Finite Group Databasebeta

← #10594951

Subgroups — #10594951

Order 576.

493 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64133C211yesmaximal, characteristic, complemented
2C3×C4×#67C211yesmaximal, characteristic
3C2×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
4Dic3×#67C211yesmaximal, characteristic
5Dic3×#67C211yesmaximal, characteristic
6#63964C211yesmaximal, characteristic, complemented
7#63964C211yesmaximal, characteristic, complemented
8#63896C211yesmaximal, characteristic, complemented
9#63893C211yesmaximal, characteristic, complemented
10D3×Dic3⋊C4C211yesmaximal, characteristic, complemented
11#63867C211yesmaximal, characteristic, complemented
12#63851C211yesmaximal, characteristic, complemented
13#63845C211yesmaximal, characteristic, complemented
14#63840C211yesmaximal, characteristic, complemented
15#63837C211yesmaximal, characteristic, complemented
16#5539—31nomaximal
17#5511—31nomaximal
18C22×C32×C4C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C22×C3×Dic3C2211yescharacteristic
22C22×C3×Dic3C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×#627C2211yescharacteristic
25C2×#627C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic
28#3757C2211yescharacteristic
29#3757C2211yescharacteristic
30C4⋊(C3⋊Dic3)C2211yescharacteristic
31#3750C2211yescharacteristic
32#3750C2211yescharacteristic
33C3×#217C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×Dic3⋊C4C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×Dic3⋊C4C2211yescharacteristic
39C3×Dic3⋊C4C2211yescharacteristic
40C3×C4×Dic3C2211yescharacteristic
41#3724C2211yescharacteristic
42#3724C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47Dic3⋊Dic3C2211yescharacteristic
48Dic3⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×C4×D4—31no—
54C2×C3×C4⋊C4—31no—
55D4×Dic3—31no—
56D4×Dic3—31no—
57C4×#67—31no—
58C4×#67—31no—
59C4×#67D311yescharacteristic
60#924—31no—
61#924—31no—
62#924—31no—
63#924—31no—
64#924—31no—
65#924—61no—
66C2×Dic3⋊C4—31no—
67C2×Dic3⋊C4D311yescharacteristic, complemented
68#897—31no—
69#896—31no—
70#896—31no—
71D3×C4⋊C4—31no—
72#890—31no—
73#889—31no—
74#889—31no—
75#888—31no—
76#888—31no—
77#887—31no—
78#878—31no—
79#878—31no—
80#878—31no—
81#878—31no—
82#870—31no—
83C4×Dic6—31no—
84C23×C32C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C2×C32×C4—41no—
87C2×C32×C4C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93#634—41no—
94C3×#67Q814yescomplemented
95C2×C3×Dic3—22no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104C3×C4×D3—41no—
105#627—22no—
106#627—41no—
107D3×Dic3—22no—
108D3×Dic3—41no—
109#554—91noSylow
110C2×C3×D4—31no—
111C22×C3×C4—21no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4C2×D311yescharacteristic
116C22×C3×C4C2×D311yescharacteristic
117C2×#67C2×D311yescharacteristic
118C22×Dic3—31no—
119C22×Dic3—31no—
120C22×Dic3C2×D311yescharacteristic
121C22×Dic3C2×D311yescharacteristic
122C2×C4×D3—31no—
123C2×C4×D3—31no—
124C2×C4×D3C2×D311yescharacteristic
125C2×Dic6—31no—
126C2×Dic6—31no—
127C2×Dic6—31no—
