Finite Group Databasebeta

← #10595017

Subgroups — #10595017

Order 576.

702 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828C211yesmaximal, characteristic, complemented
2C2×#3798C211yesmaximal, characteristic, complemented
3C3×C4×#67C212yesmaximal, complemented
4#63979C211yesmaximal, characteristic, complemented
5#63958C212yesmaximal, complemented
6#63953C211yesmaximal, characteristic, complemented
7#63913C211yesmaximal, characteristic, complemented
8#63910C212yesmaximal, complemented
9C4×#626C212yesmaximal, complemented
10C4×#624C211yesmaximal, characteristic, complemented
11#63840C211yesmaximal, characteristic, complemented
12#5548—32nomaximal
13C22×C32×C4C2211yescharacteristic
14C2×#638C2211yescharacteristic
15C2×#638C2211yescharacteristic
16#3828—24no—
17C2×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C4×C3⋊D3C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×#634C2211yescharacteristic
21C2×C3×#67C2212yes—
22C2×C3×C4×D3C2212yes—
23C2×#627C2211yescharacteristic
24C2×#626C2212yes—
25C2×#626C2212yes—
26C2×#625C2211yescharacteristic
27C2×#624C2212yes—
28#3798—24no—
29C3×#217C2212yes—
30C3×(C2×D3)⋊C4C2212yes—
31C3×Dic3⋊C4C2212yes—
32C3×C4×Dic3C2212yes—
33Dic3⋊Dic3C2212yes—
34#3722C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2211yescharacteristic
37Dic32C2211yescharacteristic
38C2×#235—32no—
39C2×#235—32no—
40C2×#235—61no—
41C3×C4×D4—32no—
42#940—32no—
43#938—32no—
44C4×#67—32no—
45C4×#67D312yescomplemented
46#895—32no—
47#888—32no—
48#885—32no—
49#885—32no—
50#884—32no—
51#884—32no—
52C4×D12—32no—
53C42×D3—32no—
54C23×C32C2311yescharacteristic
55C22×C3⋊D3C2311yescharacteristic
56C22×C3⋊D3C2311yescharacteristic
57C22×C3×D3C2312yes—
58C2×C32×C4—22no—
59C2×C32×C4C2311yescharacteristic
60C2×C32×C4C2311yescharacteristic
61#638—22no—
62#638—22no—
63C2×C3⋊Dic3C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C4⋊(C3⋊D3)—22no—
66C4×C3⋊D3—22no—
67C4×C3⋊D3D414yescomplemented
68#634—22no—
69C3×#67—24no—
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2312yes—
73C3×C4×D3—24no—
74#627—22no—
75#626—24no—
76#626—24no—
77#625—22no—
78#624—24no—
79#532—91noSylow
80C2×C3×D4—32no—
81C22×C3×C4—21no—
82C22×C3×C4—32no—
83C22×C3×C4C2×D312yes—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67—32no—
87C2×#67—32no—
88C2×#67—32no—
89C2×#67—61no—
90C2×#67—61no—
91C2×#67C2×D312yes—
92#235—68no—
93#235—68no—
94#235—68no—
95C2×D12—32no—
96C2×D12—32no—
97C2×D12—32no—
98C2×D12—61no—
99C2×C4×D3—32no—
100C2×C4×D3—32no—
101C2×C4×D3—32no—
102C2×C4×D3—32no—
103C2×C4×D3—61no—
104C2×C4×D3—61no—
105C2×C4×D3C2×D312yes—
106C2×Dic6—32no—
107C2×Dic6—32no—
108C2×Dic6—61no—
109C3×C4⋊C4—32no—
110C3×C22⋊C4—32no—
111C3×C22⋊C4—32no—
112C3×C42—32no—
113#217—32no—
114#217C2×D312yes—
115(C2×D3)⋊C4—32no—
116(C2×D3)⋊C4—32no—
117(C2×D3)⋊C4—32no—
118(C2×D3)⋊C4C2×D312yes—
119C4⋊Dic3—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4C2×D312yes—
122C4×Dic3—32no—
123C4×Dic3—32no—
124C4×Dic3C2×D312yes—
125C22×C32—21no—
126C22×C32—22no—
127C22×C32C2411yescharacteristic
128C2×C3⋊D3—21no—
129C2×C3⋊D3—22no—
130C2×C3⋊D3—22no—
131C2×C3⋊D3C2×D412yes—
132C2×C3×D3—22no—
133C2×C3×D3—22no—
134C2×C3×D3—22no—
135C32×C4—21no—
136C32×C4C2×D412yes—
137C3⋊Dic3—21no—
138C3⋊Dic3C2×D412yes—
139C3×Dic3—22no—
140C3×Dic3—22no—
141C3×Dic3#4114yescomplemented
142C2×#41—91no—
143C2×#41—91no—
144C4⋊Q8—91no—
145#127—91no—
146#124—91no—
147#124—91no—
148#121—92no—
149#121—92no—
150C4×D4—32no—
151C4×D4—92no—
152C2×C42—91no—
153C23×C3—21no—
154C23×C3—32no—
155C23×C3C22×D312yes—
156C22×D3—32no—
157C22×D3—32no—
