Finite Group Databasebeta

← #10594479

Subgroups — #10594479

Order 576.

418 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64097C211yesmaximal, characteristic, complemented
2C3×#898C211yesmaximal, characteristic
3C3×#889C211yesmaximal, characteristic, complemented
4#63896C211yesmaximal, characteristic
5#63896C211yesmaximal, characteristic
6#63878C211yesmaximal, characteristic
7#63870C211yesmaximal, characteristic, complemented
8#63867C211yesmaximal, characteristic, complemented
9#63861C211yesmaximal, characteristic
10#63859C211yesmaximal, characteristic
11#63858C211yesmaximal, characteristic
12#63846C211yesmaximal, characteristic, complemented
13#63845C211yesmaximal, characteristic, complemented
14#63844C211yesmaximal, characteristic, complemented
15#63844C211yesmaximal, characteristic, complemented
16#5701—31nomaximal
17#5696—31nomaximal
18C2×#634C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×D3×Dic3C2211yescharacteristic
21C32×C4⋊C4C2211yescharacteristic
22C4⋊(C3⋊Dic3)C2211yescharacteristic
23#3750C2211yescharacteristic
24#3750C2211yescharacteristic
25C4×C3⋊Dic3C2211yescharacteristic
26C3×(C2×D3)⋊C4C2211yescharacteristic
27C3×(C2×D3)⋊C4C2211yescharacteristic
28C3×(C2×D3)⋊C4C2211yescharacteristic
29C3×C4⋊Dic3C2211yescharacteristic
30C3×C4⋊Dic3C2211yescharacteristic
31C3×Dic3⋊C4C2211yescharacteristic
32C3×Dic3⋊C4C2211yescharacteristic
33C3×Dic3⋊C4C2211yescharacteristic
34C3×Dic3⋊C4C2211yescharacteristic
35C3×Dic3⋊C4C2211yescharacteristic
36C3×C4×Dic3C2211yescharacteristic
37C3×C4×Dic3C2211yescharacteristic
38#3724C2211yescharacteristic
39#3724C2211yescharacteristic
40#3724C2211yescharacteristic
41#3724C2211yescharacteristic
42Dic3⋊Dic3C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50Dic32C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#126—31no—
54C3×#125—31no—
55#898—31no—
56#898—31no—
57#898D311yescharacteristic
58#896—31no—
59#896—31no—
60#892—31no—
61#890—31no—
62#890—31no—
63#889—31no—
64#889—31no—
65#889—31no—
66#889—31no—
67#889—31no—
68#889D311yescharacteristic, complemented
69#888—31no—
70#888—31no—
71#888—61no—
72#887—31no—
73#887—31no—
74#879—31no—
75#879—31no—
76#879—31no—
77#878—31no—
78#878—31no—
79#877—31no—
80#876—31no—
81#870—31no—
82C4×Dic6—31no—
83C4×Dic6—31no—
84C22×C3×D3C2311yescharacteristic
85C2×C32×C4C2311yescharacteristic
86C2×C32×C4C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88C2×C3⋊Dic3C2311yescharacteristic
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92#634—22no—
93#634—22no—
94C2×C3×Dic3C2311yescharacteristic
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101D3×Dic3—22no—
102#541—91noSylow
103C22×Dic3—31no—
104C2×C4×D3—31no—
105C2×Dic6—31no—
106C2×Dic6—31no—
107C2×Dic6—31no—
108C2×Dic6—31no—
109C2×Dic6—61no—
110C2×Dic6—61no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—31no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—31no—
