Finite Group Databasebeta

← #10594906

Subgroups — #10594906

Order 576.

631 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3804C212yesmaximal, complemented
2#64139C211yesmaximal, characteristic, complemented
3C3×C4×#67C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
5#63966C212yesmaximal, complemented
6#63918C211yesmaximal, characteristic, complemented
7#63909C211yesmaximal, characteristic, complemented
8#63895C212yesmaximal, complemented
9#63894C211yesmaximal, characteristic, complemented
10#63890C212yesmaximal, complemented
11D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
12#5555—31nomaximal
13#5507—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#67C2211yescharacteristic
18C22×C3×Dic3C2212yes—
19C2×C3×C4×D3C2211yescharacteristic
20C2×#627C2212yes—
21C2×#626C2212yes—
22C2×#624C2212yes—
23#3804—28no—
24C2×D3×Dic3C2212yes—
25#3752C2212yes—
26C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×#217C2211yescharacteristic
28C3×(C2×D3)⋊C4C2211yescharacteristic
29C3×C4⋊Dic3C2211yescharacteristic
30C3×C4⋊Dic3C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32C3×Dic3⋊C4C2211yescharacteristic
33C3×C4×Dic3C2211yescharacteristic
34Dic3⋊Dic3C2212yes—
35Dic3⋊Dic3C2212yes—
36#3722C2212yes—
37#3722C2212yes—
38C2×#237—32no—
39C2×#235—32no—
40C3×C4×D4—31no—
41C2×C3×C4⋊C4—31no—
42#930—31no—
43#930—31no—
44#930—31no—
45#930—61no—
46#929—32no—
47C4×#67D311yescharacteristic, complemented
48C2×C4⋊Dic3—31no—
49C2×C4⋊Dic3D311yescharacteristic, complemented
50#897—32no—
51#896—32no—
52#895—31no—
53#894—32no—
54#893—31no—
55D3×C4⋊C4—31no—
56#888—31no—
57#886—32no—
58#886—32no—
59C4×D12—31no—
60#869—31no—
61C23×C32C2311yescharacteristic
62C22×C3⋊D3C2312yes—
63C22×C3×D3C2311yescharacteristic
64C2×C32×C4—22no—
65C2×C32×C4C2311yescharacteristic
66C2×C32×C4C2311yescharacteristic
67#638—24no—
68C2×C3⋊Dic3C2312yes—
69C4⋊(C3⋊D3)—22no—
70C3×#67D414yescomplemented
71C2×C3×Dic3—24no—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2311yescharacteristic
77C3×C4×D3—22no—
78#627—24no—
79#626—24no—
80#624—24no—
81D3×Dic3—24no—
82#547—91noSylow
83C2×C3×D4—31no—
84C22×C3×C4—21no—
85C22×C3×C4—32no—
86C22×C3×C4—31no—
87C22×C3×C4C2×D311yescharacteristic
88C22×C3×C4C2×D311yescharacteristic
89C2×#67—32no—
90C2×#67—32no—
91C2×#67—32no—
92C2×#67—62no—
93C2×#67C2×D311yescharacteristic
94C22×Dic3—32no—
95C22×Dic3C2×D312yes—
96#237—68no—
97#235—68no—
98C2×D12—32no—
99C2×D12—31no—
100C2×D12—31no—
101C2×D12—61no—
102C2×C4×D3—32no—
103C2×C4×D3—32no—
104C2×C4×D3—32no—
105C2×C4×D3C2×D311yescharacteristic
106C2×Dic6—32no—
107C2×Dic6—32no—
108C3×C4⋊C4—31no—
109C3×C4⋊C4—31no—
110C3×C4⋊C4—32no—
111C3×C4⋊C4—31no—
112C3×C22⋊C4—31no—
113C3×C22⋊C4—31no—
114C3×C42—31no—
115#217C2×D311yescharacteristic
116(C2×D3)⋊C4—32no—
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—62no—
123(C2×D3)⋊C4C2×D311yescharacteristic
124C4⋊Dic3—31no—
125C4⋊Dic3—31no—
126C4⋊Dic3—32no—
127C4⋊Dic3—32no—
128C4⋊Dic3—61no—
129C4⋊Dic3C2×D311yescharacteristic
130C4⋊Dic3C2×D311yescharacteristic
131C4⋊Dic3C2×D312yes—
132Dic3⋊C4—32no—
133Dic3⋊C4—32no—
134Dic3⋊C4C2×D311yescharacteristic
135C4×Dic3C2×D311yescharacteristic
136C22×C32—21no—
137C22×C32—21no—
138C22×C32C2×D412yes—
139C22×C32C2411yescharacteristic
140C2×C3⋊D3—22no—
141C2×C3⋊D3—22no—
142C2×C3⋊D3—22no—
143C2×C3×D3—21no—
144C2×C3×D3—21no—
145C2×C3×D3C2×D412yes—
146C32×C4—21no—
147C32×C4#4112yescomplemented
148C3⋊Dic3—22no—
149C3×Dic3—22no—
150C3×Dic3—22no—
151C3×Dic3—21no—
152C3×Dic3—21no—
153C3×Dic3C2×D412yes—
154C2×#41—92no—
155C4⋊Q8—91no—
156#123—92no—
157#123—92no—
158C22⋊Q8—92no—
159#121—91no—
160#121—91no—
161C4×D4—31no—
162C4×D4—91no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—91no—
165C23×C3—21no—
166C23×C3—31no—
167C23×C3C22×D311yescharacteristic
168C23×C3C22×D311yescharacteristic
169C22×D3—32no—
170C22×D3—32no—
171C22×D3—62no—
172C22×D3C22×D311yescharacteristic
173C3×D4—34no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—31no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—42no—
