Finite Group Databasebeta

← #10594469

Subgroups — #10594469

Order 576.

467 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64103C211yesmaximal, characteristic
2C3×#890C211yesmaximal, characteristic, complemented
3C3×#889C211yesmaximal, characteristic, complemented
4#63897C211yesmaximal, characteristic
5#63895C211yesmaximal, characteristic, complemented
6#63895C211yesmaximal, characteristic, complemented
7#63894C211yesmaximal, characteristic
8#63883C211yesmaximal, characteristic, complemented
9#63849C211yesmaximal, characteristic, complemented
10#63848C211yesmaximal, characteristic, complemented
11#63848C211yesmaximal, characteristic, complemented
12#63846C211yesmaximal, characteristic, complemented
13#63837C211yesmaximal, characteristic, complemented
14#63836C211yesmaximal, characteristic, complemented
15#63836C211yesmaximal, characteristic, complemented
16#5694—31nomaximal
17#5693—31nomaximal
18C2×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×#627C2211yescharacteristic
21C2×#624C2211yescharacteristic
22C2×#624C2211yescharacteristic
23C2×#624C2211yescharacteristic
24C32×C4⋊C4C2211yescharacteristic
25#3752C2211yescharacteristic
26#3752C2211yescharacteristic
27#3752C2211yescharacteristic
28#3750C2211yescharacteristic
29C3×C4⋊Dic3C2211yescharacteristic
30C3×C4⋊Dic3C2211yescharacteristic
31C3×C4⋊Dic3C2211yescharacteristic
32C3×C4⋊Dic3C2211yescharacteristic
33C3×Dic3⋊C4C2211yescharacteristic
34C3×Dic3⋊C4C2211yescharacteristic
35C3×Dic3⋊C4C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×Dic3⋊C4C2211yescharacteristic
39C3×C4×Dic3C2211yescharacteristic
40C3×C4×Dic3C2211yescharacteristic
41Dic3⋊Dic3C2211yescharacteristic
42Dic3⋊Dic3C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic
49#3722C2211yescharacteristic
50Dic32C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#125—31no—
54C3×#125—31no—
55#898—31no—
56#898—31no—
57#898—31no—
58#898—31no—
59#897—31no—
60#897—31no—
61#896—31no—
62#896—31no—
63#894—31no—
64#894—31no—
65#894—31no—
66#894—31no—
67#894—61no—
68#893—31no—
69#892—31no—
70#892—31no—
71#892—31no—
72#892—31no—
73D3×C4⋊C4—31no—
74D3×C4⋊C4—31no—
75#890—31no—
76#890D311yescharacteristic, complemented
77#889—31no—
78#889D311yescharacteristic, complemented
79#887—31no—
80C4×D12—31no—
81#872—31no—
82#872—31no—
83C4×Dic6—31no—
84C22×C3⋊D3C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C2×C32×C4C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C4⋊(C3⋊D3)—22no—
92C4×C3⋊D3—22no—
93C2×C3×Dic3C2311yescharacteristic
94C2×C3×Dic3C2311yescharacteristic
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101#627—22no—
102#624—22no—
103#624—22no—
104#624—22no—
105#552—91noSylow
106C2×D12—31no—
107C2×D12—31no—
108C2×D12—61no—
109C2×C4×D3—31no—
110C2×C4×D3—31no—
111C2×C4×D3—31no—
112C2×C4×D3—31no—
113C2×C4×D3—31no—
114C2×C4×D3—31no—
115C2×C4×D3—31no—
116C2×C4×D3—31no—
117C2×C4×D3—61no—
118C2×Dic6—31no—
