Finite Group Databasebeta

← #10594158

Subgroups — #10594158

Order 576.

465 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64087C211yesmaximal, characteristic, complemented
2C3×#886C211yesmaximal, characteristic
3C3×#877C211yesmaximal, characteristic, complemented
4#67⋊Dic3C211yesmaximal, characteristic
5C2×#3724C211yesmaximal, characteristic, complemented
6#63961C211yesmaximal, characteristic
7#63955C211yesmaximal, characteristic, complemented
8#63896C211yesmaximal, characteristic, complemented
9#63877C211yesmaximal, characteristic, complemented
10#63869C211yesmaximal, characteristic, complemented
11#63861C211yesmaximal, characteristic, complemented
12#63859C211yesmaximal, characteristic, complemented
13#63847C211yesmaximal, characteristic, complemented
14#63847C211yesmaximal, characteristic, complemented
15#63844C211yesmaximal, characteristic, complemented
16#5650—31nomaximal
17#5531—31nomaximal
18C22×C3⋊Dic3C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C22×C3×Dic3C2211yescharacteristic
22C22×C3×Dic3C2211yescharacteristic
23C2×D3×Dic3C2211yescharacteristic
24C32×C22⋊C4C2211yescharacteristic
25#3757C2211yescharacteristic
26C4⋊(C3⋊Dic3)C2211yescharacteristic
27#3750C2211yescharacteristic
28#3750C2211yescharacteristic
29C3×#217C2211yescharacteristic
30C3×(C2×D3)⋊C4C2211yescharacteristic
31C3×(C2×D3)⋊C4C2211yescharacteristic
32C3×C4⋊Dic3C2211yescharacteristic
33C3×C4⋊Dic3C2211yescharacteristic
34C3×Dic3⋊C4C2211yescharacteristic
35C3×Dic3⋊C4C2211yescharacteristic
36C3×C4×Dic3C2211yescharacteristic
37C3×C4×Dic3C2211yescharacteristic
38#3724C2211yescharacteristic
39#3724C2211yescharacteristic
40#3724C2211yescharacteristic
41#3724C2211yescharacteristic
42#3724C2211yescharacteristic
43#3724C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47(C2×D3)⋊Dic3C2211yescharacteristic
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50Dic32C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#123—31no—
54C3×#117—31no—
55#935—31no—
56D4×Dic3—31no—
57#929—31no—
58#926—31no—
59C2×Dic3⋊C4—31no—
60C2×Dic3⋊C4—31no—
61#898—31no—
62#896—31no—
63#890—31no—
64#889—31no—
65#889—31no—
66#889—31no—
67#889—31no—
68#887—31no—
69#886D311yescharacteristic
70#67⋊C4—31no—
71#879—31no—
72#879—31no—
73#878—31no—
74#878—31no—
75#878—31no—
76#878—61no—
77#877—31no—
78#877—31no—
79#877D311yescharacteristic, complemented
80#876—31no—
81#872—31no—
82#870—31no—
83C4×Dic6—31no—
84C23×C32C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C2×C32×C4C2311yescharacteristic
87C2×C32×C4C2311yescharacteristic
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3C2311yescharacteristic
90C2×C3⋊Dic3C2311yescharacteristic
91C2×C3⋊Dic3C2311yescharacteristic
92C2×C3⋊Dic3C2311yescharacteristic
93#634—22no—
94C3×#67—22no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C2×C3×Dic3C2311yescharacteristic
103C2×C3×Dic3C2311yescharacteristic
104D3×Dic3—22no—
105#552—91noSylow
106C2×C3×D4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C2×#67C2×D311yescharacteristic
110C22×Dic3—31no—
111C22×Dic3—31no—
112C22×Dic3—31no—
113C22×Dic3—61no—
114C22×Dic3C2×D311yescharacteristic
115C22×Dic3C2×D311yescharacteristic
116C2×C4×D3—31no—
117C2×Dic6—31no—
118C2×Dic6—31no—
119C2×Dic6—61no—
120C3×C4⋊C4—31no—
121C3×C4⋊C4—31no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—31no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4—31no—
126C3×C22⋊C4—31no—
127C3×C22⋊C4—31no—
128C3×C22⋊C4C2×D311yescharacteristic
129C3×C22⋊C4C2×D311yescharacteristic
130C3×C42—31no—
131C3×C42—31no—
132#217—31no—
133#217—31no—
134#217—31no—
135#217—31no—
136#217—31no—
137#217—61no—
138#217C2×D311yescharacteristic
139(C2×D3)⋊C4—31no—
140(C2×D3)⋊C4—31no—
141(C2×D3)⋊C4—31no—
142(C2×D3)⋊C4C2×D311yescharacteristic
143(C2×D3)⋊C4C2×D311yescharacteristic
144C4⋊Dic3—31no—
145C4⋊Dic3—31no—
146C4⋊Dic3—31no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—61no—
150C4⋊Dic3C2×D311yescharacteristic
151C4⋊Dic3C2×D311yescharacteristic
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—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—61no—
172Dic3⋊C4—61no—
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×C32—21no—
184C22×C32—21no—
185C22×C32#4112yes—
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#4112yes—
196C3×Dic3—21no—
197C3×Dic3—21no—
198C3×Dic3—21no—