184C2×C3×C4—64no—
185C2×C3×C4—62no—
186C2×C3×C4C22×D311yescharacteristic
187C2×C3×C4C22×D311yescharacteristic
188C2×C3×C4C22×D311yescharacteristic
189C2×C3×C4C22×D311yescharacteristic
190#67—64no—
191#67—64no—
192#67—64no—
193#67—124no—
194#67D1214yescomplemented
195C2×Dic3—24no—
196C2×Dic3—32no—
197C2×Dic3—32no—
198C2×Dic3—62no—
199C2×Dic3—64no—
200C2×Dic3C22×D311yescharacteristic
201C2×Dic3C22×D311yescharacteristic
202C2×Dic3C22×D312yes—
203C2×Dic3C22×D312yes—
204C2×Dic3C22×D311yescharacteristic
205D12—64no—
206D12—62no—
207D12—62no—
208D12—122no—
209C4×D3—22no—
210C4×D3—64no—
211C4×D3—64no—
212C4×D3—64no—
213Dic6—64no—
214Dic6—64no—
215C2×C32—22no—
216C2×C32C22×D411yescharacteristic
217C2×C32C2×#4111yescharacteristic
218C2×C322−1+411yescharacteristic
219C3⋊D3—42no—
220C3×D3—22no—
221#41—188no—
222C2×Q8—92no—
223C2×D4—31no—
224C2×D4—92no—
225C2×D4—92no—
226C2×D4—91no—
227C22×C4—32no—
228C22×C4—31no—
229C22×C4—92no—
230C22×C4—92no—
231C22×C4D3211yescharacteristic
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—32no—
235C4⋊C4—31no—
236C4⋊C4—92no—
237C4⋊C4—92no—
238C4⋊C4—91no—
239C22⋊C4—31no—
240C22⋊C4—31no—
241C22⋊C4—92no—
242C22⋊C4—92no—
243C22⋊C4—92no—
244C42—31no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—32no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3C2×D1212yes—
257C22×C3D3×D412yes—
258C22×C3C23×D311yescharacteristic
259C22×C3C23×D311yescharacteristic
260C2×D3—21no—
261C2×D3—21no—
262C2×D3—62no—
263C2×D3—62no—
264C2×D3—62no—
265C2×D3—62no—
266C2×D3—62no—
267C2×D3—62no—
268C2×D3—122no—
269C2×D3—122no—
270C2×D3—122no—
271C2×D3C2×D1212yes—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—22no—
275C3×C4—32no—
276C3×C4—41no—
277C3×C4—62no—
278C3×C4—62no—
279C3×C4—61no—
280C3×C4—61no—
281C3×C4#23512yescomplemented
282C3×C4#23712yescomplemented
283Dic3—22no—
284Dic3—22no—
285Dic3—21no—
286Dic3—21no—
287Dic3—62no—
288Dic3—62no—
289Dic3—122no—
290Dic3C2×D1212yes—
291C32#54711yescharacteristic, complemented, Sylow
292C23—31no—
293C23—92no—
294C23C2×D3211yescharacteristic
295Q8—184no—
296D4—34no—
297D4—184no—
298D4—184no—
299D4—182no—
300C2×C4—22no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—31no—
304C2×C4—32no—
305C2×C4—32no—
306C2×C4—64no—
307C2×C4—62no—
308C2×C4—92no—
309C2×C4—184no—
310C2×C4—184no—
311C2×C4C2×D3211yescharacteristic
312C2×C4C2×D3211yescharacteristic
313C2×C3—22no—
314C2×C3—22no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—42no—
319C2×C3—62no—
320C2×C3C22×D1211yescharacteristic
321C2×C3C2×#23511yescharacteristic
322C2×C3C2×D3×D411yescharacteristic
323C2×C3C2×#23711yescharacteristic
324C2×C3#100711yescharacteristic
325C2×C3#101011yescharacteristic
326D3—22no—
327D3—122no—
328D3—122no—
329D3—242no—
330C22—21no—
331C22—21no—
332C22—32no—
333C22—61no—
334C22—61no—
335C22—182no—
336C22—182no—
337C22—182no—
338C22D3×D1212yes—
339C22C22×D3211yescharacteristic
340C4—21no—
341C4—32no—
342C4—61no—
343C4—62no—
344C4—62no—
345C4—61no—
346C4—182no—
347C4#380412yescomplemented
348C3—21no—
349C3#550711yescharacteristic, complemented
350C3#555511yescharacteristic, complemented
351C2—22no—
352C2—62no—
353C2—362no—
354C2#6429711yescharacteristic
355C2C2×D3×D1211yescharacteristic
356C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3804C2semidirect
2. #64139C2semidirect
3. C3×C4×#67C2semidirect
4. C2×C3×C4⋊Dic3C2semidirect
5. #63966C2semidirect
6. #63918C2semidirect
7. #63909C2semidirect
8. #63895C2semidirect
9. #63894C2semidirect
10. #63890C2semidirect
11. D3×C4⋊Dic3C2semidirect
47. C4×#67D3semidirect
49. C2×C4⋊Dic3D3semidirect
70. C3×#67D4semidirect
147. C32×C4#41semidirect
194. #67D12semidirect
281. C3×C4#235semidirect
282. C3×C4#237semidirect
291. C32#547semidirect
347. C4#3804semidirect
349. C3#5507semidirect
350. C3#5555semidirect

Maximal subgroups

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

subgroupindex
C2×#38042
#641392
C3×C4×#672
C2×C3×C4⋊Dic32
#639662
#639182
#639092
#638952
#638942
#638902
D3×C4⋊Dic32
#55553
#55073