Finite Group Databasebeta

← #86135

Subgroups — #86135

Order 384.

1004 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#529C211yesmaximal, characteristic, complemented
2#5712C214yesmaximal, complemented
3#5689C212yesmaximal, complemented
4#5682C212yesmaximal, complemented
5#5665C214yesmaximal, complemented
6#5635C212yesmaximal, complemented
7#5618C212yesmaximal, complemented
8#5584C211yesmaximal, characteristic, complemented
9#5570C211yesmaximal, characteristic, complemented
10#5552C212yesmaximal, complemented
11#5551C211yesmaximal, characteristic, complemented
12D4×Dic6C211yesmaximal, characteristic, complemented
13D4×Dic6C212yesmaximal, complemented
14#5526C212yesmaximal, complemented
15#5518C212yesmaximal, complemented
16#5479C211yesmaximal, characteristic, complemented
17C2×#869C211yesmaximal, characteristic, complemented
18#3552—31nomaximal, Sylow
19C2×#235C2212yes—
20C22×Dic6C2212yes—
21C22×Dic6C2212yes—
22C3×#126C2212yes—
23C3×#125C2211yescharacteristic
24C3×#124C2211yescharacteristic
25C3×#123C2212yes—
26C3×C22⋊Q8C2211yescharacteristic
27C3×#121C2211yescharacteristic
28C3×C4×Q8C2211yescharacteristic
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2212yes—
31C3×#117C2212yes—
32C2×C3×C42C2211yescharacteristic
33#946C2212yes—
34Q8×Dic3C2212yes—
35#939C2214yes—
36#938C2212yes—
37D4×Dic3C2214yes—
38D4×Dic3C2212yes—
39#930C2211yescharacteristic
40#929C2212yes—
41C4×#67C2212yes—
42C2×C4⋊Dic3C2212yes—
43C2×C4⋊Dic3C2212yes—
44#924C2211yescharacteristic
45#924C2212yes—
46#924C2212yes—
47#898C2214yes—
48#897C2214yes—
49#897C2214yes—
50#896C2214yes—
51#892C2212yes—
52D3×C4⋊C4C2212yes—
53#890C2214yes—
54#890C2212yes—
55#889C2214yes—
56#889C2214yes—
57#888C2212yes—
58#888C2214yes—
59#886C2214yes—
60#886C2214yes—
61#885C2214yes—
62#883C2214yes—
63#67⋊C4C2214yes—
64#879C2214yes—
65#879C2214yes—
66#878C2214yes—
67#878C2214yes—
68#878C2214yes—
69#876C2212yes—
70#875C2211yescharacteristic
71#875C2212yes—
72C4×D12C2211yescharacteristic
73#872C2212yes—
74#870C2211yescharacteristic
75#869C2212yes—
76#869C2212yes—
77#869C2212yes—
78#869C2211yescharacteristic
79#869C2211yescharacteristic
80#869C2212yes—
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2212yes—
84#563—34no—
85#557—31no—
86#556—31no—
87#554—32no—
88#553—32no—
89D4×Q8—32no—
90D4×Q8—31no—
91#548—32no—
92#547—31no—
93#544—32no—
94#542—32no—
95#541—32no—
96#539—34no—
97#536—32no—
98C2×C4⋊Q8—31no—
99#529—31no—
100#529D311yescharacteristic, complemented
101C2×C3×Q8C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2312yes—
104C22×C3×C4C2312yes—
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C2×#67C2312yes—
109C2×#67C2314yes—
110C22×Dic3C2314yes—
111C22×Dic3C2314yes—
112#235—44no—
113C2×D12C2311yescharacteristic
114C2×C4×D3C2312yes—
115C2×C4×D3C2314yes—
116C2×Dic6—44no—
117C2×Dic6—44no—
118C2×Dic6C2312yes—
119C2×Dic6C2311yescharacteristic
120C2×Dic6C2312yes—
121C2×Dic6C2314yes—
122C2×Dic6C2314yes—
123C3×C4⋊C4C2311yescharacteristic
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2312yes—
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2312yes—
