Finite Group Databasebeta

← #10594171

Subgroups — #10594171

Order 576.

422 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64088C211yesmaximal, characteristic, complemented
2C3×#885C211yesmaximal, characteristic
3C3×#878C211yesmaximal, characteristic, complemented
4#63969C211yesmaximal, characteristic
5#63964C211yesmaximal, characteristic, complemented
6#63961C211yesmaximal, characteristic
7#63956C211yesmaximal, characteristic, complemented
8#63899C211yesmaximal, characteristic, complemented
9#63887C211yesmaximal, characteristic, complemented
10#63873C211yesmaximal, characteristic, complemented
11#63869C211yesmaximal, characteristic, complemented
12#63867C211yesmaximal, characteristic, complemented
13#63862C211yesmaximal, characteristic, complemented
14#63861C211yesmaximal, characteristic, complemented
15#63839C211yesmaximal, characteristic, complemented
16#5672—31nomaximal
17#5646—31nomaximal
18C2×C3×#67C2211yescharacteristic
19C22×C3×Dic3C2211yescharacteristic
20C2×C3×Dic6C2211yescharacteristic
21C2×C3×Dic6C2211yescharacteristic
22C2×#627C2211yescharacteristic
23C2×D3×Dic3C2211yescharacteristic
24C32×C22⋊C4C2211yescharacteristic
25#3757C2211yescharacteristic
26#3757C2211yescharacteristic
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28#3750C2211yescharacteristic
29#3750C2211yescharacteristic
30C4×C3⋊Dic3C2211yescharacteristic
31C3×#217C2211yescharacteristic
32C3×#217C2211yescharacteristic
33C3×(C2×D3)⋊C4C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×C4⋊Dic3C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×C4×Dic3C2211yescharacteristic
39#3724C2211yescharacteristic
40#3724C2211yescharacteristic
41#3724C2211yescharacteristic
42#3724C2211yescharacteristic
43#3724C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47Dic3⋊Dic3C2211yescharacteristic
48Dic3⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#124—31no—
54C3×C22⋊Q8—31no—
55#946—31no—
56#944—31no—
57#944—31no—
58#944—31no—
59#935—31no—
60#935—31no—
61#929—31no—
62#924—31no—
63#898—31no—
64#898—31no—
65#896—31no—
66#890—31no—
67#889—31no—
68#888—31no—
69#885D311yescharacteristic
70#883—31no—
71#879—31no—
72#879—31no—
73#879—31no—
74#879—31no—
75#879—31no—
76#879—31no—
77#879—61no—
78#878—31no—
79#878—31no—
80#878—31no—
81#878D311yescharacteristic, complemented
82#870—31no—
83#870—31no—
84C23×C32C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C2×C32×C4C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88C2×C3⋊Dic3C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C3×#67—41no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3C2311yescharacteristic
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C3×Dic6—41no—
102C3×Dic6—41no—
103#627—41no—
104D3×Dic3—41no—
105#563—91noSylow
106C2×C3×Q8—31no—
107C2×C3×Q8—31no—
108C2×C3×D4—31no—
109C22×C3×C4—31no—
110C2×#67C2×D311yescharacteristic
111C22×Dic3—31no—
112C22×Dic3C2×D311yescharacteristic
113C2×C4×D3—31no—
114C2×Dic6—31no—
115C2×Dic6—31no—
116C2×Dic6C2×D311yescharacteristic
117C2×Dic6C2×D311yescharacteristic
118C3×C4⋊C4—31no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—31no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4—31no—
123C3×C22⋊C4—31no—
124C3×C22⋊C4—31no—
125C3×C22⋊C4—31no—
126C3×C22⋊C4C2×D311yescharacteristic
127C3×C22⋊C4C2×D311yescharacteristic
128C3×C42—31no—
129#217—31no—
130#217—31no—
131#217—31no—
132#217—31no—
133#217—31no—
134#217—31no—
135#217—31no—
136#217—61no—
137#217—61no—
138#217C2×D311yescharacteristic
139#217C2×D311yescharacteristic
140(C2×D3)⋊C4—31no—
141(C2×D3)⋊C4—31no—
142(C2×D3)⋊C4—31no—
143(C2×D3)⋊C4C2×D311yescharacteristic
144(C2×D3)⋊C4C2×D311yescharacteristic
145C4⋊Dic3—31no—
146C4⋊Dic3—31no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—31no—
150C4⋊Dic3—31no—
151C4⋊Dic3—31no—
152C4⋊Dic3—61no—
153C4⋊Dic3C2×D311yescharacteristic
154Dic3⋊C4—31no—
155Dic3⋊C4—31no—
156Dic3⋊C4—31no—
157Dic3⋊C4—31no—
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—61no—
174Dic3⋊C4—61no—
175Dic3⋊C4C2×D311yescharacteristic
176Dic3⋊C4C2×D311yescharacteristic
177C4×Dic3—31no—
178C4×Dic3—31no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—61no—
182C4×Dic3C2×D311yescharacteristic
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32C2411yescharacteristic
187C2×C3×D3—21no—
188C2×C3×D3—21no—
