Finite Group Databasebeta

← #10594394

Subgroups — #10594394

Order 576.

502 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
2C3×#896C211yesmaximal, characteristic, complemented
3C3×#889C211yesmaximal, characteristic, complemented
4C4×#626C211yesmaximal, characteristic
5#63899C211yesmaximal, characteristic, complemented
6#63893C211yesmaximal, characteristic, complemented
7#63881C211yesmaximal, characteristic
8#63881C211yesmaximal, characteristic
9D3×Dic3⋊C4C211yesmaximal, characteristic, complemented
10#63871C211yesmaximal, characteristic, complemented
11#63865C211yesmaximal, characteristic, complemented
12#63865C211yesmaximal, characteristic, complemented
13#63845C211yesmaximal, characteristic, complemented
14#63839C211yesmaximal, characteristic, complemented
15#63837C211yesmaximal, characteristic, complemented
16#5690—31nomaximal
17#5636—31nomaximal
18C2×C4×C3⋊D3C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×C4×C3⋊D3C2211yescharacteristic
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×Dic6C2211yescharacteristic
23C2×#627C2211yescharacteristic
24C2×#627C2211yescharacteristic
25C2×#626C2211yescharacteristic
26C2×D3×Dic3C2211yescharacteristic
27C2×D3×Dic3C2211yescharacteristic
28C32×C4⋊C4C2211yescharacteristic
29C4⋊(C3⋊Dic3)C2211yescharacteristic
30#3750C2211yescharacteristic
31#3750C2211yescharacteristic
32C3×(C2×D3)⋊C4C2211yescharacteristic
33C3×(C2×D3)⋊C4C2211yescharacteristic
34C3×C4⋊Dic3C2211yescharacteristic
35C3×Dic3⋊C4C2211yescharacteristic
36C3×Dic3⋊C4C2211yescharacteristic
37C3×Dic3⋊C4C2211yescharacteristic
38C3×Dic3⋊C4C2211yescharacteristic
39C3×Dic3⋊C4C2211yescharacteristic
40C3×Dic3⋊C4C2211yescharacteristic
41C3×C4×Dic3C2211yescharacteristic
42#3724C2211yescharacteristic
43#3724C2211yescharacteristic
44Dic3⋊Dic3C2211yescharacteristic
45Dic3⋊Dic3C2211yescharacteristic
46Dic3⋊Dic3C2211yescharacteristic
47#3722C2211yescharacteristic
48#3722C2211yescharacteristic
49#3722C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51Dic32C2211yescharacteristic
52Dic32C2211yescharacteristic
53C3×#125—31no—
54C3×C22⋊Q8—31no—
55#946—31no—
56#944—31no—
57C4×#67—31no—
58C2×Dic3⋊C4—31no—
59#897—31no—
60#897—31no—
61#896—31no—
62#896—31no—
63#896—31no—
64#896—31no—
65#896—31no—
66#896D311yescharacteristic, complemented
67#893—31no—
68#893—31no—
69D3×C4⋊C4—31no—
70D3×C4⋊C4—31no—
71D3×C4⋊C4—31no—
72D3×C4⋊C4—61no—
73#890—31no—
74#889—31no—
75#889D311yescharacteristic, complemented
76#888—31no—
77#887—31no—
78#67⋊C4—31no—
79#67⋊C4—31no—
80#878—31no—
81#878—31no—
82C4×D12—31no—
83C4×Dic6—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—22no—
93C4×C3⋊D3—41no—
94C4×C3⋊D3—41no—
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C3×C4×D3—41no—
103C3×Dic6—41no—
104#627—22no—
105#627—41no—
106#626Q814yescomplemented
107D3×Dic3—22no—
108D3×Dic3—41no—
109#554—91noSylow
110C2×C3×Q8—31no—
111C22×C3×C4—31no—
112C2×#67—31no—
113C22×Dic3—31no—
114C22×Dic3—31no—
115C2×D12—31no—
116C2×C4×D3—31no—
117C2×C4×D3—31no—