158C22×D3—61no—
159C22×D3—61no—
160C22×D3C22×D312yes—
161C3×D4—64no—
162C2×C3×C4—24no—
163C2×C3×C4—24no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—32no—
167C2×C3×C4—32no—
168C2×C3×C4—32no—
169C2×C3×C4—64no—
170C2×C3×C4C22×D312yes—
171C2×C3×C4C22×D312yes—
172#67—24no—
173#67—64no—
174#67—64no—
175#67—64no—
176#67—64no—
177#67—64no—
178#67—64no—
179#67—64no—
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3C22×D312yes—
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D312yes—
187D12—64no—
188D12—64no—
189D12—64no—
190D12—64no—
191C4×D3—24no—
192C4×D3—38no—
193C4×D3—64no—
194C4×D3—64no—
195C4×D3—64no—
196C4×D3—64no—
197C4×D3—64no—
198Dic6—64no—
199Dic6—64no—
200Dic6—64no—
201C2×C32—41no—
202C2×C32C22×D411yescharacteristic
203C2×C32C2×#4112yes—
204C3⋊D3—22no—
205C3⋊D3—41no—
206C3×D3—42no—
207#41—184no—
208#41—184no—
209C2×Q8—91no—
210C2×Q8—91no—
211C2×D4—32no—
212C2×D4—92no—
213C2×D4—92no—
214C2×D4—91no—
215C2×D4—91no—
216C2×D4—91no—
217C2×D4—91no—
218C22×C4—32no—
219C22×C4—92no—
220C22×C4—91no—
221C22×C4—91no—
222C22×C4D3211yescharacteristic
223C4⋊C4—32no—
224C4⋊C4—92no—
225C22⋊C4—32no—
226C22⋊C4—32no—
227C22⋊C4—92no—
228C22⋊C4—92no—
229C42—32no—
230C42—91no—
231C42—91no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—21no—
237C22×C3—42no—
238C22×C3—62no—
239C22×C3—62no—
240C22×C3—62no—
241C22×C3C23×D312yes—
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—34no—
246C2×D3—62no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—62no—
253C2×D3—122no—
254C2×D3—122no—
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—22no—
258C3×C4—34no—
259C3×C4—62no—
260C3×C4—62no—
261C3×C4D3×D414yes—
262Dic3—22no—
263Dic3—22no—
264Dic3—34no—
265Dic3—62no—
266Dic3—62no—
267Dic3—62no—
268Dic3#23514yescomplemented
269C32#53211yescharacteristic, complemented, Sylow
270C23—32no—
271C23—91no—
272C23—91no—
273C23C2×D3211yescharacteristic
274Q8—182no—
275Q8—182no—
276D4—64no—
277D4—182no—
278D4—184no—
279D4—184no—
280D4—182no—
281D4—182no—
282D4—182no—
283C2×C4—22no—
284C2×C4—32no—
285C2×C4—32no—
286C2×C4—32no—
287C2×C4—64no—
288C2×C4—91no—
289C2×C4—94no—
290C2×C4—91no—
291C2×C4—184no—
292C2×C4—182no—
293C2×C4C2×D3211yescharacteristic
294C2×C4C2×D3211yescharacteristic
295C2×C3—22no—
296C2×C3—21no—
297C2×C3—42no—
298C2×C3—42no—
299C2×C3—122no—
300C2×C3C2×#23512yes—
301C2×C3C2×D3×D412yes—
302C2×C3D3×#4112yes—
303D3—42no—
304D3—64no—
305D3—122no—
306D3—122no—
307D3—122no—
308C22—22no—
309C22—21no—
310C22—62no—
311C22—62no—
312C22—62no—
313C22—92no—
314C22—182no—
315C22—182no—
316C22—181no—
317C22C22×D3211yescharacteristic
318C4—21no—
319C4—34no—
320C4—62no—
321C4—62no—
322C4—92no—
323C4—181no—
324C4#381112yes—
325C3—21no—
326C3#554812yescomplemented
327C2—41no—
328C2—122no—
329C2—182no—
330C2—361no—
331C2D3×#23512yes—
332C2C2×#381111yescharacteristic
subgroupG/H
centerC2×C4C2×D32
commutator, socle, cosocleC22×C32C24
FrattiniC22C22×D32
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3828C2semidirect
2. C2×#3798C2semidirect
3. C3×C4×#67C2semidirect
4. #63979C2semidirect
5. #63958C2semidirect
6. #63953C2semidirect
7. #63913C2semidirect
8. #63910C2semidirect
9. C4×#626C2semidirect
10. C4×#624C2semidirect
11. #63840C2semidirect
45. C4×#67D3semidirect
67. C4×C3⋊D3D4semidirect
141. C3×Dic3#41semidirect
268. Dic3#235semidirect
269. C32#532semidirect
326. C3#5548semidirect

Maximal subgroups

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

subgroupindex
C2×#38282
C2×#37982
C3×C4×#672
#639792
#639582
#639532
#639132
#639102
C4×#6262
C4×#6242
#638402
#55483