Finite Group Databasebeta

← #10593982

Subgroups — #10593982

Order 576.

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

normal subgroups only

This group has 523 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
1C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×C4×D12C211yesmaximal, characteristic, complemented
3C3×#870C211yesmaximal, characteristic, complemented
4#63916C211yesmaximal, characteristic, complemented
5#63916C211yesmaximal, characteristic, complemented
6#63910C212yesmaximal, complemented
7#63904C212yesmaximal, complemented
8#63900C211yesmaximal, characteristic, complemented
9#63861C212yesmaximal, complemented
10#63850C212yesmaximal, complemented
11#63849C211yesmaximal, characteristic, complemented
12#5695—31nomaximal
13#5558—31nomaximal
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×D12C2211yescharacteristic
18C2×C3×C4×D3C2212yes—
19C2×#626C2212yes—
20C2×#626C2212yes—
21C2×#624C2212yes—
22C32×C42C2211yescharacteristic
23#3752C2211yescharacteristic
24#3752C2211yescharacteristic
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26C3×(C2×D3)⋊C4C2212yes—
27C3×C4⋊Dic3C2211yescharacteristic
28C3×C4⋊Dic3C2211yescharacteristic
29C3×C4⋊Dic3C2211yescharacteristic
30C3×Dic3⋊C4C2212yes—
31C3×Dic3⋊C4C2212yes—
32#3724C2212yes—
33#3722C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×#125—31no—
38C3×C4×D4—31no—
39#938—31no—
40#938—31no—
41#929—32no—
42C4×#67—32no—
43#898—32no—
44#895—31no—
45#895—31no—
46#895—32no—
47#894—32no—
48#893—32no—
49#892—31no—
50#892—31no—
51D3×C4⋊C4—31no—
52D3×C4⋊C4—31no—
53#884—32no—
54#879—32no—
55C4×D12—31no—
56C4×D12—31no—
57C4×D12—61no—
58C4×D12D311yescharacteristic, complemented
59#870D311yescharacteristic, complemented
60C22×C3⋊D3C2311yescharacteristic
61C22×C3⋊D3C2311yescharacteristic
62C22×C3×D3C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C32×C4C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C4⋊(C3⋊D3)—22no—
69C4×C3⋊D3—22no—
70C4×C3⋊D3—22no—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2312yes—
74C3×D12—22no—
75C3×C4×D3—42no—
76#626—24no—
77#626—42no—
78#624—42no—
79#553—91noSylow
80C2×C3×D4—31no—
81C22×C3×C4—32no—
82C2×#67—32no—
83C2×#67—32no—
84C2×D12—32no—
85C2×D12—31no—
86C2×D12—31no—
87C2×D12—32no—
88C2×D12—61no—
89C2×D12C2×D311yescharacteristic
90C2×C4×D3—31no—
91C2×C4×D3—31no—
92C2×C4×D3—31no—
93C2×C4×D3—31no—
94C2×C4×D3—32no—
95C2×C4×D3—32no—
96C2×C4×D3—61no—
97C2×C4×D3—61no—
98C2×C4×D3C2×D312yes—
99C3×C4⋊C4—31no—
100C3×C4⋊C4—31no—
101C3×C4⋊C4—31no—
102C3×C4⋊C4—32no—
103C3×C4⋊C4—32no—
104C3×C22⋊C4—32no—
105C3×C42—21no—
106C3×C42C2×D311yescharacteristic
107C3×C42C2×D311yescharacteristic
108#217—32no—
109#217—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—31no—
113(C2×D3)⋊C4—31no—
114(C2×D3)⋊C4—31no—
115(C2×D3)⋊C4—31no—
116(C2×D3)⋊C4—32no—
117(C2×D3)⋊C4—32no—
118(C2×D3)⋊C4—61no—
119(C2×D3)⋊C4—61no—
120(C2×D3)⋊C4C2×D312yes—
121C4⋊Dic3—31no—
122C4⋊Dic3—31no—
123C4⋊Dic3—61no—
124C4⋊Dic3C2×D311yescharacteristic
125C4⋊Dic3C2×D311yescharacteristic
126C4⋊Dic3C2×D311yescharacteristic
127Dic3⋊C4—32no—
128Dic3⋊C4—32no—
129Dic3⋊C4C2×D312yes—
130Dic3⋊C4C2×D312yes—
131C4×Dic3—32no—
132C4×Dic3—32no—
133C22×C32C2411yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3—21no—
137C2×C3⋊D3—21no—
138C2×C3⋊D3—21no—
139C2×C3⋊D3#4112yes—
140C2×C3×D3—22no—
141C2×C3×D3—22no—
142C2×C3×D3—22no—
143C32×C4—21no—
144C32×C4—21no—
145C32×C4#4112yes—
146C3⋊Dic3—21no—
147C3⋊Dic3—21no—
148C3×Dic3—22no—
149C3×Dic3—22no—
150C3×Dic3—22no—
151#126—92no—
152#125—31no—
153#123—92no—
154#121—92no—
155#121—91no—
156#121—91no—
157C4×D4—31no—
158C4×D4—92no—
159C4×D4—91no—
160#117—91no—
161C2×C4⋊C4—91no—
162C23×C3—32no—
163C22×D3—31no—
164C22×D3—31no—
165C22×D3—31no—
166C22×D3—31no—
167C22×D3—61no—
168C22×D3—61no—
169C22×D3C22×D312yes—
