Finite Group Databasebeta

← #10594535

Subgroups — #10594535

Order 576.

484 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64098C211yesmaximal, characteristic, complemented
2C3×#897C211yesmaximal, characteristic, complemented
3C3×#896C211yesmaximal, characteristic, complemented
4#63902C211yesmaximal, characteristic
5#63899C211yesmaximal, characteristic, complemented
6#63893C211yesmaximal, characteristic, complemented
7D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
8#63887C211yesmaximal, characteristic, complemented
9#63887C211yesmaximal, characteristic, complemented
10#63879C211yesmaximal, characteristic
11#63879C211yesmaximal, characteristic
12D3×Dic3⋊C4C211yesmaximal, characteristic, complemented
13#63847C211yesmaximal, characteristic, complemented
14#63843C211yesmaximal, characteristic, complemented
15#63839C211yesmaximal, characteristic, complemented
16#5636—31nomaximal
17#5635—31nomaximal
18C2×C3×C4×D3C2211yescharacteristic
19C2×C3×C4×D3C2211yescharacteristic
20C2×C3×Dic6C2211yescharacteristic
21C2×C3×Dic6C2211yescharacteristic
22C2×#627C2211yescharacteristic
23C2×#625C2211yescharacteristic
24C2×D3×Dic3C2211yescharacteristic
25C2×D3×Dic3C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic
28C32×C4⋊C4C2211yescharacteristic
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30#3750C2211yescharacteristic
31#3750C2211yescharacteristic
32#3750C2211yescharacteristic
33#3750C2211yescharacteristic
34C4×C3⋊Dic3C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×C4⋊Dic3C2211yescharacteristic
40C3×C4⋊Dic3C2211yescharacteristic
41C3×Dic3⋊C4C2211yescharacteristic
42C3×Dic3⋊C4C2211yescharacteristic
43#3724C2211yescharacteristic
44#3724C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47Dic3⋊Dic3C2211yescharacteristic
48Dic3⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×C22⋊Q8—31no—
54C3×C22⋊Q8—31no—
55#946—31no—
56#946—31no—
57Q8×Dic3—31no—
58#944—31no—
59C4×#67—31no—
60C4×#67—31no—
61C2×C4⋊Dic3—31no—
62C2×Dic3⋊C4—31no—
63#897—31no—
64#897D311yescharacteristic, complemented
65#896—31no—
66#896D311yescharacteristic, complemented
67D3×C4⋊C4—31no—
68D3×C4⋊C4—31no—
69#889—31no—
70#889—31no—
71#889—31no—
72#889—31no—
73#889—61no—
74#888—31no—
75#887—31no—
76#67⋊C4—31no—
77#67⋊C4—31no—
78#67⋊C4—31no—
79#67⋊C4—31no—
80#878—31no—
81#878—31no—
82#878—31no—
83#878—31no—
84C22×C3×D3C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C2×C32×C4C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88C2×C32×C4C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93C2×C3×Dic3C2311yescharacteristic
94C2×C3×Dic3C2311yescharacteristic
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C3×C4×D3—22no—
100C3×C4×D3—41no—
101C3×Dic6—22no—
102C3×Dic6—41no—
103#627—41no—
104#625Q814yescomplemented
105D3×Dic3—22no—
106D3×Dic3—41no—
107D3×Dic3—41no—
108D3×Dic3—41no—
109#554—91noSylow
110C2×C3×Q8—31no—
111C2×C3×Q8—31no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C2×#67—31no—
115C2×#67—31no—
116C22×Dic3—31no—
117C22×Dic3—31no—
118C22×Dic3—31no—
119C22×Dic3—31no—
120C2×C4×D3—31no—
121C2×C4×D3—31no—
122C2×C4×D3—31no—