128C2×Dic6—31no—
129C2×Dic6—31no—
130C2×Dic6—31no—
131C2×Dic6—61no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C22⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C42—31no—
140#217—31no—
141#217—31no—
142#217—31no—
143#217—31no—
144#217—31no—
145#217—31no—
146#217—61no—
147#217—61no—
148#217C2×D311yescharacteristic
149(C2×D3)⋊C4—31no—
150(C2×D3)⋊C4—31no—
151(C2×D3)⋊C4C2×D311yescharacteristic
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—61no—
161Dic3⋊C4—31no—
162Dic3⋊C4—31no—
163Dic3⋊C4—31no—
164Dic3⋊C4—31no—
165Dic3⋊C4—31no—
166Dic3⋊C4—31no—
167Dic3⋊C4—31no—
168Dic3⋊C4—31no—
169Dic3⋊C4—31no—
170Dic3⋊C4—31no—
171Dic3⋊C4—31no—
172Dic3⋊C4—31no—
173Dic3⋊C4—31no—
174Dic3⋊C4—31no—
175Dic3⋊C4—61no—
176Dic3⋊C4—61no—
177Dic3⋊C4C2×D311yescharacteristic
178Dic3⋊C4C2×D311yescharacteristic
179Dic3⋊C4C2×D311yescharacteristic
180Dic3⋊C4C2×D311yescharacteristic
181Dic3⋊C4C2×D311yescharacteristic
182C4×Dic3—31no—
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3C2×D311yescharacteristic
187C22×C32—21no—
188C22×C32—21no—
189C22×C32C2×Q812yes—
190C22×C32C2411yescharacteristic
191C2×C3×D3—21no—
192C2×C3×D3—21no—
193C2×C3×D3C2×Q812yes—
194C32×C4—21no—
195C32×C4—21no—
196C3⋊Dic3—21no—
197C3⋊Dic3—21no—
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—21no—
203C3×Dic3—21no—
204C3×Dic3—21no—
205C3×Dic3C2×Q812yes—
206C3×Dic3#4112yes—
207C4⋊Q8—91no—
208#125—91no—
209#125—91no—
210C22⋊Q8—91no—
211C22⋊Q8—91no—
212C22⋊Q8—91no—
213C22⋊Q8—91no—
214C22⋊Q8—91no—
215C22⋊Q8—91no—
216C4×Q8—91no—
217C4×D4—31no—
218C4×D4—91no—
219C4×D4—91no—
220C2×C4⋊C4—31no—
221C2×C4⋊C4—91no—
222C23×C3—21no—
223C23×C3—31no—
224C23×C3C22×D311yescharacteristic
225C23×C3C22×D311yescharacteristic
226C22×D3C22×D311yescharacteristic
227C3×D4—34no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—62no—
241C2×C3×C4—121no—
242C2×C3×C4—121no—
243C2×C3×C4C22×D311yescharacteristic
244C2×C3×C4C22×D311yescharacteristic
245C2×C3×C4C22×D311yescharacteristic
246C2×C3×C4C22×D311yescharacteristic
247#67Dic614yescomplemented
248C2×Dic3—22no—
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—31no—
255C2×Dic3—31no—
256C2×Dic3—31no—
257C2×Dic3—41no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—62no—
263C2×Dic3—121no—
264C2×Dic3C22×D311yescharacteristic
265C2×Dic3C22×D311yescharacteristic
266C2×Dic3C22×D311yescharacteristic
267C2×Dic3C22×D311yescharacteristic
268C2×Dic3C22×D311yescharacteristic
269C2×Dic3C22×D311yescharacteristic
270C2×Dic3C22×D311yescharacteristic
271C4×D3—41no—
272C4×D3—62no—
273C4×D3—121no—
274Dic6—62no—
275Dic6—62no—
276Dic6—121no—
277Dic6—121no—
278Dic6—121no—
279Dic6—121no—
280Dic6—122no—
281C2×C32—22no—
282C2×C32C22×Q811yescharacteristic
283C2×C32C2×#4111yescharacteristic
284C2×C32H(5,𝔽2)11yescharacteristic
285C3×D3—22no—
286C2×Q8—91no—
287C2×Q8—91no—
288C2×Q8—91no—
289C2×D4—31no—
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—91no—
294C22×C4—91no—
295C22×C4D3211yescharacteristic
296C4⋊C4—31no—
297C4⋊C4—31no—
298C4⋊C4—31no—
299C4⋊C4—31no—
300C4⋊C4—31no—
301C4⋊C4—91no—
302C4⋊C4—91no—
