Finite Group Databasebeta

← #87033

Subgroups — #87033

Order 384.

1012 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#557C211yesmaximal, characteristic, complemented
2#5732C212yesmaximal, complemented
3#5725C211yesmaximal, characteristic
4#5724C212yesmaximal, complemented
5#5721C211yesmaximal, characteristic, complemented
6#5693C214yesmaximal, complemented
7#5692C214yesmaximal, complemented
8#5691C212yesmaximal, complemented
9#5690C212yesmaximal, complemented
10#5585C211yesmaximal, characteristic, complemented
11#5585C211yesmaximal, characteristic, complemented
12#5580C212yesmaximal, complemented
13#5578C211yesmaximal, characteristic, complemented
14#5574C212yesmaximal, complemented
15C4×D3×Q8C211yesmaximal, characteristic, complemented
16#5568C212yesmaximal, complemented
17#5568C212yesmaximal, complemented
18#3507—31nomaximal, Sylow
19C2×#239C2212yes—
20C2×D3×Q8C2211yescharacteristic
21C3×C4⋊Q8C2211yescharacteristic
22C3×C4⋊Q8C2212yes—
23C3×#125C2212yes—
24C3×#125C2214yes—
25C3×C4×Q8C2212yes—
26C3×C4×Q8C2212yes—
27C3×C4×Q8C2211yescharacteristic
28C3×C4×Q8C2211yescharacteristic
29#947C2212yes—
30#947C2212yes—
31#947C2212yes—
32#946C2211yescharacteristic
33Q8×Dic3C2212yes—
34Q8×Dic3C2211yescharacteristic
35#944C2212yes—
36#944C2211yescharacteristic
37#898C2214yes—
38#898C2214yes—
39#898C2214yes—
40#897C2214yes—
41#897C2211yescharacteristic
42#896C2214yes—
43#896C2214yes—
44#896C2212yes—
45#896C2214yes—
46#895C2212yes—
47#895C2212yes—
48#894C2214yes—
49#894C2214yes—
50#894C2214yes—
51#894C2214yes—
52#893C2212yes—
53#893C2212yes—
54#893C2214yes—
55#893C2212yes—
56#893C2212yes—
57#892C2214yes—
58D3×C4⋊C4C2214yes—
59D3×C4⋊C4C2212yes—
60D3×C4⋊C4C2211yescharacteristic
61#890C2212yes—
62#889C2214yes—
63#889C2212yes—
64#888C2211yescharacteristic
65#887C2212yes—
66#887C2212yes—
67#887C2211yescharacteristic
68#876C2214yes—
69#875C2212yes—
70#875C2212yes—
71#874C2211yescharacteristic
72C4×D12C2212yes—
73C4×D12C2214yes—
74C4×D12C2212yes—
75#872C2214yes—
76#872C2214yes—
77#872C2212yes—
78C42×D3C2212yes—
79C42×D3C2211yescharacteristic
80C4×Dic6C2212yes—
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2212yes—
84#559—31no—
85#559—31no—
86#557—31no—
87#557D311yescharacteristic, complemented
88#556—32no—
89#556—32no—
90#556—32no—
91#554—32no—
92#552—34no—
93#539—34no—
94#536—32no—
95#533—31no—
96#532—31no—
97#529—32no—
98#529—32no—
99#520—32no—
100C2×C4×Q8—31no—
101C2×C3×Q8C2312yes—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×Q8C2311yescharacteristic
104#239—28no—
105D3×Q8—24no—
106C2×D12C2312yes—
107C2×D12C2312yes—
108C2×D12C2314yes—
109C2×D12C2312yes—
110C2×C4×D3C2311yescharacteristic
111C2×C4×D3C2312yes—
112C2×C4×D3C2314yes—
113C2×C4×D3C2312yes—
114C2×C4×D3C2311yescharacteristic
115C2×C4×D3C2314yes—
116C2×C4×D3C2314yes—
117C2×C4×D3C2311yescharacteristic
118C2×Dic6C2311yescharacteristic
119C2×Dic6C2311yescharacteristic
120C2×Dic6C2312yes—
121C2×Dic6C2312yes—
122C3×C4⋊C4C2312yes—
123C3×C4⋊C4C2311yescharacteristic
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2311yescharacteristic
126C3×C4⋊C4C2312yes—