189C2×C3×D3—21no—
190C32×C4—21no—
191C32×C4—21no—
192C3⋊Dic3—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—21no—
203C4⋊Q8—91no—
204C4⋊Q8—91no—
205#126—91no—
206#126—91no—
207#126—91no—
208#126—91no—
209#125—91no—
210#125—91no—
211#124—31no—
212#123—91no—
213#123—91no—
214C22⋊Q8—31no—
215C22⋊Q8—91no—
216C22⋊Q8—91no—
217C22⋊Q8—91no—
218C23×C3—21no—
219C23×C3—31no—
220C23×C3C22×D311yescharacteristic
221C23×C3C22×D311yescharacteristic
222C22×D3C22×D311yescharacteristic
223C3×Q8—121no—
224C3×Q8—121no—
225C3×D4—121no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
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×C4—121no—
236C2×C3×C4C22×D311yescharacteristic
237C2×C3×C4C22×D311yescharacteristic
238C2×C3×C4C22×D311yescharacteristic
239C2×C3×C4C22×D311yescharacteristic
240#67—41no—
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—41no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—121no—
255C2×Dic3C22×D311yescharacteristic
256C2×Dic3C22×D311yescharacteristic
257C2×Dic3C22×D311yescharacteristic
258C2×Dic3C22×D311yescharacteristic
259C2×Dic3C22×D311yescharacteristic
260C2×Dic3C22×D311yescharacteristic
261C2×Dic3C22×D311yescharacteristic
262C4×D3—121no—
263Dic6—41no—
264Dic6—41no—
265Dic6—121no—
266Dic6—121no—
267C2×C32—41no—
268C2×C32H(5,𝔽2)11yescharacteristic
269C2×C322−1+411yescharacteristic
270C2×C322−1+411yescharacteristic
271C3×D3—41no—
272C2×Q8—31no—
273C2×Q8—31no—
274C2×Q8—91no—
275C2×D4—31no—
276C22×C4—31no—
277C22×C4—91no—
278C4⋊C4—31no—
279C4⋊C4—31no—
280C4⋊C4—31no—
281C4⋊C4—91no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—91no—
288C4⋊C4—91no—
289C4⋊C4—91no—
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—91no—
293C4⋊C4—91no—
294C22⋊C4—31no—
295C22⋊C4—31no—
296C22⋊C4—31no—
297C22⋊C4—31no—
298C22⋊C4—91no—
299C22⋊C4—91no—
300C22⋊C4—91no—
301C22⋊C4—91no—
302C22⋊C4—91no—
303C22⋊C4D3211yescharacteristic
304C42—31no—
305C42—91no—
306C42—91no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3C23×D311yescharacteristic
321C22×C3C23×D311yescharacteristic
322C2×D3—21no—
323C2×D3—21no—
324C2×D3—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—61no—
332C3×C4—61no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—61no—
338Dic3—21no—
339Dic3—21no—
340Dic3—21no—
341Dic3—21no—
342Dic3—21no—
343Dic3—21no—
344Dic3—21no—
345Dic3—61no—
346Dic3—61no—
347Dic3—61no—
348Dic3—61no—
349Dic3—61no—
350Dic3—61no—
351Dic3—61no—
352Dic3—61no—
353Dic3—121no—
354Dic3—121no—
355Dic3—121no—
356Dic3—121no—
357C32#56311yescharacteristic, complemented, Sylow
358C23—31no—
359C23C2×D3211yescharacteristic
360Q8—121no—
361Q8—121no—
362Q8—361no—
363D4—121no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—91no—
374C2×C4—91no—
375C2×C4—121no—
376C2×C4—361no—
377C2×C4C2×D3211yescharacteristic
378C2×C4C2×D3211yescharacteristic
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—81no—
385C2×C3—121no—
386C2×C3#100411yescharacteristic
387C2×C3#100411yescharacteristic
388C2×C3#100711yescharacteristic
389C2×C3#101011yescharacteristic
390C2×C3#101011yescharacteristic
391C2×C3#101011yescharacteristic
392D3—41no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—61no—
397C22—61no—
398C22—61no—
399C22C22×D3211yescharacteristic
400C4—21no—
401C4—21no—
402C4—61no—
403C4—61no—
404C4—61no—
405C4—61no—
406C4—61no—
407C4—61no—
408C4—61no—
409C4—181no—
410C4—181no—
411C4—181no—
412C4—181no—
413C3—21no—
414C3#564611yescharacteristic, complemented
415C3#567211yescharacteristic, complemented
416C2—41no—
417C2—121no—
418C2#6429711yescharacteristic
419C2#6430911yescharacteristic
420C2#6431311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC2×C3×#67C22
hypercenterC22⋊C4D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64088C2semidirect
3. C3×#878C2semidirect
5. #63964C2semidirect
7. #63956C2semidirect
8. #63899C2semidirect
9. #63887C2semidirect
10. #63873C2semidirect
11. #63869C2semidirect
12. #63867C2semidirect
13. #63862C2semidirect
14. #63861C2semidirect
15. #63839C2semidirect
81. #878D3semidirect
357. C32#563semidirect
414. C3#5646semidirect
415. C3#5672semidirect

Maximal subgroups

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

subgroupindex
#640882
C3×#8852
C3×#8782
#639692
#639642
#639612
#639562
#638992
#638872
#638732
#638692
#638672
#638622
#638612
#638392
#56723
#56463