129C3×C22⋊C4C2312yes—
130C3×C22⋊C4C2312yes—
131C3×C22⋊C4C2312yes—
132C3×C22⋊C4C2312yes—
133C3×C22⋊C4C2312yes—
134C3×C42C2312yes—
135C3×C42C2311yescharacteristic
136C3×C42C2311yescharacteristic
137C3×C42C2311yescharacteristic
138C3×C42C2311yescharacteristic
139#217C2312yes—
140#217C2314yes—
141#217C2314yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2314yes—
147(C2×D3)⋊C4C2314yes—
148C4⋊Dic3C2311yescharacteristic
149C4⋊Dic3C2311yescharacteristic
150C4⋊Dic3C2312yes—
151C4⋊Dic3C2314yes—
152C4⋊Dic3C2314yes—
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2314yes—
157C4⋊Dic3C2314yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166C4×Dic3C2312yes—
167C4×Dic3C2314yes—
168C4×Dic3C2314yes—
169C2×#41—32no—
170C22×Q8—32no—
171C22×Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—34no—
174C4⋊Q8—32no—
175C4⋊Q8—32no—
176C4⋊Q8—31no—
177C4⋊Q8—31no—
178C4⋊Q8—32no—
179C4⋊Q8—32no—
180#126—34no—
181#126—34no—
182#126—34no—
183#126—32no—
184#126C2×D312yes—
185#125—34no—
186#125—34no—
187#125—34no—
188#125—32no—
189#125—31no—
190#125C2×D311yescharacteristic
191#124—34no—
192#124—31no—
193#124—32no—
194#124C2×D311yescharacteristic
195#123—32no—
196#123—34no—
197#123—34no—
198#123—34no—
199#123C2×D312yes—
200C22⋊Q8—32no—
201C22⋊Q8—31no—
202C22⋊Q8—32no—
203C22⋊Q8—32no—
204C22⋊Q8—34no—
205C22⋊Q8—34no—
206C22⋊Q8—34no—
207C22⋊Q8—34no—
208C22⋊Q8—34no—
209C22⋊Q8—34no—
210C22⋊Q8C2×D311yescharacteristic
211#121—34no—
212#121—32no—
213#121—31no—
214#121C2×D311yescharacteristic
215C4×Q8—32no—
216C4×Q8—31no—
217C4×Q8—32no—
218C4×Q8—32no—
219C4×Q8C2×D311yescharacteristic
220C4×D4—34no—
221C4×D4—32no—
222C4×D4—32no—
223C4×D4—34no—
224C4×D4—31no—
225C4×D4C2×D311yescharacteristic
226C4×D4C2×D312yes—
227#117—32no—
228#117—32no—
229#117C2×D312yes—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C42C2×D311yescharacteristic
234C23×C3C2412yes—
235C23×C3C2411yescharacteristic
236C22×D3C2412yes—
237C3×Q8—22no—
238C3×D4—24no—
239C3×D4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—22no—
244C2×C3×C4C2412yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2412yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253#67—24no—
254#67—44no—
255C2×Dic3—44no—
256C2×Dic3—44no—
257C2×Dic3C2412yes—
258C2×Dic3C2412yes—
259C2×Dic3C2414yes—
260C2×Dic3C2412yes—
261C2×Dic3C2414yes—
262D12—22no—
263C4×D3—24no—
264C4×D3—44no—
265Dic6—22no—
266Dic6—24no—
267Dic6—24no—
268Dic6—44no—
269Dic6—44no—
270#41—124no—
271C2×Q8—31no—
272C2×Q8—32no—
273C2×Q8—32no—
274C2×Q8—34no—
275C2×Q8—34no—
276C2×Q8—124no—
277C2×Q8—124no—
278C2×Q8C22×D311yescharacteristic
279C2×D4—32no—
280C2×D4—34no—
281C2×D4—31no—
282C2×D4C22×D312yes—
283C2×D4C22×D311yescharacteristic
284C22×C4—34no—
285C22×C4—34no—
286C22×C4—32no—
287C22×C4—34no—
288C22×C4C22×D311yescharacteristic
289C22×C4C22×D311yescharacteristic
290C22×C4C22×D311yescharacteristic
291C22×C4C22×D312yes—
292C4⋊C4—32no—
293C4⋊C4—32no—
294C4⋊C4—34no—
295C4⋊C4—32no—
296C4⋊C4—34no—
297C4⋊C4—34no—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—32no—
301C4⋊C4—31no—
302C4⋊C4—34no—
