Finite Group Databasebeta

← #87034

Subgroups — #87034

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×#557C211yesmaximal, characteristic, complemented
2#5727C212yesmaximal, complemented
3#5726C211yesmaximal, characteristic, complemented
4#5724C212yesmaximal, complemented
5D3×C4⋊Q8C211yesmaximal, characteristic, complemented
6#5699C214yesmaximal, complemented
7#5695C212yesmaximal, complemented
8#5692C214yesmaximal, complemented
9#5689C212yesmaximal, complemented
10#5584C211yesmaximal, characteristic, complemented
11#5583C212yesmaximal, complemented
12#5582C212yesmaximal, complemented
13#5580C211yesmaximal, characteristic, complemented
14#5580C211yesmaximal, characteristic, complemented
15#5579C212yesmaximal, complemented
16#5576C212yesmaximal, complemented
17Q8×D12C211yesmaximal, characteristic, complemented
18#3552—31nomaximal, Sylow
19C2×#239C2212yes—
20C2×D3×Q8C2212yes—
21C2×D3×Q8C2212yes—
22C3×C4⋊Q8C2211yescharacteristic
23C3×C4⋊Q8C2212yes—
24C3×#125C2212yes—
25C3×#125C2214yes—
26C3×C4×Q8C2211yescharacteristic
27C3×C4×Q8C2212yes—
28C3×C4×Q8C2212yes—
29C3×C4×Q8C2211yescharacteristic
30#947C2211yescharacteristic
31#947C2211yescharacteristic
32#946C2212yes—
33#946C2211yescharacteristic
34#946C2211yescharacteristic
35#946C2212yes—
36#944C2212yes—
37#898C2214yes—
38#898C2214yes—
39#898C2212yes—
40#898C2212yes—
41#897C2214yes—
42#897C2214yes—
43#897C2212yes—
44#896C2214yes—
45#896C2214yes—
46#896C2214yes—
47#896C2214yes—
48#895C2214yes—
49#895C2211yescharacteristic
50#895C2212yes—
51#895C2211yescharacteristic
52#894C2212yes—
53#894C2212yes—
54#894C2214yes—
55#894C2214yes—
56#894C2214yes—
57#893C2214yes—
58#892C2212yes—
59D3×C4⋊C4C2212yes—
60D3×C4⋊C4C2212yes—
61D3×C4⋊C4C2212yes—
62#890C2211yescharacteristic
63#890C2211yescharacteristic
64#889C2214yes—
65#888C2214yes—
66#888C2212yes—
67#888C2212yes—
68#876C2214yes—
69#875C2214yes—
70#875C2212yes—
71C4×D12C2214yes—
72C4×D12C2212yes—
73C4×D12C2212yes—
74C4×D12C2211yescharacteristic
75C4×D12C2211yescharacteristic
76C4×D12C2212yes—
77#872C2212yes—
78#872C2212yes—
79C42×D3C2211yescharacteristic
80#870C2214yes—
81#869C2212yes—
82#869C2211yescharacteristic
83C4×Dic6C2212yes—
84#563—34no—
85#557D311yescharacteristic, complemented
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#529—31no—
101C2×C3×Q8C2312yes—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×Q8C2311yescharacteristic
104#239—44no—
105D3×Q8—44no—
106D3×Q8—44no—
107C2×D12C2312yes—
108C2×D12C2311yescharacteristic
109C2×D12C2311yescharacteristic
110C2×D12C2312yes—
111C2×D12C2314yes—
112C2×C4×D3C2311yescharacteristic
113C2×C4×D3C2311yescharacteristic
114C2×C4×D3C2311yescharacteristic
115C2×C4×D3C2312yes—
116C2×C4×D3C2312yes—
117C2×C4×D3C2314yes—
118C2×C4×D3C2314yes—
119C2×C4×D3C2314yes—
120C2×Dic6C2312yes—
121C2×Dic6C2314yes—
122C2×Dic6C2314yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2314yes—
126C3×C4⋊C4C2314yes—
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2311yescharacteristic
129C3×C4⋊C4C2314yes—
