Finite Group Databasebeta

← #10594504

Subgroups — #10594504

Order 576.

429 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#64106C211yesmaximal, characteristic, complemented
2C3×#898C211yesmaximal, characteristic, complemented
3C3×#889C211yesmaximal, characteristic, complemented
4#63918C211yesmaximal, characteristic, complemented
5#63896C211yesmaximal, characteristic, complemented
6#63893C211yesmaximal, characteristic, complemented
7#63890C211yesmaximal, characteristic
8#63869C211yesmaximal, characteristic, complemented
9#63868C211yesmaximal, characteristic, complemented
10#63865C211yesmaximal, characteristic, complemented
11#63861C211yesmaximal, characteristic, complemented
12#63853C211yesmaximal, characteristic
13#63852C211yesmaximal, characteristic
14#63848C211yesmaximal, characteristic, complemented
15#63840C211yesmaximal, characteristic, complemented
16#5712—31nomaximal
17#5699—31nomaximal
18C2×C4×C3⋊D3C2211yescharacteristic
19C2×#634C2211yescharacteristic
20C2×#627C2211yescharacteristic
21C2×#627C2211yescharacteristic
22C2×#626C2211yescharacteristic
23C2×D3×Dic3C2211yescharacteristic
24C32×C4⋊C4C2211yescharacteristic
25#3752C2211yescharacteristic
26#3752C2211yescharacteristic
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×(C2×D3)⋊C4C2211yescharacteristic
30C3×(C2×D3)⋊C4C2211yescharacteristic
31C3×(C2×D3)⋊C4C2211yescharacteristic
32C3×C4⋊Dic3C2211yescharacteristic
33C3×C4⋊Dic3C2211yescharacteristic
34C3×Dic3⋊C4C2211yescharacteristic
35C3×Dic3⋊C4C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×Dic3⋊C4C2211yescharacteristic
39C3×C4×Dic3C2211yescharacteristic
40C3×C4×Dic3C2211yescharacteristic
41#3724C2211yescharacteristic
42#3724C2211yescharacteristic
43Dic3⋊Dic3C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic
49#3722C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#126—31no—
54C3×#125—31no—
55#898—31no—
56#898—31no—
57#898—31no—
58#898—31no—
59#898D311yescharacteristic, complemented
60#897—31no—
61#897—31no—
62#897—31no—
63#897—61no—
64#896—31no—
65#896—31no—
66#896—31no—
67#894—31no—
68#894—31no—
69#890—31no—
70#889D311yescharacteristic, complemented
71#888—31no—
72#888—31no—
73#888—31no—
74#886—31no—
75#885—31no—
76#883—31no—
77#879—31no—
78#878—31no—
79#878—31no—
80#876—31no—
81#875—31no—
82#870—31no—
83#869—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
92C4×C3⋊D3—41no—
93#634—41no—
94C2×C3×Dic3C2311yescharacteristic
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101#627—41no—
102#627—41no—
103#626—41no—
104D3×Dic3—41no—
105#563—91noSylow
106C2×#67—31no—
107C22×Dic3—31no—
108C2×D12—31no—
109C2×C4×D3—31no—
110C2×C4×D3—31no—
111C2×C4×D3—31no—
112C2×C4×D3—61no—
113C2×Dic6—31no—
114C2×Dic6—31no—
115C2×Dic6—31no—
116C2×Dic6—31no—
117C2×Dic6—31no—
118C2×Dic6—31no—
119C2×Dic6—61no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4—31no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—31no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4—31no—
126C3×C4⋊C4—31no—
127C3×C4⋊C4—31no—
128C3×C4⋊C4C2×D311yescharacteristic
129C3×C4⋊C4C2×D311yescharacteristic
130C3×C22⋊C4—31no—
131C3×C22⋊C4—31no—
132C3×C22⋊C4—31no—
133C3×C42—31no—
134C3×C42—31no—
135#217—31no—
136#217—31no—
137(C2×D3)⋊C4—31no—
138(C2×D3)⋊C4—31no—
139(C2×D3)⋊C4—31no—
140(C2×D3)⋊C4—31no—
141(C2×D3)⋊C4—31no—
142(C2×D3)⋊C4—31no—
143(C2×D3)⋊C4—31no—
144(C2×D3)⋊C4—31no—
145(C2×D3)⋊C4—31no—
146(C2×D3)⋊C4—31no—
147(C2×D3)⋊C4—31no—
148(C2×D3)⋊C4—31no—
149(C2×D3)⋊C4—61no—
150(C2×D3)⋊C4—61no—
151(C2×D3)⋊C4C2×D311yescharacteristic
152(C2×D3)⋊C4C2×D311yescharacteristic
153(C2×D3)⋊C4C2×D311yescharacteristic
154C4⋊Dic3—31no—
155C4⋊Dic3—31no—
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—31no—
161C4⋊Dic3—31no—
162C4⋊Dic3—61no—
163C4⋊Dic3—61no—
164C4⋊Dic3C2×D311yescharacteristic
165C4⋊Dic3C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic
175Dic3⋊C4C2×D311yescharacteristic
176Dic3⋊C4C2×D311yescharacteristic
177Dic3⋊C4C2×D311yescharacteristic
178Dic3⋊C4C2×D311yescharacteristic
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3C2×D311yescharacteristic
182C4×Dic3C2×D311yescharacteristic
183C22×C32C2411yescharacteristic
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3⋊D3—21no—
187C2×C3×D3—21no—
188C2×C3×D3—21no—
189C2×C3×D3—21no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—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—31no—
