Finite Group Databasebeta

← #10595210

Subgroups — #10595210

Order 576.

526 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 130 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
1#64149C211yesmaximal, characteristic
2C3×#936C211yesmaximal, characteristic, complemented
3C3×D4×Dic3C211yesmaximal, characteristic, complemented
4#63975C212yesmaximal, complemented
5#63966C212yesmaximal, complemented
6#63953C212yesmaximal, complemented
7#63952C212yesmaximal, complemented
8C4×#627C211yesmaximal, characteristic, complemented
9#63894C211yesmaximal, characteristic, complemented
10#63864C211yesmaximal, characteristic, complemented
11#63843C211yesmaximal, characteristic, complemented
12#5681—31nomaximal
13#5559—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C2×#638C2212yes—
17C2×C4⋊(C3⋊D3)C2211yescharacteristic
18C22×C3×Dic3C2212yes—
19C2×C3×Dic6C2211yescharacteristic
20C2×#627C2211yescharacteristic
21C2×#624C2212yes—
22C4×C3⋊Dic3C2211yescharacteristic
23C3×#217C2212yes—
24C3×#217C2212yes—
25C3×#217C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4×Dic3C2211yescharacteristic
28C3×C4×Dic3C2211yescharacteristic
29#3724C2211yescharacteristic
30Dic3⋊Dic3C2212yes—
31Dic3⋊Dic3C2211yescharacteristic
32Dic3⋊Dic3C2211yescharacteristic
33#3722C2212yes—
34#3722C2212yes—
35#3722C2212yes—
36Dic32C2212yes—
37C3×#124—31no—
38C3×C4×D4—31no—
39#947—31no—
40Q8×Dic3—31no—
41#940—31no—
42#940—31no—
43#940—61no—
44#936D311yescharacteristic, complemented
45D4×Dic3D311yescharacteristic, complemented
46#929—32no—
47C4×#67—32no—
48#893—31no—
49#887—31no—
50#886—32no—
51#885—32no—
52#885—32no—
53#883—32no—
54#883—32no—
55#67⋊C4—32no—
56#875—31no—
57C4×D12—31no—
58C4×Dic6—31no—
59C4×Dic6—31no—
60C23×C32C2312yes—
61C22×C3⋊D3C2312yes—
62C32×D4—22no—
63C2×C32×C4C2311yescharacteristic
64#638—24no—
65#638—42no—
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C4⋊(C3⋊D3)—22no—
69C2×C3×Dic3—42no—
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2311yescharacteristic
76C3×Dic6—22no—
77#627—22no—
78#624—42no—
79#559—91noSylow
80C2×C3×Q8—31no—
81C2×C3×D4—21no—
82C2×C3×D4C2×D311yescharacteristic
83C2×C3×D4C2×D311yescharacteristic
84C22×C3×C4—32no—
85C2×#67—32no—
86C2×#67—32no—
87C2×#67—32no—
88C2×#67—32no—
89C2×#67—62no—
90C2×#67—62no—
91C22×Dic3C2×D312yes—
92C2×D12—31no—
93C2×D12—31no—
94C2×D12—61no—
95C2×C4×D3—32no—
96C2×C4×D3—32no—
97C2×Dic6—31no—
98C2×Dic6—31no—
99C2×Dic6C2×D311yescharacteristic
100C3×C4⋊C4—31no—
101C3×C22⋊C4—32no—
102C3×C22⋊C4—32no—
103C3×C22⋊C4—32no—
104C3×C42—31no—
105C3×C42—31no—
106#217C2×D312yes—
107#217C2×D312yes—
108#217C2×D312yes—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4—32no—
114(C2×D3)⋊C4—32no—
115C4⋊Dic3—31no—
116C4⋊Dic3—31no—
117C4⋊Dic3—32no—
118C4⋊Dic3C2×D311yescharacteristic
119Dic3⋊C4—31no—
120Dic3⋊C4—31no—
121Dic3⋊C4—31no—
122Dic3⋊C4—31no—
123Dic3⋊C4—32no—
124C4×Dic3—31no—
125C4×Dic3—31no—
126C4×Dic3—32no—
127C4×Dic3—32no—
128C4×Dic3—61no—
129C4×Dic3C2×D311yescharacteristic
130C4×Dic3C2×D311yescharacteristic
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—22no—
134C22×C32C2411yescharacteristic
135C2×C3⋊D3—22no—
136C2×C3⋊D3—22no—
137C2×C3⋊D3—22no—
138C32×C4#4112yes—
139C3⋊Dic3—21no—
140C3⋊Dic3#4112yes—
141C3×Dic3—21no—
142C3×Dic3—21no—
143C3×Dic3—22no—
144C3×Dic3—21no—
145C3×Dic3—22no—
146C3×Dic3—21no—
147#127—91no—
148#124—31no—
149#124—91no—
150#124—92no—
151#123—92no—
152#123—92no—
153C4×Q8—91no—
154C4×Q8—91no—
155C4×D4—31no—
156C4×D4—91no—
157C4×D4—92no—
158C23×C3—22no—
159C23×C3C22×D312yes—
160C23×C3C22×D312yes—
161C22×D3—32no—
162C22×D3—32no—
163C22×D3—62no—
164C3×Q8—62no—
165C3×D4—22no—
