Finite Group Databasebeta

← #87010

Subgroups — #87010

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×#556C211yesmaximal, characteristic, complemented
2#5727C211yesmaximal, characteristic, complemented
3D3×C4⋊Q8C211yesmaximal, characteristic, complemented
4#5712C214yesmaximal, complemented
5#5707C214yesmaximal, complemented
6#5677C212yesmaximal, complemented
7#5675C212yesmaximal, complemented
8#5643C212yesmaximal, complemented
9#5635C212yesmaximal, complemented
10#5583C212yesmaximal, complemented
11#5580C211yesmaximal, characteristic, complemented
12Q8×D12C212yesmaximal, complemented
13#5570C211yesmaximal, characteristic, complemented
14#5557C211yesmaximal, characteristic, complemented
15#5555C211yesmaximal, characteristic, complemented
16#5530C212yesmaximal, complemented
17#5519C212yesmaximal, complemented
18#3552—31nomaximal, Sylow
19C2×D3×Q8C2212yes—
20C2×D3×Q8C2212yes—
21C2×#237C2212yes—
22C3×C4⋊Q8C2211yescharacteristic
23C3×#126C2214yes—
24C3×#124C2212yes—
25C3×C22⋊Q8C2212yes—
26C3×C4×Q8C2211yescharacteristic
27C3×C4×Q8C2212yes—
28C3×C4×D4C2211yescharacteristic
29C3×#117C2212yes—
30#947C2211yescharacteristic
31#946C2211yescharacteristic
32#946C2212yes—
33Q8×Dic3C2212yes—
34#944C2212yes—
35#938C2211yescharacteristic
36#936C2211yescharacteristic
37#930C2212yes—
38C4×#67C2212yes—
39#926C2212yes—
40C2×C4⋊Dic3C2212yes—
41#898C2214yes—
42#898C2212yes—
43#897C2214yes—
44#897C2214yes—
45#897C2212yes—
46#897C2211yescharacteristic
47#896C2214yes—
48#895C2211yescharacteristic
49#894C2212yes—
50#893C2212yes—
51D3×C4⋊C4C2212yes—
52D3×C4⋊C4C2212yes—
53#890C2214yes—
54#890C2211yescharacteristic
55#890C2212yes—
56#890C2211yescharacteristic
57#889C2214yes—
58#888C2212yes—
59#888C2212yes—
60#888C2212yes—
61#886C2214yes—
62#885C2214yes—
63#884C2214yes—
64#883C2214yes—
65#883C2214yes—
66#883C2212yes—
67#67⋊C4C2214yes—
68#879C2212yes—
69#878C2214yes—
70#878C2214yes—
71#878C2214yes—
72#876C2214yes—
73C4×D12C2212yes—
74C4×D12C2214yes—
75C4×D12C2211yescharacteristic
76#872C2212yes—
77C42×D3C2211yescharacteristic
78#870C2214yes—
79#869C2212yes—
80#869C2214yes—
81#869C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2211yescharacteristic
84#563—34no—
85#557—31no—
86#556D311yescharacteristic, complemented
87#554—32no—
88#553—32no—
89D4×Q8—31no—
90D4×Q8—32no—
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×D4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C2×#67C2312yes—
106C2×#67C2314yes—
107C22×Dic3C2314yes—
108D3×Q8—44no—
109D3×Q8—44no—
110#237—44no—
111C2×D12C2312yes—
112C2×D12C2311yescharacteristic
113C2×C4×D3C2311yescharacteristic
114C2×C4×D3C2311yescharacteristic
115C2×C4×D3C2311yescharacteristic
116C2×C4×D3C2312yes—
117C2×C4×D3C2314yes—
118C2×C4×D3C2314yes—
119C2×Dic6C2311yescharacteristic
120C2×Dic6C2312yes—
121C2×Dic6C2314yes—
122C2×Dic6C2314yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C4⋊C4C2314yes—
126C3×C4⋊C4C2311yescharacteristic
127C3×C4⋊C4C2312yes—