119C2×Dic6—31no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—31no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—31no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4—31no—
126C3×C4⋊C4—31no—
127C3×C4⋊C4—31no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—31no—
130C3×C4⋊C4—31no—
131C3×C4⋊C4C2×D311yescharacteristic
132C3×C4⋊C4C2×D311yescharacteristic
133C3×C42—31no—
134C3×C42—31no—
135(C2×D3)⋊C4—31no—
136(C2×D3)⋊C4—31no—
137(C2×D3)⋊C4—31no—
138(C2×D3)⋊C4—31no—
139(C2×D3)⋊C4—31no—
140(C2×D3)⋊C4—31no—
141(C2×D3)⋊C4—31no—
142(C2×D3)⋊C4—31no—
143(C2×D3)⋊C4—31no—
144(C2×D3)⋊C4—31no—
145(C2×D3)⋊C4—31no—
146(C2×D3)⋊C4—31no—
147(C2×D3)⋊C4—31no—
148(C2×D3)⋊C4—31no—
149(C2×D3)⋊C4—31no—
150(C2×D3)⋊C4—31no—
151(C2×D3)⋊C4—61no—
152(C2×D3)⋊C4—61no—
153(C2×D3)⋊C4—61no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3C2×D311yescharacteristic
159C4⋊Dic3C2×D311yescharacteristic
160C4⋊Dic3C2×D311yescharacteristic
161C4⋊Dic3C2×D311yescharacteristic
162Dic3⋊C4—31no—
163Dic3⋊C4—31no—
164Dic3⋊C4—31no—
165Dic3⋊C4—31no—
166Dic3⋊C4—31no—
167Dic3⋊C4—31no—
168Dic3⋊C4—61no—
169Dic3⋊C4C2×D311yescharacteristic
170Dic3⋊C4C2×D311yescharacteristic
171Dic3⋊C4C2×D311yescharacteristic
172Dic3⋊C4C2×D311yescharacteristic
173Dic3⋊C4C2×D311yescharacteristic
174Dic3⋊C4C2×D311yescharacteristic
175C4×Dic3—31no—
176C4×Dic3—31no—
177C4×Dic3—31no—
178C4×Dic3—31no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3C2×D311yescharacteristic
182C4×Dic3C2×D311yescharacteristic
183C22×C32C2411yescharacteristic
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3#4112yes—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—21no—
193C3⋊Dic3—21no—
194C3⋊Dic3—21no—
195C3×Dic3—21no—
196C3×Dic3—21no—
197C3×Dic3—21no—
198C3×Dic3—21no—
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3#4112yes—
203#126—91no—
204#126—91no—
205#125—31no—
206#125—31no—
207#125—91no—
208#123—91no—
209#123—91no—
210C22⋊Q8—91no—
211C22⋊Q8—91no—
212C4×Q8—91no—
213C4×D4—91no—
214#117—91no—
215#117—91no—
216#117—91no—
217C2×C4⋊C4—91no—
218C22×D3—31no—
219C22×D3—31no—
220C22×D3—31no—
221C22×D3—31no—
222C22×D3—61no—
223C22×D3—61no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4C22×D311yescharacteristic
236C2×C3×C4C22×D311yescharacteristic
237C2×C3×C4C22×D311yescharacteristic
238C2×C3×C4C22×D311yescharacteristic
239C2×C3×C4C22×D311yescharacteristic
240C2×C3×C4C22×D311yescharacteristic
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3C22×D311yescharacteristic
248C2×Dic3C22×D311yescharacteristic
249C2×Dic3C22×D311yescharacteristic
250C2×Dic3C22×D311yescharacteristic
251C2×Dic3C22×D311yescharacteristic
252C2×Dic3C22×D311yescharacteristic
253C2×Dic3C22×D311yescharacteristic
254C2×Dic3C22×D311yescharacteristic
255D12—62no—
256D12—62no—
257D12—122no—
258C4×D3—62no—
259C4×D3—62no—
260C4×D3—62no—
261C4×D3—62no—
262C4×D3—62no—
263C4×D3—62no—
264C4×D3—62no—
265C4×D3—62no—
266C4×D3—122no—
267Dic6—62no—
268Dic6—62no—
269C2×C32C2×#4111yescharacteristic
270C2×C32C2×#4111yescharacteristic
271C2×C322−1+411yescharacteristic
272C3⋊D3—22no—
273C3⋊D3—41no—
274C2×Q8—91no—
