Finite Group Databasebeta

← #10594525

Subgroups — #10594525

Order 576.

615 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 140 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×#3795C211yesmaximal, characteristic, complemented
2C2×D3×Dic6C211yesmaximal, characteristic, complemented
3#64096C211yesmaximal, characteristic, complemented
4C3×#897C211yesmaximal, characteristic, complemented
5C3×#895C211yesmaximal, characteristic, complemented
6#63915C212yesmaximal, complemented
7C4×#625C211yesmaximal, characteristic, complemented
8#63899C212yesmaximal, complemented
9#63898C212yesmaximal, complemented
10#63885C212yesmaximal, complemented
11#63863C211yesmaximal, characteristic, complemented
12#5633—31nomaximal
13#5598—31nomaximal
14C2×#634C2211yescharacteristic
15C2×C3×D12C2211yescharacteristic
16C2×C3×D12C2212yes—
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×C4×D3C2211yescharacteristic
19C2×C3×Dic6C2211yescharacteristic
20C2×#627C2212yes—
21C2×#625C2212yes—
22C2×#625C2211yescharacteristic
23C2×D3×Dic3C2212yes—
24C2×D3×Dic3C2212yes—
25#3795—24no—
26D3×Dic6—24no—
27C32×C4⋊C4C2211yescharacteristic
28#3750C2212yes—
29C4×C3⋊Dic3C2211yescharacteristic
30C4×C3⋊Dic3C2212yes—
31C3×(C2×D3)⋊C4C2212yes—
32C3×(C2×D3)⋊C4C2212yes—
33C3×C4⋊Dic3C2212yes—
34#3724C2211yescharacteristic
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2211yescharacteristic
38(C2×D3)⋊Dic3C2211yescharacteristic
39C2×D3×Q8—31no—
40C2×#237—31no—
41C2×#235—31no—
42C22×Dic6—31no—
43C3×C22⋊Q8—31no—
44C3×#121—31no—
45#947—31no—
46#939—32no—
47#936—31no—
48D4×Dic3—32no—
49C4×#67—31no—
50C4×#67—31no—
51#897D311yescharacteristic, complemented
52#896—32no—
53#895D311yescharacteristic, complemented
54#893—32no—
55#887—31no—
56#887—31no—
57#887—61no—
58#885—32no—
59#885—32no—
60#884—32no—
61#878—32no—
62C22×C3×D3C2312yes—
63C22×C3×D3C2311yescharacteristic
64C22×C3×D3C2311yescharacteristic
65C2×C32×C4C2312yes—
66C2×C32×C4C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2311yescharacteristic
69C2×C3⋊Dic3C2312yes—
70#634D414yescomplemented
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C3×D12—24no—
75C3×D12—22no—
76C3×C4×D3—22no—
77C3×C4×D3—22no—
78C3×Dic6—22no—
79#627—24no—
80#625—22no—
81#625—24no—
82D3×Dic3—24no—
83D3×Dic3—24no—
84#548—91noSylow
85C2×C3×Q8—31no—
86C2×C3×D4—32no—
87C2×C3×D4—31no—
88C22×C3×C4—31no—
89C22×C3×C4—31no—
90C2×#67—31no—
91C2×#67—32no—
92C2×#67—32no—
93C2×#67—31no—
94C22×Dic3—32no—
95C22×Dic3—32no—
96D3×Q8—64no—
97#237—64no—
98#235—64no—
99C2×D12C2×D312yes—
100C2×D12C2×D311yescharacteristic
101C2×C4×D3—32no—
102C2×C4×D3—32no—
103C2×C4×D3C2×D311yescharacteristic
104C2×C4×D3C2×D311yescharacteristic
105C2×Dic6—31no—
106C2×Dic6—31no—
107C2×Dic6—32no—
108C2×Dic6—32no—
109C2×Dic6—61no—
110C2×Dic6—64no—
111C2×Dic6C2×D311yescharacteristic
112C3×C4⋊C4—21no—
113C3×C4⋊C4—32no—
114C3×C4⋊C4C2×D311yescharacteristic
115C3×C4⋊C4C2×D311yescharacteristic
116C3×C22⋊C4—32no—
117C3×C22⋊C4—32no—
118#217—32no—
119#217—31no—
120#217—32no—
121#217—31no—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—31no—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—31no—
126(C2×D3)⋊C4C2×D312yes—
127(C2×D3)⋊C4C2×D312yes—
128C4⋊Dic3C2×D312yes—
129Dic3⋊C4—32no—
130Dic3⋊C4—32no—
131Dic3⋊C4—31no—
132Dic3⋊C4—31no—
133Dic3⋊C4—62no—
134C4×Dic3—32no—
135C4×Dic3—32no—
136C4×Dic3—31no—
137C4×Dic3—31no—
138C4×Dic3—62no—
139C4×Dic3—61no—
140C22×C32C2411yescharacteristic
141C2×C3×D3—21no—
142C2×C3×D3—21no—
143C2×C3×D3—22no—
144C2×C3×D3—21no—
145C2×C3×D3—21no—
146C2×C3×D3—22no—
147C2×C3×D3—21no—
148C2×C3×D3—22no—
149C2×C3×D3#4112yescomplemented
150C32×C4—22no—
151C32×C4C2×D412yes—
152C3⋊Dic3—21no—
153C3⋊Dic3—21no—
154C3⋊Dic3C2×D414yes—
155C3×Dic3—21no—
156C3×Dic3—22no—
157C3×Dic3—21no—
158C2×#41—91no—
159C22×Q8—91no—
160#124—92no—
161#124—91no—
162C22⋊Q8—31no—
163C22⋊Q8—92no—
164#121—31no—
165#121—92no—
166C4×Q8—91no—
167C4×D4—92no—
168C4×D4—91no—
169C23×C3—31no—
170C23×C3—32no—
171C23×C3—31no—
172C22×D3C22×D312yes—
173C22×D3C22×D311yescharacteristic