166C3×D4—22no—
167C3×D4—42no—
168C2×C3×C4—21no—
169C2×C3×C4—32no—
170C2×C3×C4—32no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—122no—
176C2×C3×C4C22×D311yescharacteristic
177C2×C3×C4C22×D311yescharacteristic
178#67—64no—
179#67—64no—
180#67—124no—
181#67—122no—
182#67—122no—
183#67—124no—
184C2×Dic3—31no—
185C2×Dic3—31no—
186C2×Dic3—31no—
187C2×Dic3—31no—
188C2×Dic3—42no—
189C2×Dic3—61no—
190C2×Dic3—61no—
191C2×Dic3C22×D312yes—
192C2×Dic3C22×D312yes—
193C2×Dic3C22×D311yescharacteristic
194C2×Dic3C22×D311yescharacteristic
195C2×Dic3C22×D311yescharacteristic
196C2×Dic3C22×D311yescharacteristic
197D12—62no—
198D12—62no—
199D12—122no—
200C4×D3—122no—
201C4×D3—122no—
202Dic6—22no—
203Dic6—62no—
204Dic6—62no—
205C2×C32—42no—
206C2×C32C2×#4111yescharacteristic
207C2×C32C2×#4111yescharacteristic
208C2×C32H(5,𝔽2)11yescharacteristic
209C3⋊D3—42no—
210C2×Q8—31no—
211C2×Q8—91no—
212C2×D4—92no—
213C2×D4—92no—
214C2×D4—91no—
215C2×D4D3211yescharacteristic
216C22×C4—32no—
217C22×C4—92no—
218C4⋊C4—31no—
219C4⋊C4—91no—
220C4⋊C4—91no—
221C4⋊C4—91no—
222C4⋊C4—92no—
223C22⋊C4—32no—
224C22⋊C4—32no—
225C22⋊C4—32no—
226C22⋊C4—92no—
227C22⋊C4—92no—
228C22⋊C4—92no—
229C42—31no—
230C42—31no—
231C42—92no—
232C42—91no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—21no—
240C22×C3—42no—
241C22×C3—42no—
242C22×C3—42no—
243C22×C3C23×D311yescharacteristic
244C22×C3C23×D311yescharacteristic
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—122no—
252C2×D3—122no—
253C2×D3—122no—
254C3×C4—22no—
255C3×C4—62no—
256C3×C4—61no—
257C3×C4—61no—
258C3×C4—61no—
259C3×C4—62no—
260C3×C4—61no—
261C3×C4#23512yes—
262C3×C4#23712yes—
263Dic3—22no—
264Dic3—22no—
265Dic3—21no—
266Dic3—21no—
267Dic3—21no—
268Dic3—21no—
269Dic3—32no—
270Dic3—32no—
271Dic3—62no—
272Dic3—62no—
273Dic3—61no—
274Dic3—61no—
275C32#55911yescharacteristic, complemented, Sylow
276C23—92no—
277C23C2×D3212yes—
278Q8—62no—
279Q8—182no—
280D4—22no—
281D4—184no—
282D4—182no—
283D4—362no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—32no—
289C2×C4—32no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—122no—
293C2×C4—362no—
294C2×C4C2×D3211yescharacteristic
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—42no—
299C2×C3—42no—
300C2×C3—82no—
301C2×C3C2×#23511yescharacteristic
302C2×C3C2×#23711yescharacteristic
303C2×C3#100411yescharacteristic
304C2×C3D3×#4111yescharacteristic
305C2×C3D3×#4111yescharacteristic
306C2×C3#100911yescharacteristic
307D3—122no—
308D3—122no—
309D3—242no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—182no—
314C22—182no—
315C22—182no—
316C22C22×D3211yescharacteristic
317C4—61no—
318C4—61no—
319C4—61no—
320C4—62no—
321C4—61no—
322C4—62no—
323C4—92no—
324C4—181no—
325C4#380412yes—
326C3—21no—
327C3#555911yescharacteristic, complemented
328C3#568111yescharacteristic, complemented
329C2—42no—
330C2—362no—
331C2#6431211yescharacteristic
332C2#6431411yescharacteristic
333C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
prime radical#64149C2
hypercenterC2×D4D32
hypocenterC32#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#936C2semidirect
3. C3×D4×Dic3C2semidirect
4. #63975C2semidirect
5. #63966C2semidirect
6. #63953C2semidirect
7. #63952C2semidirect
8. C4×#627C2semidirect
9. #63894C2semidirect
10. #63864C2semidirect
11. #63843C2semidirect
44. #936D3semidirect
45. D4×Dic3D3semidirect
275. C32#559semidirect
327. C3#5559semidirect
328. C3#5681semidirect

Maximal subgroups

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

subgroupindex
#641492
C3×#9362
C3×D4×Dic32
#639752
#639662
#639532
#639522
C4×#6272
#638942
#638642
#638432
#56813
#55593