Finite Group Databasebeta

← #86993

Subgroups — #86993

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×#556C211yesmaximal, characteristic, complemented
2#5728C211yesmaximal, characteristic, complemented
3#5721C211yesmaximal, characteristic, complemented
4#5708C214yesmaximal, complemented
5#5701C214yesmaximal, complemented
6#5680C212yesmaximal
7#5671C212yesmaximal, complemented
8#5632C212yesmaximal, complemented
9#5625C212yesmaximal, complemented
10#5578C212yesmaximal, complemented
11C4×D3×Q8C211yesmaximal, characteristic, complemented
12#5569C212yesmaximal, complemented
13#5567C211yesmaximal, characteristic, complemented
14#5557C211yesmaximal, characteristic
15D4×Dic6C211yesmaximal, characteristic, complemented
16#5529C212yesmaximal, complemented
17#5516C212yesmaximal, complemented
18#3523—31nomaximal, Sylow
19C2×D3×Q8C2211yescharacteristic
20C22×Dic6C2212yes—
21C3×C4⋊Q8C2211yescharacteristic
22C3×#126C2214yes—
23C3×#124C2212yes—
24C3×C22⋊Q8C2212yes—
25C3×C4×Q8C2211yescharacteristic
26C3×C4×Q8C2212yes—
27C3×C4×D4C2211yescharacteristic
28C3×#117C2212yes—
29#946C2212yes—
30Q8×Dic3C2212yes—
31Q8×Dic3C2212yes—
32Q8×Dic3C2211yescharacteristic
33#944C2212yes—
34#944C2212yes—
35#938C2211yescharacteristic
36#936C2211yescharacteristic
37D4×Dic3C2212yes—
38C4×#67C2212yes—
39#926C2212yes—
40#924C2212yes—
41#897C2212yes—
42#897C2211yescharacteristic
43#896C2214yes—
44#892C2212yes—
45D3×C4⋊C4C2212yes—
46D3×C4⋊C4C2211yescharacteristic
47#890C2212yes—
48#890C2214yes—
49#890C2212yes—
50#890C2211yescharacteristic
51#889C2214yes—
52#889C2214yes—
53#889C2214yes—
54#888C2214yes—
55#888C2212yes—
56#887C2212yes—
57#887C2212yes—
58#887C2212yes—
59#887C2211yescharacteristic
60#887C2212yes—
61#883C2214yes—
62#883C2212yes—
63#67⋊C4C2214yes—
64#879C2214yes—
65#879C2214yes—
66#879C2214yes—
67#879C2212yes—
68#878C2214yes—
69#878C2214yes—
70#877C2214yes—
71#877C2214yes—
72#872C2214yes—
73C42×D3C2212yes—
74C42×D3C2211yescharacteristic
75#870C2212yes—
76#870C2214yes—
77#869C2211yescharacteristic
78C4×Dic6C2212yes—
79C4×Dic6C2212yes—
80C4×Dic6C2214yes—
81C4×Dic6C2214yes—
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2211yescharacteristic
84#557—31no—
85#557—31no—
86#557—32no—
87#556—32no—
88#556D311yescharacteristic, complemented
89#552—34no—
90#552—32no—
91D4×Q8—32no—
92D4×Q8—31no—
93#541—34no—
94#541—32no—
95#533—31no—
96#533—32no—
97#529—32no—
98#529—31no—
99#519—32no—
100C2×C4×Q8—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C2×#67C2312yes—
106C22×Dic3C2314yes—
107D3×Q8—24no—
108C2×C4×D3C2311yescharacteristic
109C2×C4×D3C2312yes—
110C2×C4×D3C2312yes—
111C2×C4×D3C2311yescharacteristic
112C2×C4×D3C2311yescharacteristic
113C2×Dic6—28no—
114C2×Dic6C2311yescharacteristic
115C2×Dic6C2314yes—
116C2×Dic6C2312yes—
117C2×Dic6C2312yes—
118C2×Dic6C2311yescharacteristic
119C3×C4⋊C4C2311yescharacteristic
120C3×C4⋊C4C2312yes—
121C3×C4⋊C4C2312yes—
122C3×C4⋊C4C2311yescharacteristic
123C3×C4⋊C4C2314yes—
124C3×C4⋊C4C2312yes—
125C3×C22⋊C4C2314yes—
126C3×C22⋊C4C2312yes—
127C3×C22⋊C4C2314yes—
128C3×C42C2311yescharacteristic
129C3×C42C2312yes—
130C3×C42C2314yes—
131#217C2312yes—
132#217C2312yes—
133#217C2312yes—
134#217C2314yes—
135(C2×D3)⋊C4C2312yes—
136(C2×D3)⋊C4C2314yes—
137C4⋊Dic3C2312yes—
138C4⋊Dic3C2312yes—
139C4⋊Dic3C2312yes—
140C4⋊Dic3C2311yescharacteristic
141C4⋊Dic3C2312yes—
142C4⋊Dic3C2311yescharacteristic
143C4⋊Dic3C2312yes—
144C4⋊Dic3C2314yes—
145C4⋊Dic3C2314yes—
146Dic3⋊C4C2312yes—
147Dic3⋊C4C2314yes—
148Dic3⋊C4C2312yes—
149Dic3⋊C4C2314yes—
150Dic3⋊C4C2314yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2314yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2312yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2314yes—
158C4×Dic3C2312yes—
159C4×Dic3C2312yes—
160C4×Dic3C2311yescharacteristic
161C4×Dic3C2314yes—