123C2×C4×D3—31no—
124C2×C4×D3C2×D311yescharacteristic
125C2×C4×D3C2×D311yescharacteristic
126C2×Dic6—31no—
127C2×Dic6—31no—
128C2×Dic6C2×D311yescharacteristic
129C2×Dic6C2×D311yescharacteristic
130C3×C4⋊C4—21no—
131C3×C4⋊C4—31no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4C2×D311yescharacteristic
136C3×C4⋊C4C2×D311yescharacteristic
137C3×C22⋊C4—31no—
138C3×C22⋊C4—31no—
139C3×C22⋊C4—31no—
140C3×C22⋊C4—31no—
141#217—31no—
142#217—31no—
143(C2×D3)⋊C4—31no—
144(C2×D3)⋊C4—31no—
145(C2×D3)⋊C4C2×D311yescharacteristic
146(C2×D3)⋊C4C2×D311yescharacteristic
147(C2×D3)⋊C4C2×D311yescharacteristic
148(C2×D3)⋊C4C2×D311yescharacteristic
149C4⋊Dic3—31no—
150C4⋊Dic3—31no—
151C4⋊Dic3—31no—
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3—31no—
155C4⋊Dic3—61no—
156C4⋊Dic3C2×D311yescharacteristic
157C4⋊Dic3C2×D311yescharacteristic
158Dic3⋊C4—31no—
159Dic3⋊C4—31no—
160Dic3⋊C4—31no—
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—61no—
175Dic3⋊C4—61no—
176Dic3⋊C4—61no—
177Dic3⋊C4—61no—
178Dic3⋊C4C2×D311yescharacteristic
179Dic3⋊C4C2×D311yescharacteristic
180C4×Dic3—31no—
181C4×Dic3—31no—
182C4×Dic3—31no—
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3—61no—
187C22×C32C2411yescharacteristic
188C2×C3×D3—21no—
189C2×C3×D3—21no—
190C2×C3×D3—21no—
191C2×C3×D3—21no—
192C2×C3×D3C2×Q812yes—
193C2×C3×D3C2×Q812yes—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—21no—
197C3⋊Dic3—21no—
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3⋊Dic3C2×Q812yes—
201C3×Dic3—21no—
202C3×Dic3—21no—
203C3×Dic3—21no—
204C3×Dic3—21no—
205C3×Dic3—21no—
206C3×Dic3#4112yes—
207C4⋊Q8—91no—
208#125—91no—
209#125—91no—
210C22⋊Q8—31no—
211C22⋊Q8—31no—
212C22⋊Q8—91no—
213C22⋊Q8—91no—
214C22⋊Q8—91no—
215C22⋊Q8—91no—
216C4×Q8—91no—
217C4×D4—91no—
218C4×D4—91no—
219C4×D4—91no—
220C2×C4⋊C4—91no—
221C2×C4⋊C4—91no—
222C23×C3—31no—
223C23×C3—31no—
224C22×D3C22×D311yescharacteristic
225C22×D3C22×D311yescharacteristic
226C3×Q8—62no—
227C3×Q8—121no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
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—62no—
238C2×C3×C4—121no—
239C2×C3×C4C22×D311yescharacteristic
240C2×C3×C4C22×D311yescharacteristic
241C2×C3×C4C22×D311yescharacteristic
242C2×C3×C4C22×D311yescharacteristic
243C2×C3×C4C22×D311yescharacteristic
244C2×C3×C4C22×D311yescharacteristic
245#67—34no—
246#67—34no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—31no—
255C2×Dic3—61no—
256C2×Dic3—61no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259C2×Dic3—62no—
260C2×Dic3—121no—
261C2×Dic3—121no—
262C2×Dic3—121no—
263C2×Dic3C22×D311yescharacteristic
264C2×Dic3C22×D311yescharacteristic
265C2×Dic3C22×D311yescharacteristic
266C2×Dic3C22×D311yescharacteristic
267C2×Dic3C22×D311yescharacteristic
268C2×Dic3C22×D311yescharacteristic
269C4×D3—22no—
270C4×D3—41no—
271C4×D3—62no—
272C4×D3—121no—
273C4×D3—121no—
274C4×D3—121no—
275Dic6—22no—
276Dic6—41no—
277Dic6—121no—
278Dic6—121no—
279C2×C32C22×Q811yescharacteristic
280C2×C32C2×#4111yescharacteristic
281C2×C32H(5,𝔽2)11yescharacteristic
282C3×D3—22no—
283C3×D3—22no—
284C2×Q8—31no—
285C2×Q8—31no—
286C2×Q8—91no—