128C3×C4⋊C4C2311yescharacteristic
129C3×C22⋊C4C2314yes—
130C3×C22⋊C4C2314yes—
131C3×C22⋊C4C2312yes—
132C3×C42C2311yescharacteristic
133C3×C42C2312yes—
134C3×C42C2314yes—
135#217C2312yes—
136#217C2312yes—
137#217C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2312yes—
140(C2×D3)⋊C4C2312yes—
141(C2×D3)⋊C4C2314yes—
142(C2×D3)⋊C4C2312yes—
143(C2×D3)⋊C4C2312yes—
144(C2×D3)⋊C4C2312yes—
145(C2×D3)⋊C4C2314yes—
146C4⋊Dic3C2311yescharacteristic
147C4⋊Dic3C2312yes—
148C4⋊Dic3C2314yes—
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2314yes—
151C4⋊Dic3C2314yes—
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2311yescharacteristic
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2314yes—
164C4×Dic3C2314yes—
165C4×Dic3C2312yes—
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—32no—
176C4⋊Q8—31no—
177C4⋊Q8—32no—
178C4⋊Q8—34no—
179C4⋊Q8C2×D311yescharacteristic
180#126—32no—
181#126—34no—
182#126—32no—
183#126—34no—
184#126C2×D314yes—
185#125—31no—
186#125—31no—
187#125—34no—
188#125—34no—
189#125—32no—
190#125—34no—
191#124—31no—
192#124—31no—
193#124—34no—
194#124C2×D312yes—
195#123—32no—
196#123—32no—
197#123—34no—
198#123—34no—
199#123—34no—
200C22⋊Q8—31no—
201C22⋊Q8—32no—
202C22⋊Q8—31no—
203C22⋊Q8—34no—
204C22⋊Q8—34no—
205C22⋊Q8—32no—
206C22⋊Q8—34no—
207C22⋊Q8—34no—
208C22⋊Q8—34no—
209C22⋊Q8—34no—
210C22⋊Q8C2×D312yes—
211#121—31no—
212#121—32no—
213#121—31no—
214#121—34no—
215C4×Q8—32no—
216C4×Q8—31no—
217C4×Q8—32no—
218C4×Q8C2×D311yescharacteristic
219C4×Q8C2×D312yes—
220C4×D4—32no—
221C4×D4—32no—
222C4×D4—34no—
223C4×D4—31no—
224C4×D4—32no—
225C4×D4—34no—
226C4×D4C2×D311yescharacteristic
227#117—32no—
228#117—32no—
229#117C2×D312yes—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C42—31no—
234C23×C3C2412yes—
235C22×D3C2412yes—
236C22×D3C2411yescharacteristic
237C3×Q8—24no—
238C3×Q8—22no—
239C3×D4—22no—
240C2×C3×C4—24no—
241C2×C3×C4C2412yes—
242C2×C3×C4C2411yescharacteristic
243C2×C3×C4C2412yes—
244C2×C3×C4C2414yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248#67—24no—
249#67—44no—
250C2×Dic3—44no—
251C2×Dic3C2412yes—
252C2×Dic3C2411yescharacteristic
253C2×Dic3C2411yescharacteristic
254C2×Dic3C2411yescharacteristic
255C2×Dic3C2414yes—
256C2×Dic3C2412yes—
257C2×Dic3C2412yes—
258D12—24no—
259D12—22no—
260C4×D3—22no—
261C4×D3—24no—
262C4×D3—22no—
263C4×D3—22no—
264C4×D3—44no—
265C4×D3—44no—
266Dic6—24no—
267Dic6—22no—
268Dic6—44no—
269Dic6—44no—
270#41—124no—
271C2×Q8—31no—
272C2×Q8—32no—
273C2×Q8—34no—
274C2×Q8—34no—
275C2×Q8—124no—
276C2×Q8—124no—
277C2×Q8C22×D311yescharacteristic
278C2×Q8C22×D312yes—
279C2×D4—32no—
280C2×D4—34no—
281C2×D4—32no—
282C2×D4—31no—
283C2×D4C22×D311yescharacteristic
284C22×C4—34no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—32no—
289C22×C4—34no—
290C22×C4—34no—
291C22×C4C22×D312yes—
292C4⋊C4—31no—
293C4⋊C4—32no—
294C4⋊C4—34no—
295C4⋊C4—34no—
296C4⋊C4—34no—
297C4⋊C4—32no—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—32no—