115C3×C4⋊C4—31no—
116C3×C4⋊C4—31no—
117C3×C4⋊C4—31no—
118C3×C4⋊C4—31no—
119C3×C4⋊C4C2×D311yescharacteristic
120C3×C4⋊C4C2×D311yescharacteristic
121C3×C22⋊C4—31no—
122C3×C22⋊C4—31no—
123C3×C22⋊C4—31no—
124C3×C42—31no—
125C3×C42—31no—
126#217—31no—
127#217—31no—
128#217—31no—
129(C2×D3)⋊C4—31no—
130(C2×D3)⋊C4—31no—
131(C2×D3)⋊C4—31no—
132(C2×D3)⋊C4C2×D311yescharacteristic
133(C2×D3)⋊C4C2×D311yescharacteristic
134(C2×D3)⋊C4C2×D311yescharacteristic
135C4⋊Dic3—31no—
136C4⋊Dic3—31no—
137C4⋊Dic3—31no—
138C4⋊Dic3—31no—
139C4⋊Dic3—31no—
140C4⋊Dic3—31no—
141C4⋊Dic3—31no—
142C4⋊Dic3—61no—
143C4⋊Dic3C2×D311yescharacteristic
144C4⋊Dic3C2×D311yescharacteristic
145Dic3⋊C4—31no—
146Dic3⋊C4—31no—
147Dic3⋊C4—31no—
148Dic3⋊C4—31no—
149Dic3⋊C4—31no—
150Dic3⋊C4—31no—
151Dic3⋊C4—31no—
152Dic3⋊C4—31no—
153Dic3⋊C4—31no—
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—61no—
163Dic3⋊C4—61no—
164Dic3⋊C4C2×D311yescharacteristic
165Dic3⋊C4C2×D311yescharacteristic
166Dic3⋊C4C2×D311yescharacteristic
167Dic3⋊C4C2×D311yescharacteristic
168Dic3⋊C4C2×D311yescharacteristic
169C4×Dic3—31no—
170C4×Dic3—31no—
171C4×Dic3—31no—
172C4×Dic3—31no—
173C4×Dic3—31no—
174C4×Dic3—31no—
175C4×Dic3—31no—
176C4×Dic3—31no—
177C4×Dic3—61no—
178C4×Dic3C2×D311yescharacteristic
179C4×Dic3C2×D311yescharacteristic
180C22×C32C2411yescharacteristic
181C2×C3×D3—21no—
182C2×C3×D3—21no—
183C2×C3×D3—21no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—21no—
187C3⋊Dic3—21no—
188C3⋊Dic3—21no—
189C3⋊Dic3—21no—
190C3⋊Dic3#4112yes—
191C3×Dic3—21no—
192C3×Dic3—21no—
193C3×Dic3—21no—
194C3×Dic3—21no—
195C3×Dic3—21no—
196C3×Dic3—21no—
197C3×Dic3—21no—
198C4⋊Q8—91no—
199#126—31no—
200#126—91no—
201#126—91no—
202#126—91no—
203#125—31no—
204#125—91no—
205#125—91no—
206#125—91no—
207#125—91no—
208C22⋊Q8—91no—
209C22⋊Q8—91no—
210C4×Q8—91no—
211C4×Q8—91no—
212#117—91no—
213C23×C3—31no—
214C22×D3C22×D311yescharacteristic
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4C22×D311yescharacteristic
226C2×C3×C4C22×D311yescharacteristic
227C2×C3×C4C22×D311yescharacteristic
228C2×C3×C4C22×D311yescharacteristic
229C2×C3×C4C22×D311yescharacteristic
230C2×C3×C4C22×D311yescharacteristic
231C2×Dic3—31no—
232C2×Dic3—31no—
233C2×Dic3—31no—
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—31no—
237C2×Dic3—31no—
238C2×Dic3—31no—
239C2×Dic3—61no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C2×Dic3—61no—
243C2×Dic3—62no—
244C2×Dic3C22×D311yescharacteristic
245C2×Dic3C22×D311yescharacteristic
246C2×Dic3C22×D311yescharacteristic
247C2×Dic3C22×D311yescharacteristic
248C2×Dic3C22×D311yescharacteristic
249C2×Dic3C22×D311yescharacteristic
250C2×Dic3C22×D311yescharacteristic
251C4×D3—62no—
252Dic6—62no—
253Dic6—62no—
254Dic6—62no—
255Dic6—62no—
256Dic6—122no—
257Dic6—122no—
258C2×C32C2×#4111yescharacteristic
259C2×C322−1+411yescharacteristic