287C2×D4—91no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—91no—
291C22×C4—91no—
292C22×C4—91no—
293C22×C4—91no—
294C4⋊C4—31no—
295C4⋊C4—31no—
296C4⋊C4—31no—
297C4⋊C4—31no—
298C4⋊C4—91no—
299C4⋊C4—91no—
300C4⋊C4—91no—
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⋊C4D3211yescharacteristic
310C22⋊C4—31no—
311C22⋊C4—31no—
312C22⋊C4—31no—
313C22⋊C4—31no—
314C22⋊C4—91no—
315C22⋊C4—91no—
316C42—91no—
317C42—91no—
318C42—91no—
319C22×C3—21no—
320C22×C3—32no—
321C22×C3—32no—
322C22×C3—61no—
323C22×C3—61no—
324C22×C3—61no—
325C22×C3—61no—
326C22×C3C23×D311yescharacteristic
327C22×C3C23×D311yescharacteristic
328C2×D3—21no—
329C2×D3—21no—
330C2×D3—21no—
331C2×D3—21no—
332C2×D3D3×Q812yes—
333C2×D3D3×Q812yes—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—32no—
341C3×C4—41no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—61no—
347C3×C4—61no—
348C3×C4—61no—
349Dic3—21no—
350Dic3—21no—
351Dic3—21no—
352Dic3—21no—
353Dic3—21no—
354Dic3—32no—
355Dic3—32no—
356Dic3—62no—
357Dic3—61no—
358Dic3—61no—
359Dic3—61no—
360Dic3—61no—
361Dic3—61no—
362Dic3—61no—
363Dic3—121no—
364Dic3—121no—
365Dic3—121no—
366Dic3#23712yes—
367C32#55411yescharacteristic, complemented, Sylow
368C23—31no—
369C23—31no—
370Q8—62no—
371Q8—121no—
372Q8—361no—
373D4—94no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—62no—
381C2×C4—91no—
382C2×C4—91no—
383C2×C4—91no—
384C2×C4—91no—
385C2×C4—121no—
386C2×C4—182no—
387C2×C4—361no—
388C2×C4—361no—
389C2×C4—361no—
390C2×C4C2×D3211yescharacteristic
391C2×C4C2×D3211yescharacteristic
392C2×C4C2×D3211yescharacteristic
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—62no—
397C2×C3—62no—
398C2×C3C2×#23711yescharacteristic
399C2×C3#100411yescharacteristic
400C2×C3C2×D3×Q811yescharacteristic
401C2×C3C2×D3×Q811yescharacteristic
402C2×C3D3×#4111yescharacteristic
403C2×C3#100911yescharacteristic
404D3—22no—
405D3—22no—
406C22—32no—
407C22—32no—
408C22—61no—
409C22—61no—
410C22—61no—
411C22—61no—
412C22C22×D3211yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—32no—
417C4—61no—
418C4—61no—
419C4—61no—
420C4—61no—
421C4—61no—
422C4—92no—
423C4—181no—
424C4—181no—
425C4—181no—
426C3—21no—
427C3#563511yescharacteristic, complemented
428C3#563611yescharacteristic, complemented
429C2—62no—
430C2—62no—
431C2#6430711yescharacteristic
432C2D3×#23711yescharacteristic
433C2D32×Q811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#625C22
hypercenterC4⋊C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64098C2semidirect
2. C3×#897C2semidirect
3. C3×#896C2semidirect
5. #63899C2semidirect
6. #63893C2semidirect
7. D3×C4⋊Dic3C2semidirect
8. #63887C2semidirect
9. #63887C2semidirect
12. D3×Dic3⋊C4C2semidirect
13. #63847C2semidirect
14. #63843C2semidirect
15. #63839C2semidirect
64. #897D3semidirect
66. #896D3semidirect
104. #625Q8semidirect
367. C32#554semidirect
427. C3#5635semidirect
428. C3#5636semidirect

Maximal subgroups

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

subgroupindex
#640982
C3×#8972
C3×#8962
#639022
#638992
#638932
D3×C4⋊Dic32
#638872
#638792
D3×Dic3⋊C42
#638472
#638432
#638392
#56363
#56353