303C4⋊C4—91no—
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4—91no—
308C4⋊C4—91no—
309C4⋊C4—91no—
310C4⋊C4—91no—
311C4⋊C4—91no—
312C22⋊C4—31no—
313C22⋊C4—31no—
314C22⋊C4—91no—
315C22⋊C4—91no—
316C22⋊C4—91no—
317C22⋊C4—91no—
318C42—31no—
319C42—91no—
320C42—91no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—32no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—61no—
331C22×C3—61no—
332C22×C3C2×Dic612yes—
333C22×C3D3×Q812yes—
334C22×C3C23×D311yescharacteristic
335C22×C3C23×D311yescharacteristic
336C2×D3—21no—
337C2×D3—21no—
338C2×D3C2×Dic612yes—
339C3×C4—22no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—32no—
345C3×C4—32no—
346C3×C4—41no—
347C3×C4—61no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—61no—
352Dic3—21no—
353Dic3—21no—
354Dic3—21no—
355Dic3—21no—
356Dic3—21no—
357Dic3—61no—
358Dic3—61no—
359Dic3—61no—
360Dic3—61no—
361Dic3—61no—
362Dic3—61no—
363Dic3—61no—
364Dic3—61no—
365Dic3—121no—
366Dic3—121no—
367Dic3—121no—
368Dic3—121no—
369Dic3C2×Dic612yes—
370Dic3#23512yes—
371C32#55411yescharacteristic, complemented, Sylow
372C23—31no—
373C23C2×D3211yescharacteristic
374Q8—182no—
375Q8—361no—
376Q8—361no—
377D4—34no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—41no—
386C2×C4—62no—
387C2×C4—91no—
388C2×C4—91no—
389C2×C4—91no—
390C2×C4—91no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—182no—
394C2×C4—361no—
395C2×C4C2×D3211yescharacteristic
396C2×C4C2×D3211yescharacteristic
397C2×C3—22no—
398C2×C3—22no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—42no—
403C2×C3—62no—
404C2×C3C22×Dic611yescharacteristic
405C2×C3C2×#23511yescharacteristic
406C2×C3#100411yescharacteristic
407C2×C3#100411yescharacteristic
408C2×C3C2×D3×Q811yescharacteristic
409C2×C3D3×#4111yescharacteristic
410D3—22no—
411C22—21no—
412C22—21no—
413C22—32no—
414C22—61no—
415C22—61no—
416C22D3×Dic612yes—
417C22C22×D3211yescharacteristic
418C4—21no—
419C4—21no—
420C4—32no—
421C4—32no—
422C4—61no—
423C4—61no—
424C4—61no—
425C4—61no—
426C4—61no—
427C4—181no—
428C4—181no—
429C4—181no—
430C4—181no—
431C3—21no—
432C3#551111yescharacteristic, complemented
433C3#553911yescharacteristic, complemented
434C2—22no—
435C2—62no—
436C2C2×D3×Dic611yescharacteristic
437C2D3×#23511yescharacteristic
438C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×C3×#67C22
hypercenterC22×C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64133C2semidirect
3. C2×C3×Dic3⋊C4C2semidirect
6. #63964C2semidirect
7. #63964C2semidirect
8. #63896C2semidirect
9. #63893C2semidirect
10. D3×Dic3⋊C4C2semidirect
11. #63867C2semidirect
12. #63851C2semidirect
13. #63845C2semidirect
14. #63840C2semidirect
15. #63837C2semidirect
67. C2×Dic3⋊C4D3semidirect
94. C3×#67Q8semidirect
247. #67Dic6semidirect
371. C32#554semidirect
432. C3#5511semidirect
433. C3#5539semidirect

Maximal subgroups

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

subgroupindex
#641332
C3×C4×#672
C2×C3×Dic3⋊C42
Dic3×#672
#639642
#638962
#638932
D3×Dic3⋊C42
#638672
#638512
#638452
#638402
#638372
#55393
#55113