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—21no—
203#126—91no—
204#126—91no—
205#125—91no—
206#125—91no—
207#125—91no—
208#123—31no—
209#123—91no—
210C22⋊Q8—91no—
211C22⋊Q8—91no—
212C4×Q8—91no—
213C4×D4—91no—
214#117—31no—
215#117—91no—
216#117—91no—
217C2×C4⋊C4—91no—
218C23×C3—21no—
219C23×C3—31no—
220C23×C3C22×D311yescharacteristic
221C23×C3C22×D311yescharacteristic
222C22×D3C22×D311yescharacteristic
223C3×D4—62no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—31no—
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—62no—
234C2×C3×C4—62no—
235C2×C3×C4C22×D311yescharacteristic
236C2×C3×C4C22×D311yescharacteristic
237C2×C3×C4C22×D311yescharacteristic
238C2×C3×C4C22×D311yescharacteristic
239#67—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—62no—
257C2×Dic3—122no—
258C2×Dic3C22×D311yescharacteristic
259C2×Dic3C22×D311yescharacteristic
260C2×Dic3C22×D311yescharacteristic
261C2×Dic3C22×D311yescharacteristic
262C2×Dic3C22×D311yescharacteristic
263C2×Dic3C22×D311yescharacteristic
264C2×Dic3C22×D311yescharacteristic
265C4×D3—62no—
266Dic6—62no—
267Dic6—62no—
268Dic6—122no—
269C2×C32—22no—
270C2×C32C2×#4111yescharacteristic
271C2×C32C2×#4111yescharacteristic
272C2×C322−1+411yescharacteristic
273C3×D3—41no—
274C2×Q8—91no—
275C2×D4—31no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—91no—
279C22×C4—91no—
280C4⋊C4—31no—
281C4⋊C4—31no—
282C4⋊C4—31no—
283C4⋊C4—31no—
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—
294C4⋊C4—91no—
295C4⋊C4—91no—
296C22⋊C4—31no—
297C22⋊C4—31no—
298C22⋊C4—31no—
299C22⋊C4—91no—
300C22⋊C4—91no—
301C22⋊C4—91no—
302C22⋊C4—91no—
303C22⋊C4D3211yescharacteristic
304C42—31no—
305C42—31no—
306C42—91no—
307C42—91no—
308C42—91no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—22no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—61no—
318C22×C3—61no—
319C22×C3—61no—
320C22×C3#23512yes—
321C22×C3#23712yes—
322C22×C3C23×D311yescharacteristic
323C22×C3C23×D311yescharacteristic
324C2×D3—21no—
325C2×D3—21no—
326C2×D3—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—61no—
334C3×C4—61no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—61no—
339C3×C4—61no—
340Dic3—21no—
341Dic3—21no—
342Dic3—21no—
343Dic3—21no—
344Dic3—21no—
345Dic3—21no—
346Dic3—21no—
347Dic3—32no—
348Dic3—32no—
349Dic3—62no—
350Dic3—61no—
351Dic3—61no—
352Dic3—61no—
353Dic3—61no—
354Dic3—61no—
355Dic3—61no—
356Dic3—121no—
357Dic3—121no—
358Dic3—121no—
359C32#55211yescharacteristic, complemented, Sylow
360C23—31no—
361C23C2×D3211yescharacteristic
362Q8—182no—
363D4—62no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—91no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—91no—
377C2×C4—182no—
378C2×C4—182no—
379C2×C4C2×D3211yescharacteristic
380C2×C4C2×D3211yescharacteristic
381C2×C3—22no—
382C2×C3—22no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—42no—
387C2×C3—121no—
388C2×C3C2×#23511yescharacteristic
389C2×C3C2×#23711yescharacteristic
390C2×C3D3×#4111yescharacteristic
391C2×C3D3×#4111yescharacteristic
392C2×C3#101011yescharacteristic
393C2×C3#101011yescharacteristic
394D3—41no—
395C22—21no—
396C22—21no—
397C22—61no—
398C22—61no—
399C22—61no—
400C22#379512yes—
401C22C22×D3211yescharacteristic
402C4—21no—
403C4—21no—
404C4—61no—
405C4—61no—
406C4—61no—
407C4—61no—
408C4—61no—
409C4—61no—
410C4—61no—
411C4—92no—
412C4—181no—
413C4—181no—
414C4—181no—
415C3—21no—
416C3#553111yescharacteristic, complemented
417C3#565011yescharacteristic, complemented
418C2—22no—
419C2—121no—
420C2C2×#379511yescharacteristic
421C2#6430911yescharacteristic
422C2#6431211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
prime radicalC2×C3×#67C22
hypercenterC22⋊C4D32
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64087C2semidirect
3. C3×#877C2semidirect
5. C2×#3724C2semidirect
7. #63955C2semidirect
8. #63896C2semidirect
9. #63877C2semidirect
10. #63869C2semidirect
11. #63861C2semidirect
12. #63859C2semidirect
13. #63847C2semidirect
14. #63847C2semidirect
15. #63844C2semidirect
79. #877D3semidirect
359. C32#552semidirect
416. C3#5531semidirect
417. C3#5650semidirect

Maximal subgroups

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

subgroupindex
#640872
C3×#8862
C3×#8772
#67⋊Dic32
C2×#37242
#639612
#639552
#638962
#638772
#638692
#638612
#638592
#638472
#638442
#56503
#55313