174C22×D3C22×D311yescharacteristic
175C3×Q8—62no—
176C3×D4—64no—
177C3×D4—62no—
178C2×C3×C4—21no—
179C2×C3×C4—22no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—32no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4C22×D312yes—
186C2×C3×C4C22×D312yes—
187C2×C3×C4C22×D311yescharacteristic
188C2×C3×C4C22×D311yescharacteristic
189#67—64no—
190#67—64no—
191#67—62no—
192#67—62no—
193C2×Dic3—32no—
194C2×Dic3—32no—
195C2×Dic3—31no—
196C2×Dic3—31no—
197C2×Dic3—31no—
198C2×Dic3—31no—
199C2×Dic3—61no—
200C2×Dic3—62no—
201C2×Dic3—61no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204C2×Dic3C22×D311yescharacteristic
205C2×Dic3C22×D311yescharacteristic
206C2×Dic3C22×D312yes—
207D12—24no—
208D12—22no—
209C4×D3—22no—
210C4×D3—22no—
211C4×D3—64no—
212C4×D3—64no—
213Dic6—22no—
214Dic6—34no—
215Dic6—34no—
216Dic6—64no—
217Dic6—64no—
218Dic6—64no—
219C2×C32C22×D411yescharacteristic
220C2×C32C2×#4111yescharacteristic
221C2×C322−1+411yescharacteristic
222C3×D3—22no—
223C3×D3—41no—
224C3×D3—42no—
225#41—184no—
226C2×Q8—31no—
227C2×Q8—92no—
228C2×Q8—91no—
229C2×Q8—184no—
230C2×D4—32no—
231C2×D4—31no—
232C2×D4—91no—
233C2×D4—92no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—92no—
237C22×C4—92no—
238C4⋊C4—32no—
239C4⋊C4—91no—
240C4⋊C4—92no—
241C4⋊C4D3211yescharacteristic
242C22⋊C4—32no—
243C22⋊C4—32no—
244C22⋊C4—91no—
245C22⋊C4—91no—
246C22⋊C4—92no—
247C22⋊C4—92no—
248C42—92no—
249C42—91no—
250C22×C3—21no—
251C22×C3—32no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—62no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3—61no—
260C22×C3C23×D311yescharacteristic
261C22×C3C23×D311yescharacteristic
262C2×D3—21no—
263C2×D3—21no—
264C2×D3—21no—
265C2×D3—22no—
266C2×D3—22no—
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—22no—
270C2×D3#23712yescomplemented
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4—61no—
276C3×C4—62no—
277C3×C4—61no—
278C3×C4D3×D412yes—
279C3×C4D3×D412yes—
280Dic3—22no—
281Dic3—21no—
282Dic3—21no—
283Dic3—34no—
284Dic3—34no—
285Dic3—64no—
286Dic3—62no—
287Dic3—61no—
288Dic3—61no—
289Dic3—61no—
290Dic3—61no—
291Dic3—121no—
292C32#54811yescharacteristic, complemented, Sylow
293C23—31no—
294C23—31no—
295C23—32no—
296Q8—62no—
297Q8—94no—
298Q8—184no—
299D4—64no—
300D4—62no—
301D4—184no—
302D4—182no—
303C2×C4—31no—
304C2×C4—32no—
305C2×C4—31no—
306C2×C4—62no—
307C2×C4—62no—
308C2×C4—92no—
309C2×C4—91no—
310C2×C4—91no—
311C2×C4—184no—
312C2×C4—184no—
313C2×C4C2×D3212yes—
314C2×C4C2×D3211yescharacteristic
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—62no—
319C2×C3—122no—
320C2×C3—121no—
321C2×C3C2×D3×D411yescharacteristic
322C2×C3C2×D3×D411yescharacteristic
323C2×C3C2×#23711yescharacteristic
324C2×C3#100711yescharacteristic
325C2×C3D3×#4111yescharacteristic
326C2×C3#101011yescharacteristic
327D3—22no—
328D3—41no—
329D3—42no—
330C22—32no—
331C22—61no—
332C22—61no—
333C22—61no—
334C22—62no—
335C22—62no—
336C22—61no—
337C22—61no—
338C22—62no—
339C22C22×D3211yescharacteristic
340C4—22no—
341C4—61no—
342C4—61no—
343C4—62no—
344C4—94no—
345C4—181no—
346C4—181no—
347C4#380212yes—
348C3—21no—
349C3#559811yescharacteristic, complemented
350C3#563311yescharacteristic, complemented
351C2—62no—
352C2—121no—
353C2—122no—
354C2C2×#380211yescharacteristic
355C2D3×#23711yescharacteristic
356C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3795C2semidirect
2. C2×D3×Dic6C2semidirect
3. #64096C2semidirect
4. C3×#897C2semidirect
5. C3×#895C2semidirect
6. #63915C2semidirect
7. C4×#625C2semidirect
8. #63899C2semidirect
9. #63898C2semidirect
10. #63885C2semidirect
11. #63863C2semidirect
51. #897D3semidirect
53. #895D3semidirect
70. #634D4semidirect
149. C2×C3×D3#41semidirect
270. C2×D3#237semidirect
292. C32#548semidirect
349. C3#5598semidirect
350. C3#5633semidirect

Maximal subgroups

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

subgroupindex
C2×#37952
C2×D3×Dic62
#640962
C3×#8972
C3×#8952
#639152
C4×#6252
#638992
#638982
#638852
#638632
#56333
#55983