275C2×D4—91no—
276C22×C4—91no—
277C22×C4—91no—
278C22×C4—91no—
279C22×C4—91no—
280C4⋊C4—31no—
281C4⋊C4—31no—
282C4⋊C4—31no—
283C4⋊C4—31no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—31no—
287C4⋊C4—31no—
288C4⋊C4—31no—
289C4⋊C4—31no—
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—91no—
293C4⋊C4—91no—
294C4⋊C4—91no—
295C4⋊C4D3211yescharacteristic
296C22⋊C4—91no—
297C22⋊C4—91no—
298C22⋊C4—91no—
299C22⋊C4—91no—
300C22⋊C4—91no—
301C22⋊C4—91no—
302C22⋊C4—91no—
303C22⋊C4—91no—
304C42—31no—
305C42—31no—
306C42—91no—
307C42—91no—
308C42—91no—
309C22×C3—21no—
310C22×C3C23×D311yescharacteristic
311C22×C3C23×D311yescharacteristic
312C2×D3—32no—
313C2×D3—32no—
314C2×D3—62no—
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—61no—
318C2×D3—61no—
319C2×D3—61no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—121no—
329C2×D3—121no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—32no—
337C3×C4—41no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—61no—
347Dic3—21no—
348Dic3—21no—
349Dic3—21no—
350Dic3—21no—
351Dic3—21no—
352Dic3—21no—
353Dic3—21no—
354Dic3—61no—
355Dic3—61no—
356Dic3—61no—
357Dic3—61no—
358Dic3—121no—
359Dic3—121no—
360Dic3#23912yes—
361C32#55211yescharacteristic, complemented, Sylow
362C23—91no—
363C23—91no—
364Q8—182no—
365D4—182no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—182no—
377C2×C4—182no—
378C2×C4—182no—
379C2×C4—182no—
380C2×C4C2×D3211yescharacteristic
381C2×C4C2×D3211yescharacteristic
382C2×C4C2×D3211yescharacteristic
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3C2×#23911yescharacteristic
387C2×C3#100711yescharacteristic
388C2×C3D3×#4111yescharacteristic
389C2×C3D3×#4111yescharacteristic
390C2×C3D3×#4111yescharacteristic
391C2×C3#101011yescharacteristic
392D3—62no—
393D3—62no—
394D3—122no—
395D3—121no—
396D3—121no—
397D3—241no—
398C22—92no—
399C22—181no—
400C22—181no—
401C22—181no—
402C22—181no—
403C22—181no—
404C22C22×D3211yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—32no—
409C4—61no—
410C4—61no—
411C4—61no—
412C4—61no—
413C4—61no—
414C4—61no—
415C4—61no—
416C4—181no—
417C4—181no—
418C3—21no—
419C3#569311yescharacteristic, complemented
420C3#569411yescharacteristic, complemented
421C2—182no—
422C2—361no—
423C2#6429711yescharacteristic
424C2#6431211yescharacteristic
425C2D3×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC4⋊C4D32
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#890C2semidirect
3. C3×#889C2semidirect
5. #63895C2semidirect
6. #63895C2semidirect
8. #63883C2semidirect
9. #63849C2semidirect
10. #63848C2semidirect
11. #63848C2semidirect
12. #63846C2semidirect
13. #63837C2semidirect
14. #63836C2semidirect
15. #63836C2semidirect
76. #890D3semidirect
78. #889D3semidirect
361. C32#552semidirect
419. C3#5693semidirect
420. C3#5694semidirect

Maximal subgroups

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

subgroupindex
#641032
C3×#8902
C3×#8892
#638972
#638952
#638942
#638832
#638492
#638482
#638462
#638372
#638362
#56943
#56933