—122no—
337D3—122no—
338D3—242no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—32no—
343C22—61no—
344C22—61no—
345C22—182no—
346C22—182no—
347C22—182no—
348C22C22×D3211yescharacteristic
349C4—32no—
350C4—62no—
351C4—62no—
352C4—61no—
353C4—61no—
354C4—92no—
355C4—92no—
356C4#380412yescomplemented
357C3—21no—
358C3#554811yescharacteristic, complemented
359C3#559611yescharacteristic, complemented
360C2—42no—
361C2—62no—
362C2—362no—
363C2D32×D411yescharacteristic
364C2#6431211yescharacteristic
365C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#3804C2semidirect
2. #64149C2semidirect
3. C3×#938C2semidirect
4. C3×D4×Dic3C2semidirect
5. #63971C2semidirect
6. #63953C2semidirect
7. #63952C2semidirect
8. #63919C2semidirect
9. #63909C2semidirect
10. #63894C2semidirect
11. C4×D3×Dic3C2semidirect
46. #938D3semidirect
47. D4×Dic3D3semidirect
83. D3×Dic3D4semidirect
148. C32×C4#41semidirect
150. C3⋊Dic3#41semidirect
285. C3×C4#235semidirect
286. C3×C4#237semidirect
298. C32#532semidirect
356. C4#3804semidirect
358. C3#5548semidirect
359. C3#5596semidirect

Maximal subgroups

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

subgroupindex
C2×#38042
#641492
C3×#9382
C3×D4×Dic32
#639712
#639532
#639522
#639192
#639092
#638942
C4×D3×Dic32
#55963
#55483