118C2×C4×D3—31no—
119C2×C4×D3—31no—
120C2×C4×D3—31no—
121C2×C4×D3—31no—
122C2×C4×D3—31no—
123C2×C4×D3—31no—
124C2×C4×D3—61no—
125C2×C4×D3—61no—
126C2×C4×D3—61no—
127C2×C4×D3C2×D311yescharacteristic
128C2×Dic6—31no—
129C2×Dic6—31no—
130C2×Dic6—31no—
131C2×Dic6—31no—
132C2×Dic6C2×D311yescharacteristic
133C3×C4⋊C4—21no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—31no—
141C3×C4⋊C4C2×D311yescharacteristic
142C3×C4⋊C4C2×D311yescharacteristic
143C3×C22⋊C4—31no—
144C3×C22⋊C4—31no—
145C3×C42—31no—
146#217—31no—
147(C2×D3)⋊C4—31no—
148(C2×D3)⋊C4—31no—
149(C2×D3)⋊C4—31no—
150(C2×D3)⋊C4—31no—
151(C2×D3)⋊C4—31no—
152(C2×D3)⋊C4—31no—
153(C2×D3)⋊C4—31no—
154(C2×D3)⋊C4C2×D311yescharacteristic
155(C2×D3)⋊C4C2×D311yescharacteristic
156C4⋊Dic3—31no—
157C4⋊Dic3—31no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—31no—
161C4⋊Dic3—61no—
162C4⋊Dic3C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic
177Dic3⋊C4C2×D311yescharacteristic
178Dic3⋊C4C2×D311yescharacteristic
179Dic3⋊C4C2×D311yescharacteristic
180Dic3⋊C4C2×D311yescharacteristic
181Dic3⋊C4C2×D311yescharacteristic
182C4×Dic3—31no—
183C4×Dic3—31no—
184C4×Dic3—31no—
185C4×Dic3—31no—
186C4×Dic3C2×D311yescharacteristic
187C22×C32C2411yescharacteristic
188C2×C3⋊D3—21no—
189C2×C3⋊D3—21no—
190C2×C3⋊D3C2×Q812yes—
191C2×C3×D3—21no—
192C2×C3×D3—21no—
193C2×C3×D3C2×Q812yes—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—21no—
197C3⋊Dic3—21no—
198C3⋊Dic3—21no—
199C3⋊Dic3#4112yes—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—21no—
203C3×Dic3—21no—
204C3×Dic3—21no—
205C3×Dic3—21no—
206C3×Dic3C2×Q812yes—
207C4⋊Q8—91no—
208#125—31no—
209#125—91no—
210C22⋊Q8—31no—
211C22⋊Q8—91no—
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—
223C22×D3—31no—
224C22×D3—31no—
225C22×D3—61no—
226C22×D3C22×D311yescharacteristic
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—31no—
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—
246C2×Dic3—31no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C2×Dic3—61no—
255C2×Dic3—62no—
256C2×Dic3—121no—
257C2×Dic3C22×D311yescharacteristic
258C2×Dic3C22×D311yescharacteristic
259C2×Dic3C22×D311yescharacteristic
260C2×Dic3C22×D311yescharacteristic
261C2×Dic3C22×D311yescharacteristic
262C2×Dic3C22×D311yescharacteristic
263C2×Dic3C22×D311yescharacteristic
264D12—34no—
265C4×D3—41no—
266C4×D3—62no—
267C4×D3—62no—
268C4×D3—62no—
269C4×D3—64no—
270C4×D3—121no—
271C4×D3—121no—
272C4×D3—121no—
273C4×D3—121no—
274C4×D3—121no—
275C4×D3—122no—
276C4×D3—122no—
277Dic6—41no—
278Dic6—62no—
279Dic6—62no—
280Dic6—121no—
281Dic6—121no—
282C2×C32C22×Q811yescharacteristic
283C2×C32C2×#4111yescharacteristic
284C2×C32H(5,𝔽2)11yescharacteristic
285C3⋊D3—22no—
286C3×D3—22no—
287C2×Q8—31no—
288C2×Q8—91no—
289C2×Q8—91no—
290C2×D4—91no—
291C22×C4—31no—
292C22×C4—91no—
293C22×C4—91no—
294C22×C4—91no—
295C22×C4—91no—
296C22×C4—91no—
297C4⋊C4—31no—
298C4⋊C4—31no—
299C4⋊C4—31no—
300C4⋊C4—31no—
301C4⋊C4—31no—
302C4⋊C4—31no—
303C4⋊C4—31no—
304C4⋊C4—91no—
