Finite Group Databasebeta

← #10594910

Subgroups — #10594910

Order 576.

667 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×#3804C212yesmaximal, complemented
2#64139C211yesmaximal, characteristic, complemented
3C3×#930C211yesmaximal, characteristic, complemented
4C2×C3×C4×Dic3C211yesmaximal, characteristic, complemented
5#63978C212yesmaximal, complemented
6#63918C211yesmaximal, characteristic, complemented
7#63911C211yesmaximal, characteristic, complemented
8#63894C211yesmaximal, characteristic, complemented
9Dic3×D12C211yesmaximal, characteristic, complemented
10#63880C212yesmaximal, complemented
11#63852C212yesmaximal, complemented
12#5596—31nomaximal
13#5477—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#67C2212yes—
18C22×C3×Dic3C2211yescharacteristic
19C22×C3×Dic3C2211yescharacteristic
20C2×C3×D12C2211yescharacteristic
21C2×#627C2212yes—
22C2×#626C2212yes—
23C2×#626C2212yes—
24C2×#624C2212yes—
25#3804—28no—
26C2×D3×Dic3C2212yes—
27#3752C2212yes—
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×(C2×D3)⋊C4C2212yes—
30C3×C4⋊Dic3C2211yescharacteristic
31C3×C4×Dic3C2211yescharacteristic
32C3×C4×Dic3C2211yescharacteristic
33C3×C4×Dic3C2211yescharacteristic
34C3×C4×Dic3C2211yescharacteristic
35Dic3⋊Dic3C2212yes—
36#3722C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38C2×#237—32no—
39C2×#235—32no—
40C3×#121—31no—
41C2×C3×C42—31no—
42#940—31no—
43#939—32no—
44D4×Dic3—31no—
45#930—31no—
46#930—31no—
47#930—32no—
48#930—61no—
49#930D311yescharacteristic, complemented
50C2×C4×Dic3D311yescharacteristic, complemented
51#893—31no—
52#888—31no—
53#885—32no—
54#67⋊C4—32no—
55#875—32no—
56#874—31no—
57C4×D12—31no—
58C4×D12—32no—
59C4×D12—31no—
60#869—31no—
61C23×C32C2311yescharacteristic
62C22×C3⋊D3C2312yes—
63C22×C3×D3C2312yes—
64C2×C32×C4—22no—
65C2×C32×C4C2311yescharacteristic
66C2×C32×C4C2311yescharacteristic
67#638—24no—
68C2×C3⋊Dic3C2312yes—
69C4⋊(C3⋊D3)—22no—
70C3×#67—24no—
71C2×C3×Dic3—22no—
72C2×C3×Dic3D414yescomplemented
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2311yescharacteristic
78C3×D12—22no—
79#627—24no—
80#626—24no—
81#626—24no—
82#624—24no—
83D3×Dic3—24no—
84#532—91noSylow
85C2×C3×D4—32no—
86C2×C3×D4—31no—
87C22×C3×C4—21no—
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C22×C3×C4C2×D311yescharacteristic
91C22×C3×C4C2×D311yescharacteristic
92C2×#67—32no—
93C2×#67—32no—
94C2×#67—32no—
95C2×#67—32no—
96C2×#67—62no—
97C2×#67C2×D312yes—
98C22×Dic3—32no—
99C22×Dic3C2×D311yescharacteristic
100C22×Dic3C2×D311yescharacteristic
101#237—68no—
102#235—68no—
103C2×D12—32no—
104C2×D12—32no—
105C2×D12—31no—
106C2×D12—31no—
107C2×D12—61no—
108C2×D12C2×D311yescharacteristic
109C2×C4×D3—32no—
110C2×C4×D3—32no—
111C2×C4×D3—32no—
112C2×Dic6—32no—
113C2×Dic6—32no—
114C3×C4⋊C4—31no—
115C3×C22⋊C4—32no—
116C3×C42—31no—
117C3×C42—31no—
118C3×C42—31no—
119C3×C42—31no—
120#217—32no—
121(C2×D3)⋊C4—32no—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—32no—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—32no—
126(C2×D3)⋊C4—62no—
127(C2×D3)⋊C4C2×D312yes—
128C4⋊Dic3—31no—
129C4⋊Dic3—31no—
130C4⋊Dic3—32no—
131C4⋊Dic3—61no—
132C4⋊Dic3C2×D311yescharacteristic
133Dic3⋊C4—32no—
134C4×Dic3C2×D311yescharacteristic
135C4×Dic3C2×D311yescharacteristic
136C4×Dic3C2×D311yescharacteristic
137C4×Dic3C2×D311yescharacteristic
138C22×C32—21no—
139C22×C32—21no—
140C22×C32C2×D412yes—
141C22×C32C2411yescharacteristic
142C2×C3⋊D3—22no—
143C2×C3⋊D3—22no—
144C2×C3⋊D3—22no—
145C2×C3×D3—22no—
146C2×C3×D3—22no—
147C2×C3×D3—22no—
148C32×C4—21no—
149C32×C4#4112yescomplemented
150C3⋊Dic3—22no—
151C3×Dic3—22no—
152C3×Dic3—21no—
153C3×Dic3C2×D412yes—
154C3×Dic3C2×D412yes—
155C3×Dic3#4112yescomplemented
156C2×#41—92no—
157C4⋊Q8—91no—
158#127—91no—
159#124—92no—
160#121—31no—
161#121—92no—
162#121—91no—
163C4×D4—92no—
164C4×D4—91no—
165C4×D4—91no—
166C2×C42—31no—
167C23×C3—21no—
168C23×C3—32no—
169C23×C3C22×D311yescharacteristic
170C23×C3C22×D311yescharacteristic
171C22×D3—32no—
172C22×D3—32no—
173C22×D3—62no—
174C22×D3C22×D312yes—
175C3×D4—64no—
