Finite Group Databasebeta

← #86266

Subgroups — #86266

Order 384.

992 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#533C211yesmaximal, characteristic, complemented
2#5732C211yesmaximal, characteristic, complemented
3#5727C211yesmaximal, characteristic, complemented
4#5721C211yesmaximal, characteristic, complemented
5#5721C211yesmaximal, characteristic, complemented
6#5696C214yesmaximal, complemented
7#5642C212yesmaximal, complemented
8#5632C212yesmaximal, complemented
9#5624C212yesmaximal
10#5622C212yesmaximal, complemented
11#5583C212yesmaximal, complemented
12#5572C212yesmaximal, complemented
13#5569C212yesmaximal, complemented
14#5568C212yesmaximal, complemented
15#5531C214yesmaximal, complemented
16#5479C211yesmaximal, characteristic
17C2×C4×Dic6C211yesmaximal, characteristic, complemented
18#3523—31nomaximal, Sylow
19C2×D3×Q8C2212yes—
20C22×Dic6C2211yescharacteristic
21C3×C4⋊Q8C2211yescharacteristic
22C3×C4⋊Q8C2211yescharacteristic
23C3×#125C2212yes—
24C3×C22⋊Q8C2212yes—
25C3×C22⋊Q8C2212yes—
26C3×C4×Q8C2212yes—
27C3×C4×Q8C2212yes—
28C3×#117C2212yes—
29C2×C3×C42C2211yescharacteristic
30#947C2212yes—
31#946C2212yes—
32Q8×Dic3C2212yes—
33Q8×Dic3C2212yes—
34Q8×Dic3C2212yes—
35#944C2212yes—
36#944C2212yes—
37#930C2211yescharacteristic
38#929C2212yes—
39C4×#67C2212yes—
40C2×C4⋊Dic3C2211yescharacteristic
41#924C2211yescharacteristic
42C2×Dic3⋊C4C2212yes—
43C2×C4×Dic3C2212yes—
44#898C2214yes—
45#898C2214yes—
46#898C2214yes—
47#897C2212yes—
48#896C2214yes—
49#896C2214yes—
50#893C2212yes—
51#892C2212yes—
52#892C2214yes—
53#890C2214yes—
54#890C2212yes—
55#890C2212yes—
56#889C2214yes—
57#889C2214yes—
58#889C2214yes—
59#888C2214yes—
60#888C2212yes—
61#887C2212yes—
62#887C2214yes—
63#887C2212yes—
64#886C2214yes—
65#67⋊C4C2214yes—
66#878C2214yes—
67#877C2214yes—
68#876C2212yes—
69#876C2214yes—
70#875C2211yescharacteristic
71C4×D12C2211yescharacteristic
72#872C2212yes—
73#872C2214yes—
74#870C2211yescharacteristic
75#870C2214yes—
76C4×Dic6C2212yes—
77C4×Dic6C2211yescharacteristic
78C4×Dic6C2211yescharacteristic
79C4×Dic6C2211yescharacteristic
80C4×Dic6C2212yes—
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2214yes—
84#557—31no—
85#557—31no—
86#557—32no—
87#556—31no—
88#556—32no—
89#552—32no—
90#552—34no—
91D4×Q8—31no—
92D4×Q8—32no—
93#541—34no—
94#541—32no—
95#533—32no—
96#533D311yescharacteristic, complemented
97#529—32no—
98#529—31no—
99#519—32no—
100C2×C4×Q8—31no—
101C2×C3×Q8C2312yes—
102C2×C3×Q8C2312yes—
103C22×C3×C4C2311yescharacteristic
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C2×#67C2312yes—
107C22×Dic3C2312yes—
108C22×Dic3C2312yes—
109D3×Q8—28no—
110C2×D12C2311yescharacteristic
111C2×C4×D3C2312yes—
112C2×C4×D3C2314yes—
113C2×Dic6—24no—
114C2×Dic6C2312yes—
115C2×Dic6C2311yescharacteristic
116C2×Dic6C2311yescharacteristic
117C2×Dic6C2314yes—
118C2×Dic6C2311yescharacteristic
119C3×C4⋊C4C2312yes—
120C3×C4⋊C4C2314yes—
121C3×C4⋊C4C2312yes—
122C3×C4⋊C4C2314yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C22⋊C4C2314yes—
