Finite Group Databasebeta

← #10594907

Subgroups — #10594907

Order 576.

630 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3805C212yesmaximal, complemented
2C2×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C3×#930C211yesmaximal, characteristic, complemented
4C3×C4×#67C211yesmaximal, characteristic, complemented
5#63962C212yesmaximal, complemented
6#63918C211yesmaximal, characteristic, complemented
7#63908C211yesmaximal, characteristic, complemented
8#63893C212yesmaximal, complemented
9Dic3×D12C211yesmaximal, characteristic, complemented
10#63891C212yesmaximal, complemented
11D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
12#5611—31nomaximal
13#5555—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C22×C3⋊Dic3C2212yes—
16C2×C3×#67C2212yes—
17C2×C3×#67C2211yescharacteristic
18C2×C3×D12C2211yescharacteristic
19C2×C3×C4×D3C2211yescharacteristic
20C2×#627C2212yes—
21C2×#625C2212yes—
22#3805—28no—
23C2×D3×Dic3C2212yes—
24C2×D3×Dic3C2212yes—
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26C4⋊(C3⋊Dic3)C2212yes—
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28C3×#217C2211yescharacteristic
29C3×(C2×D3)⋊C4C2211yescharacteristic
30C3×(C2×D3)⋊C4C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32C3×Dic3⋊C4C2211yescharacteristic
33C3×C4×Dic3C2211yescharacteristic
34Dic3⋊Dic3C2212yes—
35Dic3⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38C2×#237—32no—
39C2×#237—32no—
40C3×#121—31no—
41C3×C4×D4—31no—
42#938—31no—
43#935—32no—
44D4×Dic3—31no—
45#930—31no—
46#930D311yescharacteristic, complemented
47C4×#67D311yescharacteristic, complemented
48C2×C4⋊Dic3—31no—
49C2×C4⋊Dic3—31no—
50C2×C4⋊Dic3—31no—
51C2×C4⋊Dic3—61no—
52#897—32no—
53D3×C4⋊C4—31no—
54#888—31no—
55#886—32no—
56#886—32no—
57#883—32no—
58#878—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
67C2×C3⋊Dic3—24no—
68C2×C3⋊Dic3C2312yes—
69C2×C3⋊Dic3C2312yes—
70C3×#67—24no—
71C3×#67D414yescomplemented
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C3×D12—22no—
77C3×C4×D3—22no—
78#627—24no—
79#625—24no—
80D3×Dic3—24no—
81D3×Dic3—24no—
82#547—91noSylow
83C2×C3×D4—31no—
84C2×C3×D4—32no—
85C2×C3×D4—31no—
86C22×C3×C4—21no—
87C22×C3×C4—31no—
88C22×C3×C4C2×D311yescharacteristic
89C22×C3×C4C2×D311yescharacteristic
90C2×#67—32no—
91C2×#67—32no—
92C2×#67C2×D312yes—
93C2×#67C2×D311yescharacteristic
94C22×Dic3—32no—
95C22×Dic3—32no—
96C22×Dic3—32no—
97C22×Dic3—32no—
98C22×Dic3—62no—
99#237—68no—
100#237—68no—
101C2×D12C2×D311yescharacteristic
102C2×C4×D3—32no—
103C2×C4×D3—32no—
104C2×C4×D3C2×D311yescharacteristic
105C2×Dic6—32no—
106C2×Dic6—32no—
107C3×C4⋊C4—31no—
108C3×C4⋊C4—31no—
109C3×C22⋊C4—31no—
110C3×C22⋊C4—32no—
111C3×C22⋊C4—31no—
112C3×C42—31no—
113#217—32no—
114#217—32no—
115#217C2×D311yescharacteristic
116(C2×D3)⋊C4—32no—
117(C2×D3)⋊C4—32no—
118(C2×D3)⋊C4C2×D311yescharacteristic
119(C2×D3)⋊C4C2×D312yes—
120C4⋊Dic3—31no—
121C4⋊Dic3—31no—
122C4⋊Dic3—32no—
123C4⋊Dic3—32no—
124C4⋊Dic3—31no—
125C4⋊Dic3—31no—
126C4⋊Dic3—32no—
127C4⋊Dic3—32no—
128C4⋊Dic3—61no—
129C4⋊Dic3—61no—
130C4⋊Dic3—62no—
131C4⋊Dic3C2×D311yescharacteristic
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—21no—
141C2×C3×D3—21no—
142C2×C3×D3—22no—
143C2×C3×D3—22no—
144C2×C3×D3—22no—
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—31no—
160#121—91no—
161C4×D4—31no—
162C4×D4—91no—
163C2×C4⋊C4—91no—
164C2×C4⋊C4—91no—
165C23×C3—21no—
166C23×C3—31no—
167C23×C3—32no—
168C23×C3C22×D311yescharacteristic
169C23×C3C22×D311yescharacteristic
170C22×D3C22×D312yes—
171C22×D3C22×D311yescharacteristic
172C3×D4—34no—
173C3×D4—64no—
174C3×D4—62no—
175C2×C3×C4—24no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—31no—
181C2×C3×C4—32no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—62no—
185C2×C3×C4C22×D311yescharacteristic
186C2×C3×C4C22×D311yescharacteristic
187C2×C3×C4C22×D311yescharacteristic