176C3×D4—62no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—34no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—32no—
186C2×C3×C4—31no—
187C2×C3×C4—42no—
188C2×C3×C4—62no—
189C2×C3×C4C22×D311yescharacteristic
190C2×C3×C4C22×D311yescharacteristic
191C2×C3×C4C22×D311yescharacteristic
192C2×C3×C4C22×D311yescharacteristic
193#67—24no—
194#67—64no—
195#67—64no—
196#67—64no—
197#67—64no—
198#67—124no—
199C2×Dic3—22no—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—62no—
203C2×Dic3—64no—
204C2×Dic3D1214yescomplemented
205C2×Dic3C22×D311yescharacteristic
206C2×Dic3C22×D312yes—
207C2×Dic3C22×D311yescharacteristic
208C2×Dic3C22×D311yescharacteristic
209C2×Dic3C22×D311yescharacteristic
210D12—22no—
211D12—64no—
212D12—64no—
213D12—62no—
214D12—62no—
215D12—122no—
216C4×D3—64no—
217C4×D3—64no—
218C4×D3—64no—
219Dic6—64no—
220Dic6—64no—
221C2×C32—22no—
222C2×C32C22×D411yescharacteristic
223C2×C32C2×#4111yescharacteristic
224C2×C32C2×#4111yescharacteristic
225C3⋊D3—42no—
226C3×D3—42no—
227#41—188no—
228C2×Q8—92no—
229C2×D4—32no—
230C2×D4—31no—
231C2×D4—92no—
232C2×D4—92no—
233C2×D4—92no—
234C2×D4—91no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—92no—
238C22×C4—92no—
239C22×C4D3211yescharacteristic
240C4⋊C4—31no—
241C4⋊C4—92no—
242C4⋊C4—91no—
243C22⋊C4—32no—
244C22⋊C4—92no—
245C22⋊C4—92no—
246C22⋊C4—92no—
247C42—31no—
248C42—31no—
249C42—31no—
250C42—31no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—62no—
260C22×C3—62no—
261C22×C3—62no—
262C22×C3C2×D1212yes—
263C22×C3D3×D412yes—
264C22×C3C23×D311yescharacteristic
265C22×C3C23×D311yescharacteristic
266C2×D3—22no—
267C2×D3—22no—
268C2×D3—22no—
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—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—22no—
281C3×C4—32no—
282C3×C4—32no—
283C3×C4—32no—
284C3×C4—41no—
285C3×C4—62no—
286C3×C4—61no—
287C3×C4#23512yescomplemented
288C3×C4#23712yescomplemented
289Dic3—21no—
290Dic3—22no—
291Dic3—62no—
292Dic3—62no—
293Dic3—122no—
294Dic3C2×D1212yes—
295Dic3C2×D1212yes—
296Dic3#23512yescomplemented
297C32#53211yescharacteristic, complemented, Sylow
298C23—32no—
299C23—92no—
300C23C2×D3211yescharacteristic
301Q8—184no—
302D4—64no—
303D4—62no—
304D4—184no—
305D4—184no—
306D4—184no—
307D4—182no—
308C2×C4—22no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—34no—
313C2×C4—31no—
314C2×C4—32no—
315C2×C4—62no—
316C2×C4—92no—
317C2×C4—184no—
318C2×C4—184no—
319C2×C4C2×D3211yescharacteristic
320C2×C4C2×D3211yescharacteristic
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—42no—
327C2×C3—122no—
328C2×C3C22×D1211yescharacteristic
329C2×C3C2×#23511yescharacteristic
330C2×C3C2×#23511yescharacteristic
331C2×C3C2×D3×D411yescharacteristic
332C2×C3C2×#23711yescharacteristic
333C2×C3D3×#4111yescharacteristic
334D3—42no—
335D3—122no—
336D3—122no—
337D3—242no—
338C22—21no—
339C22—21no—
340C22—62no—
341C22—62no—
342C22—62no—
343C22—182no—
344C22—182no—
345C22—182no—
346C22D3×D1212yes—
347C22C22×D3211yescharacteristic
348C4—21no—
349C4—32no—
350C4—32no—
351C4—32no—
352C4—62no—
353C4—61no—
354C4—182no—
355C4#380412yescomplemented
356C3—21no—
357C3#547711yescharacteristic, complemented
358C3#559611yescharacteristic, complemented
359C2—22no—
360C2—122no—
361C2—362no—
362C2C2×D3×D1211yescharacteristic
363C2D3×#23511yescharacteristic
364C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#532
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×#930C2semidirect
4. C2×C3×C4×Dic3C2semidirect
5. #63978C2semidirect
6. #63918C2semidirect
7. #63911C2semidirect
8. #63894C2semidirect
9. Dic3×D12C2semidirect
10. #63880C2semidirect
11. #63852C2semidirect
49. #930D3semidirect
50. C2×C4×Dic3D3semidirect
72. C2×C3×Dic3D4semidirect
149. C32×C4#41semidirect
155. C3×Dic3#41semidirect
204. C2×Dic3D12semidirect
287. C3×C4#235semidirect
288. C3×C4#237semidirect
296. Dic3#235semidirect
297. C32#532semidirect
355. C4#3804semidirect
357. C3#5477semidirect
358. C3#5596semidirect

Maximal subgroups

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

subgroupindex
C2×#38042
#641392
C3×#9302
C2×C3×C4×Dic32
#639782
#639182
#639112
#638942
Dic3×D122
#638802
#638522
#55963
#54773