170C3×D4—62no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—122no—
178C2×C3×C4C22×D311yescharacteristic
179C2×C3×C4C22×D311yescharacteristic
180C2×C3×C4C22×D311yescharacteristic
181C2×C3×C4C22×D311yescharacteristic
182C2×C3×C4C22×D311yescharacteristic
183C2×C3×C4C22×D311yescharacteristic
184#67—64no—
185#67—122no—
186C2×Dic3—31no—
187C2×Dic3—31no—
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192C2×Dic3C22×D312yes—
193C2×Dic3C22×D312yes—
194C2×Dic3C22×D312yes—
195D12—22no—
196D12—62no—
197D12—62no—
198D12—64no—
199D12—64no—
200D12—122no—
201C4×D3—42no—
202C4×D3—62no—
203C4×D3—62no—
204C4×D3—62no—
205C4×D3—62no—
206C4×D3—122no—
207C4×D3—122no—
208C4×D3—122no—
209C4×D3—122no—
210C2×C32C2×#4111yescharacteristic
211C2×C32C2×#4111yescharacteristic
212C2×C32H(5,𝔽2)11yescharacteristic
213C3⋊D3—22no—
214C3⋊D3—41no—
215C3×D3—42no—
216C2×D4—31no—
217C2×D4—92no—
218C2×D4—92no—
219C2×D4—91no—
220C22×C4—32no—
221C22×C4—92no—
222C22×C4—91no—
223C22×C4—91no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—31no—
227C4⋊C4—32no—
228C4⋊C4—32no—
229C4⋊C4—92no—
230C4⋊C4—91no—
231C22⋊C4—32no—
232C22⋊C4—92no—
233C22⋊C4—92no—
234C22⋊C4—92no—
235C22⋊C4—91no—
236C22⋊C4—91no—
237C42—92no—
238C42D3211yescharacteristic
239C22×C3—21no—
240C22×C3—62no—
241C22×C3—62no—
242C22×C3—62no—
243C22×C3C23×D311yescharacteristic
244C22×C3C23×D311yescharacteristic
245C2×D3—22no—
246C2×D3—22no—
247C2×D3—22no—
248C2×D3—32no—
249C2×D3—32no—
250C2×D3—62no—
251C2×D3—61no—
252C2×D3—61no—
253C2×D3—61no—
254C2×D3—61no—
255C2×D3—62no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—61no—
259C2×D3—61no—
260C2×D3—61no—
261C2×D3—61no—
262C2×D3—121no—
263C2×D3—121no—
264C2×D3—121no—
265C2×D3—121no—
266C3×C4—22no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—22no—
272C3×C4—41no—
273C3×C4—62no—
274C3×C4—62no—
275C3×C4—62no—
276C3×C4#23512yes—
277C3×C4#23912yes—
278Dic3—22no—
279Dic3—22no—
280Dic3—22no—
281Dic3—61no—
282Dic3—61no—
283Dic3—61no—
284Dic3—61no—
285Dic3—121no—
286Dic3—121no—
287C32#55311yescharacteristic, complemented, Sylow
288C23—32no—
289C23—91no—
290C23—91no—
291D4—62no—
292D4—182no—
293D4—184no—
294D4—362no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—91no—
299C2×C4—91no—
300C2×C4—122no—
301C2×C4—182no—
302C2×C4—182no—
303C2×C4—362no—
304C2×C4C2×D3211yescharacteristic
305C2×C4C2×D3211yescharacteristic
306C2×C4C2×D3211yescharacteristic
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—122no—
311C2×C3C2×#23511yescharacteristic
312C2×C3#100411yescharacteristic
313C2×C3C2×#23911yescharacteristic
314C2×C3D3×#4111yescharacteristic
315C2×C3D3×#4111yescharacteristic
316C2×C3#100911yescharacteristic
317D3—42no—
318D3—62no—
319D3—62no—
320D3—122no—
321D3—122no—
322D3—121no—
323D3—121no—
324C22—62no—
325C22—62no—
326C22—62no—
327C22—92no—
328C22—181no—
329C22—181no—
330C22—181no—
331C22—181no—
332C22—181no—
333C22C22×D3211yescharacteristic
334C4—21no—
335C4—21no—
336C4—62no—
337C4—62no—
338C4—62no—
339C4—181no—
340C4—181no—
341C4#379912yes—
342C3—21no—
343C3#555811yescharacteristic, complemented
344C3#569511yescharacteristic, complemented
345C2—122no—
346C2—182no—
347C2—361no—
348C2C2×#379911yescharacteristic
349C2#6430611yescharacteristic
350C2#6430711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊D3)C2semidirect
2. C3×C4×D12C2semidirect
3. C3×#870C2semidirect
4. #63916C2semidirect
5. #63916C2semidirect
6. #63910C2semidirect
7. #63904C2semidirect
8. #63900C2semidirect
9. #63861C2semidirect
10. #63850C2semidirect
11. #63849C2semidirect
58. C4×D12D3semidirect
59. #870D3semidirect
287. C32#553semidirect
343. C3#5558semidirect
344. C3#5695semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D3)2
C3×C4×D122
C3×#8702
#639162
#639102
#639042
#639002
#638612
#638502
#638492
#56953
#55583