188C2×C3×C4C22×D311yescharacteristic
189#67—24no—
190#67—64no—
191#67—64no—
192#67D1214yescomplemented
193C2×Dic3—32no—
194C2×Dic3—32no—
195C2×Dic3—32no—
196C2×Dic3—32no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—64no—
200C2×Dic3—64no—
201C2×Dic3—64no—
202C2×Dic3—64no—
203C2×Dic3—124no—
204C2×Dic3C22×D311yescharacteristic
205C2×Dic3C22×D311yescharacteristic
206C2×Dic3C22×D312yes—
207C2×Dic3C22×D311yescharacteristic
208D12—22no—
209C4×D3—22no—
210C4×D3—64no—
211C4×D3—64no—
212Dic6—64no—
213Dic6—64no—
214C2×C32—22no—
215C2×C32C22×D411yescharacteristic
216C2×C32C2×#4111yescharacteristic
217C2×C322−1+411yescharacteristic
218C3×D3—22no—
219C3×D3—42no—
220#41—188no—
221C2×Q8—92no—
222C2×D4—31no—
223C2×D4—32no—
224C2×D4—31no—
225C2×D4—92no—
226C22×C4—31no—
227C22×C4—92no—
228C22×C4—92no—
229C22×C4—92no—
230C22×C4D3211yescharacteristic
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—92no—
234C4⋊C4—92no—
235C4⋊C4—91no—
236C4⋊C4—91no—
237C4⋊C4—92no—
238C22⋊C4—31no—
239C22⋊C4—31no—
240C22⋊C4—32no—
241C22⋊C4—92no—
242C22⋊C4—92no—
243C42—31no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—22no—
252C22×C3—32no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—62no—
256C22×C3—62no—
257C22×C3—62no—
258C22×C3C2×D1212yes—
259C22×C3D3×D412yes—
260C22×C3C23×D311yescharacteristic
261C22×C3C23×D311yescharacteristic
262C2×D3—21no—
263C2×D3—21no—
264C2×D3—22no—
265C2×D3—22no—
266C2×D3—22no—
267C2×D3C2×D1212yes—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—22no—
272C3×C4—32no—
273C3×C4—62no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4#23712yescomplemented
277C3×C4#23712yescomplemented
278Dic3—22no—
279Dic3—21no—
280Dic3—21no—
281Dic3—62no—
282Dic3—62no—
283Dic3—62no—
284Dic3—62no—
285Dic3—122no—
286Dic3—122no—
287Dic3C2×D1212yes—
288C32#54711yescharacteristic, complemented, Sylow
289C23—31no—
290C23—32no—
291C23C2×D3211yescharacteristic
292Q8—184no—
293D4—34no—
294D4—64no—
295D4—62no—
296D4—184no—
297C2×C4—22no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—32no—
302C2×C4—62no—
303C2×C4—92no—
304C2×C4—92no—
305C2×C4—184no—
306C2×C4—184no—
307C2×C4—184no—
308C2×C4C2×D3211yescharacteristic
309C2×C4C2×D3211yescharacteristic
310C2×C3—24no—
311C2×C3—22no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—62no—
317C2×C3—122no—
318C2×C3C22×D1211yescharacteristic
319C2×C3C2×D3×D411yescharacteristic
320C2×C3C2×#23711yescharacteristic
321C2×C3C2×#23711yescharacteristic
322C2×C3#101011yescharacteristic
323C2×C3#101011yescharacteristic
324D3—22no—
325D3—42no—
326C22—21no—
327C22—21no—
328C22—32no—
329C22—61no—
330C22—61no—
331C22—62no—
332C22—62no—
333C22—62no—
334C22D3×D1212yes—
335C22C22×D3211yescharacteristic
336C4—21no—
337C4—32no—
338C4—61no—
339C4—62no—
340C4—61no—
341C4—182no—
342C4—182no—
343C4#380512yescomplemented
344C3—21no—
345C3#555511yescharacteristic, complemented
346C3#561111yescharacteristic, complemented
347C2—22no—
348C2—62no—
349C2—122no—
350C2#6429811yescharacteristic
351C2C2×D3×D1211yescharacteristic
352C2C2×#380511yescharacteristic
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×#3805C2semidirect
2. C2×C4⋊(C3⋊Dic3)C2semidirect
3. C3×#930C2semidirect
4. C3×C4×#67C2semidirect
5. #63962C2semidirect
6. #63918C2semidirect
7. #63908C2semidirect
8. #63893C2semidirect
9. Dic3×D12C2semidirect
10. #63891C2semidirect
11. D3×C4⋊Dic3C2semidirect
46. #930D3semidirect
47. C4×#67D3semidirect
71. C3×#67D4semidirect
147. C32×C4#41semidirect
192. #67D12semidirect
276. C3×C4#237semidirect
277. C3×C4#237semidirect
288. C32#547semidirect
343. C4#3805semidirect
345. C3#5555semidirect
346. C3#5611semidirect

Maximal subgroups

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

subgroupindex
C2×#38052
C2×C4⋊(C3⋊Dic3)2
C3×#9302
C3×C4×#672
#639622
#639182
#639082
#638932
Dic3×D122
#638912
D3×C4⋊Dic32
#56113
#55553