127C3×C4⋊C4C2312yes—
128C3×C4⋊C4C2314yes—
129C3×C4⋊C4C2314yes—
130C3×C4⋊C4C2314yes—
131C3×C42C2311yescharacteristic
132C3×C42C2312yes—
133C3×C42C2312yes—
134C3×C42C2314yes—
135(C2×D3)⋊C4C2312yes—
136(C2×D3)⋊C4C2314yes—
137(C2×D3)⋊C4C2314yes—
138(C2×D3)⋊C4C2314yes—
139(C2×D3)⋊C4C2314yes—
140(C2×D3)⋊C4C2312yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2314yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2314yes—
145C4⋊Dic3C2311yescharacteristic
146C4⋊Dic3C2311yescharacteristic
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2314yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2312yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160C4×Dic3C2312yes—
161C4×Dic3C2311yescharacteristic
162C4×Dic3C2311yescharacteristic
163C4×Dic3C2312yes—
164C4×Dic3C2312yes—
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2312yes—
167C4×Dic3C2314yes—
168C2×#41—32no—
169C22×Q8—31no—
170C4⋊Q8—31no—
171C4⋊Q8—32no—
172C4⋊Q8—31no—
173C4⋊Q8C2×D311yescharacteristic
174C4⋊Q8C2×D312yes—
175#127—31no—
176#126—34no—
177#126—34no—
178#126—34no—
179#126—34no—
180#125—32no—
181#125—32no—
182#125—34no—
183#125C2×D312yes—
184#125C2×D314yes—
185#124—32no—
186#124—32no—
187#124—32no—
188#124—32no—
189#124—32no—
190#123—34no—
191#123—34no—
192#123—34no—
193#123—34no—
194C22⋊Q8—31no—
195C22⋊Q8—31no—
196C22⋊Q8—32no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201#121—32no—
202#121—32no—
203C4×Q8—31no—
204C4×Q8—32no—
205C4×Q8—32no—
206C4×Q8—31no—
207C4×Q8—32no—
208C4×Q8—32no—
209C4×Q8—31no—
210C4×Q8—32no—
211C4×Q8—31no—
212C4×Q8C2×D311yescharacteristic
213C4×Q8C2×D312yes—
214C4×Q8C2×D312yes—
215C4×Q8C2×D311yescharacteristic
216C4×D4—34no—
217C4×D4—32no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—32no—
221C4×D4—34no—
222C4×D4—32no—
223C4×D4—32no—
224#117—34no—
225#117—32no—
226#117—34no—
227#117—34no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—34no—
231C2×C42—32no—
232C2×C42—31no—
233C22×D3C2414yes—
234C22×D3C2411yescharacteristic
235C3×Q8—22no—
236C3×Q8—24no—
237C3×Q8—22no—
238C2×C3×C4C2411yescharacteristic
239C2×C3×C4C2414yes—
240C2×C3×C4C2412yes—
241C2×C3×C4C2412yes—
242C2×C3×C4C2412yes—
243C2×C3×C4C2412yes—
244C2×C3×C4C2411yescharacteristic
245C2×C3×C4C2411yescharacteristic
246C2×Dic3C2411yescharacteristic
247C2×Dic3C2412yes—
248C2×Dic3C2412yes—
249C2×Dic3C2412yes—
250C2×Dic3C2411yescharacteristic
251C2×Dic3C2412yes—
252C2×Dic3C2411yescharacteristic
253D12—28no—
254D12—24no—
255D12—24no—
256D12—42no—
257C4×D3—24no—
258C4×D3—22no—
259C4×D3—28no—
260C4×D3—41no—
261C4×D3—44no—
262C4×D3—44no—
263C4×D3—42no—
264C4×D3#4114yes—
265Dic6—22no—
266Dic6—22no—
267Dic6—24no—
268Dic6—42no—
269#41—68no—
270C2×Q8—31no—
271C2×Q8—32no—
272C2×Q8—31no—
273C2×Q8—32no—
274C2×Q8—64no—
275C2×Q8C22×D311yescharacteristic
276C2×Q8C22×D312yes—
277C2×Q8C22×D311yescharacteristic
278C2×D4—32no—
279C2×D4—34no—
280C2×D4—32no—
281C2×D4—32no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—32no—
285C22×C4—34no—
286C22×C4—32no—
287C22×C4—31no—
288C22×C4—34no—
289C22×C4—34no—
290C4⋊C4—32no—
291C4⋊C4—32no—
292C4⋊C4—31no—
293C4⋊C4—34no—
294C4⋊C4—32no—