126C3×C42C2311yescharacteristic
127C3×C42C2311yescharacteristic
128C3×C42C2311yescharacteristic
129C3×C42C2311yescharacteristic
130C3×C42C2314yes—
131#217C2312yes—
132(C2×D3)⋊C4C2312yes—
133(C2×D3)⋊C4C2312yes—
134(C2×D3)⋊C4C2314yes—
135(C2×D3)⋊C4C2312yes—
136(C2×D3)⋊C4C2312yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2314yes—
139C4⋊Dic3C2312yes—
140C4⋊Dic3C2311yescharacteristic
141C4⋊Dic3C2311yescharacteristic
142C4⋊Dic3C2311yescharacteristic
143C4⋊Dic3C2314yes—
144C4⋊Dic3C2311yescharacteristic
145C4⋊Dic3C2314yes—
146C4⋊Dic3C2314yes—
147Dic3⋊C4C2312yes—
148Dic3⋊C4C2312yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160C4×Dic3C2312yes—
161C4×Dic3C2312yes—
162C4×Dic3C2312yes—
163C4×Dic3C2312yes—
164C4×Dic3C2312yes—
165C4×Dic3C2314yes—
166C4×Dic3C2314yes—
167C4×Dic3C2314yes—
168C22×Q8—32no—
169C22×Q8—31no—
170C4⋊Q8—32no—
171C4⋊Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—34no—
174C4⋊Q8C2×D311yescharacteristic
175C4⋊Q8C2×D311yescharacteristic
176#126—34no—
177#126—34no—
178#126—34no—
179#126—32no—
180#126—34no—
181#125—32no—
182#125—34no—
183#125—34no—
184#125—34no—
185#125—34no—
186#125—32no—
187#125—31no—
188#125—34no—
189#125C2×D312yes—
190#124—32no—
191#124—31no—
192#123—32no—
193#123—34no—
194C22⋊Q8—32no—
195C22⋊Q8—31no—
196C22⋊Q8—34no—
197C22⋊Q8—34no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8C2×D312yes—
201C22⋊Q8C2×D312yes—
202#121—31no—
203C4×Q8—32no—
204C4×Q8—32no—
205C4×Q8—32no—
206C4×Q8—34no—
207C4×Q8—32no—
208C4×Q8—32no—
209C4×Q8—31no—
210C4×Q8—32no—
211C4×Q8—31no—
212C4×Q8—31no—
213C4×Q8—32no—
214C4×Q8—31no—
215C4×Q8—34no—
216C4×Q8—31no—
217C4×Q8C2×D312yes—
218C4×Q8C2×D312yes—
219C4×D4—32no—
220C4×D4—32no—
221C4×D4—34no—
222C4×D4—31no—
223#117—34no—
224#117—32no—
225#117—34no—
226#117—32no—
227#117—34no—
228#117C2×D312yes—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C42—32no—
232C2×C42C2×D311yescharacteristic
233C23×C3C2411yescharacteristic
234C22×D3C2412yes—
235C3×Q8—24no—
236C3×Q8—24no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4C2411yescharacteristic
241C2×C3×C4C2414yes—
242C2×C3×C4C2414yes—
243C2×C3×C4C2411yescharacteristic
244C2×C3×C4C2411yescharacteristic
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248#67—24no—
249C2×Dic3—24no—
250C2×Dic3—24no—
251C2×Dic3C2412yes—
252C2×Dic3C2412yes—
253C2×Dic3C2412yes—
254C2×Dic3C2412yes—
255C2×Dic3C2412yes—
256C2×Dic3C2412yes—
257C2×Dic3C2412yes—
258D12—22no—
259C4×D3—24no—
260C4×D3—28no—
261Dic6—22no—
262Dic6—22no—
263Dic6—24no—
264Dic6—28no—
265Dic6#4114yes—
266C2×Q8—31no—
267C2×Q8—31no—
268C2×Q8—34no—
269C2×Q8—32no—
270C2×Q8—31no—
271C2×Q8—64no—
272C2×Q8—68no—
273C2×Q8C22×D312yes—
274C2×Q8C22×D312yes—
275C2×D4—32no—
276C2×D4—31no—
277C22×C4—32no—
278C22×C4—32no—
279C22×C4—34no—
280C22×C4—32no—
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C4⋊C4—32no—
285C4⋊C4—31no—
286C4⋊C4—32no—
287C4⋊C4—31no—
288C4⋊C4—32no—
289C4⋊C4—32no—
290C4⋊C4—32no—
291C4⋊C4—34no—