260C2×C322−1+411yescharacteristic
261C3×D3—41no—
262C2×Q8—91no—
263C2×Q8—91no—
264C22×C4—91no—
265C4⋊C4—31no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—31no—
269C4⋊C4—31no—
270C4⋊C4—31no—
271C4⋊C4—31no—
272C4⋊C4—91no—
273C4⋊C4—91no—
274C4⋊C4—91no—
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—91no—
279C4⋊C4—91no—
280C4⋊C4—91no—
281C4⋊C4—91no—
282C4⋊C4—91no—
283C4⋊C4—91no—
284C4⋊C4D3211yescharacteristic
285C22⋊C4—31no—
286C22⋊C4—31no—
287C22⋊C4—31no—
288C22⋊C4—91no—
289C22⋊C4—91no—
290C22⋊C4—91no—
291C42—31no—
292C42—31no—
293C42—91no—
294C42—91no—
295C42—91no—
296C42—91no—
297C22×C3—21no—
298C22×C3—61no—
299C22×C3—61no—
300C22×C3—61no—
301C22×C3C23×D311yescharacteristic
302C22×C3C23×D311yescharacteristic
303C2×D3—21no—
304C2×D3—21no—
305C2×D3—21no—
306C3×C4—22no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—61no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—61no—
320C3×C4—61no—
321C3×C4—61no—
322Dic3—21no—
323Dic3—21no—
324Dic3—21no—
325Dic3—21no—
326Dic3—21no—
327Dic3—21no—
328Dic3—21no—
329Dic3—32no—
330Dic3—32no—
331Dic3—62no—
332Dic3—62no—
333Dic3—61no—
334Dic3—61no—
335Dic3—61no—
336Dic3—61no—
337Dic3—61no—
338Dic3—61no—
339Dic3—121no—
340Dic3—121no—
341C32#54111yescharacteristic, complemented, Sylow
342C23—31no—
343Q8—182no—
344Q8—182no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—91no—
353C2×C4—91no—
354C2×C4—91no—
355C2×C4—91no—
356C2×C4—182no—
357C2×C4C2×D3211yescharacteristic
358C2×C4C2×D3211yescharacteristic
359C2×C4C2×D3211yescharacteristic
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—121no—
364C2×C3#100711yescharacteristic
365C2×C3D3×#4111yescharacteristic
366C2×C3D3×#4111yescharacteristic
367C2×C3#101011yescharacteristic
368C2×C3#101011yescharacteristic
369C2×C3#101011yescharacteristic
370D3—41no—
371C22—61no—
372C22—61no—
373C22—61no—
374C22C22×D3211yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—61no—
379C4—61no—
380C4—61no—
381C4—61no—
382C4—61no—
383C4—61no—
384C4—61no—
385C4—92no—
386C4—181no—
387C4—181no—
388C4—181no—
389C3—21no—
390C3#569611yescharacteristic, complemented
391C3#570111yescharacteristic, complemented
392C2—121no—
393C2#6429811yescharacteristic
394C2#6431211yescharacteristic
395C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC22×C3×D3C23
hypercenterC4⋊C4D32
hypocenterC32#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64097C2semidirect
3. C3×#889C2semidirect
7. #63870C2semidirect
8. #63867C2semidirect
12. #63846C2semidirect
13. #63845C2semidirect
14. #63844C2semidirect
15. #63844C2semidirect
68. #889D3semidirect
341. C32#541semidirect
390. C3#5696semidirect
391. C3#5701semidirect

Maximal subgroups

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

subgroupindex
#640972
C3×#8982
C3×#8892
#638962
#638782
#638702
#638672
#638612
#638592
#638582
#638462
#638452
#638442
#57013
#56963