206#126—91no—
207#126—91no—
208#126—91no—
209#125—31no—
210#125—91no—
211#124—91no—
212#123—91no—
213#123—91no—
214C22⋊Q8—91no—
215C22⋊Q8—91no—
216C22⋊Q8—91no—
217C22⋊Q8—91no—
218C23×C3—31no—
219C22×D3—31no—
220C22×D3—31no—
221C22×D3—61no—
222C22×D3C22×D311yescharacteristic
223C2×C3×C4—21no—
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×C4C22×D311yescharacteristic
234C2×C3×C4C22×D311yescharacteristic
235C2×C3×C4C22×D311yescharacteristic
236C2×C3×C4C22×D311yescharacteristic
237C2×C3×C4C22×D311yescharacteristic
238C2×C3×C4C22×D311yescharacteristic
239#67—121no—
240C2×Dic3—31no—
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—31no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—121no—
250C2×Dic3C22×D311yescharacteristic
251C2×Dic3C22×D311yescharacteristic
252C2×Dic3C22×D311yescharacteristic
253C2×Dic3C22×D311yescharacteristic
254C2×Dic3C22×D311yescharacteristic
255C2×Dic3C22×D311yescharacteristic
256C2×Dic3C22×D311yescharacteristic
257D12—121no—
258C4×D3—121no—
259C4×D3—121no—
260C4×D3—121no—
261C4×D3—122no—
262Dic6—121no—
263Dic6—121no—
264Dic6—121no—
265Dic6—121no—
266Dic6—121no—
267Dic6—121no—
268Dic6—122no—
269C2×C32H(5,𝔽2)11yescharacteristic
270C2×C322−1+411yescharacteristic
271C2×C322−1+411yescharacteristic
272C3⋊D3—41no—
273C3×D3—41no—
274C2×Q8—91no—
275C2×Q8—91no—
276C2×Q8—91no—
277C2×D4—91no—
278C22×C4—91no—
279C22×C4—91no—
280C4⋊C4—31no—
281C4⋊C4—31no—
282C4⋊C4—31no—
283C4⋊C4—31no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—31no—
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⋊C4D3211yescharacteristic
296C22⋊C4—31no—
297C22⋊C4—31no—
298C22⋊C4—31no—
299C22⋊C4—91no—
300C22⋊C4—91no—
301C22⋊C4—91no—
302C22⋊C4—91no—
303C22⋊C4—91no—
304C22⋊C4—91no—
305C22⋊C4—91no—
306C42—31no—
307C42—31no—
308C42—91no—
309C22×C3—21no—
310C22×C3—61no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3C23×D311yescharacteristic
314C22×C3C23×D311yescharacteristic
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—21no—
318C2×D3—62no—
319C2×D3—61no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—61no—
325C2×D3—121no—
326C2×D3—121no—
327C3×C4—22no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—61no—
341C3×C4—61no—
342C3×C4—61no—
343Dic3—21no—
344Dic3—21no—
345Dic3—21no—
346Dic3—21no—
347Dic3—21no—
348Dic3—21no—
349Dic3—21no—
350Dic3—61no—
351Dic3—61no—
352Dic3—61no—
353Dic3—61no—
354Dic3—61no—
355Dic3—61no—
356Dic3—121no—
357Dic3—121no—
358Dic3—121no—
359C32#56311yescharacteristic, complemented, Sylow
360C23—31no—
361C23—91no—
362Q8—361no—
363Q8—361no—
364Q8—361no—
365D4—361no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—91no—
374C2×C4—91no—
375C2×C4—91no—
376C2×C4—361no—
377C2×C4—361no—
378C2×C4C2×D3211yescharacteristic
379C2×C4C2×D3211yescharacteristic
380C2×C4C2×D3211yescharacteristic
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—121no—
385C2×C3#100711yescharacteristic
386C2×C3#100911yescharacteristic
387C2×C3#100911yescharacteristic
388C2×C3#101011yescharacteristic
389C2×C3#101011yescharacteristic
390C2×C3#101011yescharacteristic
391D3—41no—
392D3—122no—
393D3—121no—
394D3—121no—
395C22—61no—
396C22—61no—
397C22—61no—
398C22—181no—
399C22—181no—
400C22—181no—
401C22C22×D3211yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—61no—
406C4—61no—
407C4—61no—
408C4—61no—
409C4—61no—
410C4—61no—
411C4—61no—
412C4—181no—
413C4—181no—
414C4—181no—
415C3—21no—
416C3#569911yescharacteristic, complemented
417C3#571211yescharacteristic, complemented
418C2—121no—
419C2—361no—
420C2#6430811yescharacteristic
421C2#6430911yescharacteristic
422C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#626C22
hypercenterC4⋊C4D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64106C2semidirect
2. C3×#898C2semidirect
3. C3×#889C2semidirect
4. #63918C2semidirect
5. #63896C2semidirect
6. #63893C2semidirect
8. #63869C2semidirect
9. #63868C2semidirect
10. #63865C2semidirect
11. #63861C2semidirect
14. #63848C2semidirect
15. #63840C2semidirect
59. #898D3semidirect
70. #889D3semidirect
359. C32#563semidirect
416. C3#5699semidirect
417. C3#5712semidirect

Maximal subgroups

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

subgroupindex
#641062
C3×#8982
C3×#8892
#639182
#638962
#638932
#638902
#638692
#638682
#638652
#638612
#638532
#638522
#638482
#638402
#57123
#56993