303C4⋊C4—34no—
304C4⋊C4—32no—
305C4⋊C4—31no—
306C4⋊C4—34no—
307C4⋊C4—34no—
308C4⋊C4—34no—
309C4⋊C4—32no—
310C4⋊C4C22×D311yescharacteristic
311C4⋊C4C22×D311yescharacteristic
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D312yes—
314C4⋊C4C22×D312yes—
315C4⋊C4C22×D312yes—
316C22⋊C4—32no—
317C22⋊C4—34no—
318C22⋊C4—34no—
319C22⋊C4—34no—
320C22⋊C4—32no—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4—34no—
324C22⋊C4—32no—
325C22⋊C4C22×D312yes—
326C22⋊C4C22×D312yes—
327C22⋊C4C22×D312yes—
328C22⋊C4C22×D312yes—
329C22⋊C4C22×D312yes—
330C42—32no—
331C42—34no—
332C42—34no—
333C42C22×D311yescharacteristic
334C42C22×D311yescharacteristic
335C42C22×D311yescharacteristic
336C42C22×D311yescharacteristic
337C42C22×D312yes—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C32−1+412yes—
344C22×C3C2511yescharacteristic
345C2×D3—22no—
346C2×D3—22no—
347C2×D3—22no—
348C3×C4—21no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—22no—
354C3×C4—22no—
355C3×C4H(5,𝔽2)12yes—
356C3×C42−1+412yes—
357Dic3—22no—
358Dic3—24no—
359Dic3—24no—
360Dic3—22no—
361Dic3—22no—
362C23—32no—
363C23C23×D312yes—
364C23C23×D311yescharacteristic
365Q8—22no—
366Q8—62no—
367Q8—64no—
368Q8—64no—
369Q8—124no—
370Q8—124no—
371D4—24no—
372D4—22no—
373D4—64no—
374D4—62no—
375D4—124no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—32no—
383C2×C4—34no—
384C2×C4—34no—
385C2×C4—64no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4—124no—
389C2×C4C23×D312yes—
390C2×C4C23×D312yes—
391C2×C4C23×D311yescharacteristic
392C2×C4C23×D311yescharacteristic
393C2×C4C23×D311yescharacteristic
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D312yes—
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D311yescharacteristic
398C2×C3—22no—
399C2×C3—42no—
400C2×C3C2×H(5,𝔽2)11yescharacteristic
401C2×C3C2×2−1+411yescharacteristic
402C2×C3C2×2−1+411yescharacteristic
403D3—42no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—62no—
410C22—62no—
411C22—62no—
412C22#101012yes—
413C22C24×D311yescharacteristic
414C4—22no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—22no—
420C4—22no—
421C4—64no—
422C4—62no—
423C4—64no—
424C4—62no—
425C4—62no—
426C4#100912yes—
427C4#101012yes—
428C3#355211yescharacteristic, complemented, Sylow
429C2—22no—
430C2—42no—
431C2—122no—
432C2C2×#100911yescharacteristic
433C2C2×#101011yescharacteristic
434C2C2×#101011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
hypercenter#529D3
hypocenterC3#3552
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. #5712C2semidirect
3. #5689C2semidirect
4. #5682C2semidirect
5. #5665C2semidirect
6. #5635C2semidirect
7. #5618C2semidirect
8. #5584C2semidirect
9. #5570C2semidirect
10. #5552C2semidirect
11. #5551C2semidirect
12. D4×Dic6C2semidirect
13. D4×Dic6C2semidirect
14. #5526C2semidirect
15. #5518C2semidirect
16. #5479C2semidirect
17. C2×#869C2semidirect
100. #529D3semidirect
428. C3#3552semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#57122
#56892
#56822
#56652
#56352
#56182
#55842
#55702
#55522
#55512
D4×Dic62
#55262
#55182
#54792
C2×#8692
#35523