292C4⋊C4—32no—
293C4⋊C4—32no—
294C4⋊C4—34no—
295C4⋊C4—34no—
296C4⋊C4—34no—
297C4⋊C4—31no—
298C4⋊C4—34no—
299C4⋊C4—32no—
300C4⋊C4—31no—
301C4⋊C4—34no—
302C4⋊C4—34no—
303C4⋊C4—32no—
304C4⋊C4—34no—
305C4⋊C4C22×D312yes—
306C4⋊C4C22×D314yes—
307C4⋊C4C22×D312yes—
308C4⋊C4C22×D312yes—
309C4⋊C4C22×D312yes—
310C4⋊C4C22×D314yes—
311C22⋊C4—32no—
312C22⋊C4—32no—
313C22⋊C4—34no—
314C22⋊C4—32no—
315C22⋊C4—34no—
316C22⋊C4—32no—
317C22⋊C4—32no—
318C22⋊C4—34no—
319C22⋊C4C22×D314yes—
320C42—32no—
321C42—32no—
322C42—32no—
323C42—32no—
324C42—34no—
325C42—34no—
326C42—34no—
327C42—32no—
328C42C22×D311yescharacteristic
329C42C22×D311yescharacteristic
330C42C22×D311yescharacteristic
331C42C22×D311yescharacteristic
332C42C22×D314yes—
333C22×C3—21no—
334C22×C3—21no—
335C22×C32−1+412yes—
336C22×C3C2511yescharacteristic
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—22no—
340C3×C4—21no—
341C3×C4—24no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—24no—
346C3×C4C2×#4112yes—
347C3×C42−1+412yes—
348Dic3—22no—
349Dic3—22no—
350Dic3—22no—
351Dic3—22no—
352Dic3—22no—
353Dic3—22no—
354Dic3C2×#4114yes—
355C23—32no—
356C23C23×D311yescharacteristic
357Q8—24no—
358Q8—24no—
359Q8—34no—
360Q8—64no—
361Q8—68no—
362Q8—62no—
363Q8—62no—
364D4—64no—
365D4—62no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—32no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—32no—
375C2×C4—32no—
376C2×C4—68no—
377C2×C4—64no—
378C2×C4—64no—
379C2×C4—64no—
380C2×C4C23×D311yescharacteristic
381C2×C4C23×D314yes—
382C2×C4C23×D314yes—
383C2×C4C23×D311yescharacteristic
384C2×C4C23×D311yescharacteristic
385C2×C4C23×D311yescharacteristic
386C2×C4C23×D311yescharacteristic
387C2×C4C23×D311yescharacteristic
388C2×C3—22no—
389C2×C3C22×#4111yescharacteristic
390C2×C3C2×2−1+411yescharacteristic
391C2×C3C2×2−1+411yescharacteristic
392D3—42no—
393C22—21no—
394C22—21no—
395C22—62no—
396C22—62no—
397C22—62no—
398C22#100712yes—
399C22C24×D311yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—24no—
404C4—21no—
405C4—24no—
406C4—34no—
407C4—62no—
408C4—62no—
409C4—62no—
410C4—62no—
411C4—62no—
412C4—62no—
413C4D3×#4112yes—
414C4#101012yes—
415C3#352311yescharacteristic, complemented, Sylow
416C2—22no—
417C2—122no—
418C2C2×#100711yescharacteristic
419C2C2×D3×#4111yescharacteristic
420C2C2×#101011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#533C2
prime radical#930C22
hypercenter#533D3
hypocenterC3#3523
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
2. #5732C2semidirect
3. #5727C2semidirect
4. #5721C2semidirect
5. #5721C2semidirect
6. #5696C2semidirect
7. #5642C2semidirect
8. #5632C2semidirect
10. #5622C2semidirect
11. #5583C2semidirect
12. #5572C2semidirect
13. #5569C2semidirect
14. #5568C2semidirect
15. #5531C2semidirect
17. C2×C4×Dic6C2semidirect
96. #533D3semidirect
415. C3#3523semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#57322
#57272
#57212
#56962
#56422
#56322
#56242
#56222
#55832
#55722
#55692
#55682
#55312
#54792
C2×C4×Dic62
#35233