130C3×C4⋊C4C2312yes—
131C3×C4⋊C4C2312yes—
132C3×C42C2312yes—
133C3×C42C2311yescharacteristic
134C3×C42C2312yes—
135C3×C42C2314yes—
136(C2×D3)⋊C4C2312yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2312yes—
140(C2×D3)⋊C4C2312yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2314yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2314yes—
149(C2×D3)⋊C4C2314yes—
150C4⋊Dic3C2311yescharacteristic
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2314yes—
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2312yes—
156C4⋊Dic3C2312yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2314yes—
165C4×Dic3C2314yes—
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C2×#41—32no—
170C22×Q8—32no—
171C22×Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—32no—
174C4⋊Q8—32no—
175C4⋊Q8—34no—
176C4⋊Q8—31no—
177C4⋊Q8—32no—
178C4⋊Q8C2×D311yescharacteristic
179C4⋊Q8C2×D312yes—
180#126—32no—
181#126—32no—
182#126—34no—
183#126—34no—
184#126—34no—
185#125—31no—
186#125—31no—
187#125—34no—
188#125—34no—
189#125C2×D314yes—
190#125C2×D312yes—
191#124—31no—
192#124—31no—
193#124—32no—
194#124—34no—
195#123—32no—
196#123—32no—
197#123—34no—
198#123—34no—
199#123—34no—
200C22⋊Q8—31no—
201C22⋊Q8—31no—
202C22⋊Q8—32no—
203C22⋊Q8—32no—
204C22⋊Q8—34no—
205C22⋊Q8—34no—
206C22⋊Q8—34no—
207C22⋊Q8—34no—
208C22⋊Q8—32no—
209C22⋊Q8—34no—
210C22⋊Q8—34no—
211#121—31no—
212#121—31no—
213#121—34no—
214#121—32no—
215C4×Q8—32no—
216C4×Q8C2×D311yescharacteristic
217C4×Q8C2×D311yescharacteristic
218C4×Q8C2×D312yes—
219C4×Q8C2×D312yes—
220C4×D4—34no—
221C4×D4—31no—
222C4×D4—31no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—34no—
226C4×D4—32no—
227#117—32no—
228#117—32no—
229#117—32no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C42—31no—
234C22×D3C2412yes—
235C22×D3C2412yes—
236C22×D3C2411yescharacteristic
237C3×Q8—22no—
238C3×Q8—24no—
239C3×Q8—22no—
240C2×C3×C4C2412yes—
241C2×C3×C4C2411yescharacteristic
242C2×C3×C4C2412yes—
243C2×C3×C4C2414yes—
244C2×C3×C4C2412yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248C2×Dic3C2411yescharacteristic
249C2×Dic3C2412yes—
250C2×Dic3C2411yescharacteristic
251C2×Dic3C2411yescharacteristic
252C2×Dic3C2414yes—
253C2×Dic3C2412yes—
254D12—24no—
255D12—24no—
256D12—22no—
257D12—22no—
258D12—44no—
259C4×D3—22no—
260C4×D3—22no—
261C4×D3—24no—
262C4×D3—22no—
263C4×D3—24no—
264C4×D3—44no—
265C4×D3—44no—
266C4×D3—44no—
267Dic6—24no—
268Dic6—44no—
269Dic6—44no—
270#41—124no—
271C2×Q8—32no—
272C2×Q8—34no—
273C2×Q8—34no—
274C2×Q8—124no—
275C2×Q8—124no—
276C2×Q8C22×D311yescharacteristic
277C2×Q8C22×D311yescharacteristic
278C2×Q8C22×D312yes—
279C2×D4—32no—
280C2×D4—32no—
281C2×D4—31no—
282C2×D4—31no—
283C2×D4—34no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—32no—
288C22×C4—32no—
289C22×C4—34no—
290C22×C4—34no—
291C22×C4—34no—
292C4⋊C4—31no—
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—32no—
302C4⋊C4—31no—
303C4⋊C4—34no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—34no—
307C4⋊C4C22×D312yes—