301C4⋊C4—32no—
302C4⋊C4—34no—
303C4⋊C4—32no—
304C4⋊C4—34no—
305C4⋊C4—32no—
306C4⋊C4—31no—
307C4⋊C4—34no—
308C4⋊C4—34no—
309C4⋊C4—32no—
310C4⋊C4C22×D312yes—
311C4⋊C4C22×D312yes—
312C4⋊C4C22×D311yescharacteristic
313C4⋊C4C22×D311yescharacteristic
314C4⋊C4C22×D314yes—
315C4⋊C4C22×D312yes—
316C22⋊C4—32no—
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⋊C4C22×D314yes—
328C22⋊C4C22×D314yes—
329C22⋊C4C22×D312yes—
330C42—31no—
331C42—31no—
332C42—31no—
333C42—32no—
334C42—34no—
335C42C22×D311yescharacteristic
336C42C22×D312yes—
337C42C22×D314yes—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3C2511yescharacteristic
342C2×D3—21no—
343C2×D3—21no—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—22no—
347C2×D32−1+412yes—
348C3×C4—24no—
349C3×C4—21no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4H(5,𝔽2)12yes—
354C3×C42−1+412yes—
355Dic3—22no—
356Dic3—24no—
357Dic3—21no—
358Dic3—22no—
359Dic3—21no—
360Dic3—21no—
361Dic3—22no—
362C23—31no—
363C23—32no—
364C23C23×D312yes—
365Q8—24no—
366Q8—22no—
367Q8—62no—
368Q8—64no—
369Q8—124no—
370Q8—124no—
371D4—22no—
372D4—64no—
373D4—64no—
374D4—62no—
375D4—124no—
376C2×C4—24no—
377C2×C4—32no—
378C2×C4—31no—
379C2×C4—31no—
380C2×C4—31no—
381C2×C4—32no—
382C2×C4—34no—
383C2×C4—32no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—64no—
387C2×C4—62no—
388C2×C4—124no—
389C2×C4—124no—
390C2×C4—124no—
391C2×C4C23×D312yes—
392C2×C4C23×D311yescharacteristic
393C2×C4C23×D312yes—
394C2×C4C23×D312yes—
395C2×C4C23×D314yes—
396C2×C4C23×D311yescharacteristic
397C2×C4C23×D311yescharacteristic
398C2×C3—42no—
399C2×C3C2×H(5,𝔽2)11yescharacteristic
400C2×C3C2×2−1+411yescharacteristic
401C2×C3C2×2−1+411yescharacteristic
402D3—22no—
403D3—42no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—32no—
408C22—61no—
409C22—61no—
410C22—62no—
411C22—62no—
412C22—62no—
413C22C24×D311yescharacteristic
414C4—24no—
415C4—22no—
416C4—22no—
417C4—21no—
418C4—22no—
419C4—62no—
420C4—61no—
421C4—64no—
422C4—62no—
423C4—61no—
424C4—61no—
425C4—62no—
426C4#100912yes—
427C4#101012yes—
428C3#355211yescharacteristic, complemented, Sylow
429C2—42no—
430C2—62no—
431C2—122no—
432C2C2×#100911yescharacteristic
433C2C2×#101011yescharacteristic
434C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
hypercenter#556D3
hypocenterC3#3552
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. C3×#556C2semidirect
2. #5727C2semidirect
3. D3×C4⋊Q8C2semidirect
4. #5712C2semidirect
5. #5707C2semidirect
6. #5677C2semidirect
7. #5675C2semidirect
8. #5643C2semidirect
9. #5635C2semidirect
10. #5583C2semidirect
11. #5580C2semidirect
12. Q8×D12C2semidirect
13. #5570C2semidirect
14. #5557C2semidirect
15. #5555C2semidirect
16. #5530C2semidirect
17. #5519C2semidirect
86. #556D3semidirect
428. C3#3552semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#57272
D3×C4⋊Q82
#57122
#57072
#56772
#56752
#56432
#56352
#55832
#55802
Q8×D122
#55702
#55572
#55552
#55302
#55192
#35523