295C4⋊C4—32no—
296C4⋊C4—32no—
297C4⋊C4—32no—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—34no—
301C4⋊C4—31no—
302C4⋊C4—32no—
303C4⋊C4—34no—
304C4⋊C4—32no—
305C4⋊C4C22×D311yescharacteristic
306C4⋊C4C22×D314yes—
307C4⋊C4C22×D312yes—
308C4⋊C4C22×D314yes—
309C4⋊C4C22×D311yescharacteristic
310C4⋊C4C22×D312yes—
311C4⋊C4C22×D312yes—
312C4⋊C4C22×D312yes—
313C4⋊C4C22×D314yes—
314C22⋊C4—34no—
315C22⋊C4—32no—
316C22⋊C4—32no—
317C22⋊C4—34no—
318C22⋊C4—34no—
319C22⋊C4—34no—
320C22⋊C4—34no—
321C22⋊C4—34no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C42—32no—
325C42—32no—
326C42—31no—
327C42—31no—
328C42—32no—
329C42—31no—
330C42—34no—
331C42—32no—
332C42C22×D312yes—
333C42C22×D311yescharacteristic
334C42C22×D314yes—
335C42C22×D312yes—
336C22×C3C2511yescharacteristic
337C2×D3—21no—
338C2×D3—21no—
339C2×D3—24no—
340C2×D3—24no—
341C2×D3—24no—
342C2×D3C2×#4112yes—
343C3×C4—22no—
344C3×C4—24no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—22no—
348C3×C4—21no—
349C3×C4C2×#4112yes—
350C3×C42−1+412yes—
351Dic3—22no—
352Dic3—22no—
353Dic3—21no—
354Dic3—21no—
355Dic3—22no—
356Dic3—22no—
357Dic3C2×#4112yes—
358C23—31no—
359C23—34no—
360Q8—22no—
361Q8—22no—
362Q8—24no—
363Q8—64no—
364Q8—62no—
365Q8—62no—
366Q8—122no—
367D4—68no—
368D4—64no—
369D4—64no—
370D4—122no—
371C2×C4—31no—
372C2×C4—34no—
373C2×C4—31no—
374C2×C4—32no—
375C2×C4—32no—
376C2×C4—32no—
377C2×C4—31no—
378C2×C4—32no—
379C2×C4—68no—
380C2×C4—64no—
381C2×C4—62no—
382C2×C4—122no—
383C2×C4—124no—
384C2×C4—124no—
385C2×C4—121no—
386C2×C4C23×D312yes—
387C2×C4C23×D312yes—
388C2×C4C23×D314yes—
389C2×C4C23×D311yescharacteristic
390C2×C4C23×D312yes—
391C2×C4C23×D312yes—
392C2×C4C23×D311yescharacteristic
393C2×C4C23×D311yescharacteristic
394C2×C3C22×#4111yescharacteristic
395C2×C3C2×2−1+411yescharacteristic
396C2×C3#58511yescharacteristic
397D3—22no—
398D3—44no—
399C22—32no—
400C22—61no—
401C22—61no—
402C22—64no—
403C22—64no—
404C22—64no—
405C22C24×D311yescharacteristic
406C4—22no—
407C4—24no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—21no—
412C4—32no—
413C4—61no—
414C4—62no—
415C4—62no—
416C4—62no—
417C4—62no—
418C4—61no—
419C4#100712yes—
420C4D3×#4112yes—
421C3#350711yescharacteristic, complemented, Sylow
422C2—62no—
423C2—124no—
424C2C2×#100711yescharacteristic
425C2C2×D3×#4111yescharacteristic
426C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#557C2
prime radical#5725C2
hypercenter#557D3
hypocenterC3#3507
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C3×#557C2semidirect
2. #5732C2semidirect
4. #5724C2semidirect
5. #5721C2semidirect
6. #5693C2semidirect
7. #5692C2semidirect
8. #5691C2semidirect
9. #5690C2semidirect
10. #5585C2semidirect
11. #5585C2semidirect
12. #5580C2semidirect
13. #5578C2semidirect
14. #5574C2semidirect
15. C4×D3×Q8C2semidirect
16. #5568C2semidirect
17. #5568C2semidirect
87. #557D3semidirect
421. C3#3507semidirect

Maximal subgroups

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

subgroupindex
C3×#5572
#57322
#57252
#57242
#57212
#56932
#56922
#56912
#56902
#55852
#55802
#55782
#55742
C4×D3×Q82
#55682
#35073