308C4⋊C4C22×D312yes—
309C4⋊C4C22×D311yescharacteristic
310C4⋊C4C22×D314yes—
311C4⋊C4C22×D311yescharacteristic
312C4⋊C4C22×D314yes—
313C4⋊C4C22×D314yes—
314C4⋊C4C22×D312yes—
315C4⋊C4C22×D312yes—
316C22⋊C4—34no—
317C22⋊C4—32no—
318C22⋊C4—32no—
319C22⋊C4—32no—
320C22⋊C4—32no—
321C22⋊C4—32no—
322C22⋊C4—32no—
323C22⋊C4—34no—
324C22⋊C4—34no—
325C22⋊C4—32no—
326C22⋊C4—32no—
327C22⋊C4—34no—
328C22⋊C4—32no—
329C22⋊C4—32no—
330C42—31no—
331C42—31no—
332C42—31no—
333C42—34no—
334C42C22×D311yescharacteristic
335C42C22×D312yes—
336C42C22×D312yes—
337C42C22×D314yes—
338C22×C3C2511yescharacteristic
339C2×D3—21no—
340C2×D3—21no—
341C2×D3—22no—
342C2×D3—22no—
343C2×D3—22no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—22no—
347C2×D32−1+412yes—
348C3×C4—22no—
349C3×C4—24no—
350C3×C4—21no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—22no—
354C3×C4H(5,𝔽2)12yes—
355C3×C42−1+412yes—
356Dic3—24no—
357Dic3—21no—
358Dic3—22no—
359Dic3—21no—
360Dic3—22no—
361Dic3—21no—
362C23—31no—
363C23—32no—
364C23—32no—
365Q8—22no—
366Q8—22no—
367Q8—24no—
368Q8—64no—
369Q8—124no—
370Q8—124no—
371D4—64no—
372D4—64no—
373D4—62no—
374D4—62no—
375D4—124no—
376C2×C4—31no—
377C2×C4—32no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—34no—
381C2×C4—32no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—64no—
385C2×C4—62no—
386C2×C4—64no—
387C2×C4—124no—
388C2×C4—124no—
389C2×C4—124no—
390C2×C4C23×D312yes—
391C2×C4C23×D311yescharacteristic
392C2×C4C23×D312yes—
393C2×C4C23×D312yes—
394C2×C4C23×D312yes—
395C2×C4C23×D314yes—
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D311yescharacteristic
398C2×C3C2×H(5,𝔽2)11yescharacteristic
399C2×C3C2×2−1+411yescharacteristic
400C2×C3C2×2−1+411yescharacteristic
401D3—22no—
402D3—42no—
403D3—42no—
404C22—32no—
405C22—61no—
406C22—61no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22—62no—
412C22—62no—
413C22C24×D311yescharacteristic
414C4—22no—
415C4—24no—
416C4—22no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—61no—
421C4—62no—
422C4—62no—
423C4—61no—
424C4—64no—
425C4—61no—
426C4#100712yes—
427C4#100912yes—
428C3#355211yescharacteristic, complemented, Sylow
429C2—62no—
430C2—122no—
431C2—122no—
432C2C2×#100711yescharacteristic
433C2C2×#100911yescharacteristic
434C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#557C2
hypercenter#557D3
hypocenterC3#3552
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. #5727C2semidirect
3. #5726C2semidirect
4. #5724C2semidirect
5. D3×C4⋊Q8C2semidirect
6. #5699C2semidirect
7. #5695C2semidirect
8. #5692C2semidirect
9. #5689C2semidirect
10. #5584C2semidirect
11. #5583C2semidirect
12. #5582C2semidirect
13. #5580C2semidirect
14. #5580C2semidirect
15. #5579C2semidirect
16. #5576C2semidirect
17. Q8×D12C2semidirect
85. #557D3semidirect
428. C3#3552semidirect

Maximal subgroups

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

subgroupindex
C3×#5572
#57272
#57262
#57242
D3×C4⋊Q82
#56992
#56952
#56922
#56892
#55842
#55832
#55822
#55802
#55792
#55762
Q8×D122
#35523