305C4⋊C4—91no—
306C4⋊C4—91no—
307C4⋊C4—91no—
308C4⋊C4—91no—
309C4⋊C4—91no—
310C4⋊C4—91no—
311C4⋊C4—91no—
312C4⋊C4D3211yescharacteristic
313C22⋊C4—31no—
314C22⋊C4—31no—
315C22⋊C4—91no—
316C22⋊C4—91no—
317C22⋊C4—91no—
318C22⋊C4—91no—
319C42—31no—
320C42—91no—
321C42—91no—
322C22×C3—21no—
323C22×C3—32no—
324C22×C3—61no—
325C22×C3—61no—
326C22×C3C23×D311yescharacteristic
327C22×C3C23×D311yescharacteristic
328C2×D3—21no—
329C2×D3—21no—
330C2×D3—32no—
331C2×D3—32no—
332C2×D3—62no—
333C2×D3—62no—
334C2×D3—62no—
335C2×D3—61no—
336C2×D3—61no—
337C2×D3—61no—
338C2×D3—61no—
339C2×D3D3×Q812yes—
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—32no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—61no—
351C3×C4—61no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—61no—
355C3×C4—61no—
356Dic3—21no—
357Dic3—21no—
358Dic3—21no—
359Dic3—21no—
360Dic3—21no—
361Dic3—21no—
362Dic3—32no—
363Dic3—32no—
364Dic3—62no—
365Dic3—61no—
366Dic3—61no—
367Dic3—61no—
368Dic3—61no—
369Dic3—121no—
370Dic3—121no—
371Dic3D3×Q812yes—
372C32#55411yescharacteristic, complemented, Sylow
373C23—31no—
374C23—91no—
375Q8—121no—
376Q8—182no—
377Q8—361no—
378D4—94no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—31no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—91no—
387C2×C4—91no—
388C2×C4—91no—
389C2×C4—121no—
390C2×C4—182no—
391C2×C4—182no—
392C2×C4—361no—
393C2×C4—361no—
394C2×C4—361no—
395C2×C4C2×D3211yescharacteristic
396C2×C4C2×D3211yescharacteristic
397C2×C4C2×D3211yescharacteristic
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—62no—
402C2×C3#100411yescharacteristic
403C2×C3C2×D3×Q811yescharacteristic
404C2×C3C2×D3×Q811yescharacteristic
405C2×C3D3×#4111yescharacteristic
406C2×C3D3×#4111yescharacteristic
407C2×C3#100911yescharacteristic
408D3—22no—
409D3—64no—
410D3—62no—
411D3—62no—
412C22—32no—
413C22—61no—
414C22—61no—
415C22—92no—
416C22—181no—
417C22—181no—
418C22C22×D3211yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—32no—
423C4—61no—
424C4—61no—
425C4—61no—
426C4—61no—
427C4—61no—
428C4—61no—
429C4—92no—
430C4—181no—
431C4—181no—
432C3—21no—
433C3#563611yescharacteristic, complemented
434C3#569011yescharacteristic, complemented
435C2—62no—
436C2—182no—
437C2#6430611yescharacteristic
438C2#6431411yescharacteristic
439C2D32×Q811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#626C22
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. C4⋊C4×C3⋊D3C2semidirect
2. C3×#896C2semidirect
3. C3×#889C2semidirect
5. #63899C2semidirect
6. #63893C2semidirect
9. D3×Dic3⋊C4C2semidirect
10. #63871C2semidirect
11. #63865C2semidirect
12. #63865C2semidirect
13. #63845C2semidirect
14. #63839C2semidirect
15. #63837C2semidirect
66. #896D3semidirect
75. #889D3semidirect
106. #626Q8semidirect
372. C32#554semidirect
433. C3#5636semidirect
434. C3#5690semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊D32
C3×#8962
C3×#8892
C4×#6262
#638992
#638932
#638812
D3×Dic3⋊C42
#638712
#638652
#638452
#638392
#638372
#56903
#56363