162C4×Dic3C2314yes—
163C4×Dic3C2314yes—
164C4×Dic3C2312yes—
165C4×Dic3C2311yescharacteristic
166C4×Dic3C2312yes—
167C4×Dic3C2311yescharacteristic
168C22×Q8—31no—
169C22×Q8—32no—
170C4⋊Q8—32no—
171C4⋊Q8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—34no—
174C4⋊Q8—31no—
175C4⋊Q8C2×D311yescharacteristic
176#126—34no—
177#126—32no—
178#126—34no—
179#126—34no—
180#126C2×D314yes—
181#125—31no—
182#125—34no—
183#125—32no—
184#125—34no—
185#125—34no—
186#125—34no—
187#125—32no—
188#125—32no—
189#125—34no—
190#124—31no—
191#124C2×D312yes—
192#123—32no—
193#123—34no—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—31no—
197C22⋊Q8—34no—
198C22⋊Q8—32no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201C22⋊Q8C2×D312yes—
202#121—31no—
203C4×Q8—31no—
204C4×Q8—32no—
205C4×Q8—32no—
206C4×Q8—31no—
207C4×Q8—32no—
208C4×Q8—32no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—31no—
212C4×Q8—32no—
213C4×Q8—34no—
214C4×Q8—34no—
215C4×Q8—31no—
216C4×Q8—32no—
217C4×Q8C2×D311yescharacteristic
218C4×Q8C2×D312yes—
219C4×D4—32no—
220C4×D4—32no—
221C4×D4—34no—
222C4×D4C2×D311yescharacteristic
223#117—32no—
224#117—32no—
225#117—34no—
226#117—34no—
227#117—34no—
228#117C2×D312yes—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C42—32no—
232C2×C42—31no—
233C23×C3C2412yes—
234C22×D3C2411yescharacteristic
235C3×Q8—22no—
236C3×Q8—24no—
237C3×D4—22no—
238C2×C3×C4—24no—
239C2×C3×C4C2412yes—
240C2×C3×C4C2414yes—
241C2×C3×C4C2411yescharacteristic
242C2×C3×C4C2412yes—
243C2×C3×C4C2412yes—
244C2×C3×C4C2411yescharacteristic
245C2×C3×C4C2411yescharacteristic
246#67—24no—
247C2×Dic3—28no—
248C2×Dic3C2411yescharacteristic
249C2×Dic3C2411yescharacteristic
250C2×Dic3C2412yes—
251C2×Dic3C2412yes—
252C2×Dic3C2412yes—
253C2×Dic3C2414yes—
254C2×Dic3C2411yescharacteristic
255C2×Dic3C2412yes—
256C4×D3—24no—
257C4×D3—22no—
258C4×D3—22no—
259C4×D3—22no—
260C4×D3—24no—
261Dic6—24no—
262Dic6—22no—
263Dic6—28no—
264Dic6—24no—
265Dic6#4114yes—
266C2×Q8—31no—
267C2×Q8—31no—
268C2×Q8—32no—
269C2×Q8—34no—
270C2×Q8—32no—
271C2×Q8—68no—
272C2×Q8—64no—
273C2×Q8C22×D312yes—
274C2×Q8C22×D311yescharacteristic
275C2×D4—32no—
276C2×D4C22×D311yescharacteristic
277C22×C4—34no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4—32no—
281C22×C4—32no—
282C22×C4—31no—
283C22×C4C22×D312yes—
284C4⋊C4—32no—
285C4⋊C4—32no—
286C4⋊C4—31no—
287C4⋊C4—32no—
288C4⋊C4—32no—
289C4⋊C4—34no—
290C4⋊C4—34no—
291C4⋊C4—32no—
292C4⋊C4—34no—
293C4⋊C4—32no—
294C4⋊C4—34no—
295C4⋊C4—32no—
296C4⋊C4—34no—
297C4⋊C4—32no—
298C4⋊C4—34no—
299C4⋊C4—32no—
300C4⋊C4—31no—
301C4⋊C4—34no—
302C4⋊C4—34no—
303C4⋊C4—32no—
304C4⋊C4—34no—
305C4⋊C4C22×D311yescharacteristic
306C4⋊C4C22×D312yes—
307C4⋊C4C22×D312yes—
308C4⋊C4C22×D312yes—
309C4⋊C4C22×D311yescharacteristic
310C4⋊C4C22×D314yes—
311C22⋊C4—32no—
312C22⋊C4—32no—
313C22⋊C4—34no—
314C22⋊C4—32no—
315C22⋊C4—32no—
316C22⋊C4—34no—
317C22⋊C4C22×D314yes—
318C22⋊C4C22×D314yes—
319C22⋊C4C22×D312yes—
320C42—32no—
321C42—31no—
322C42—32no—
323C42—31no—
324C42—32no—
325C42—31no—
326C42—34no—
327C42—34no—
328C42—34no—
329C42—32no—
330C42C22×D311yescharacteristic
331C42C22×D312yes—
332C42C22×D314yes—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3C2511yescharacteristic
337C2×D3—21no—
338C2×D3—21no—
339C2×D32−1+412yes—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—21no—
344C3×C4—24no—
345C3×C4C2×#4112yes—
346C3×C42−1+412yes—
347Dic3—21no—
348Dic3—24no—
349Dic3—22no—
350Dic3—21no—
351Dic3—21no—
352Dic3—22no—
353Dic3—22no—
354Dic3C2×#4114yes—
355C23—31no—
356C23C23×D312yes—
357Q8—22no—
358Q8—24no—
359Q8—34no—
360Q8—68no—
361Q8—64no—
362Q8—64no—
363Q8—62no—
364D4—22no—
365D4—64no—
366C2×C4—24no—
367C2×C4—32no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—31no—
374C2×C4—34no—
375C2×C4—64no—
376C2×C4—64no—
377C2×C4—62no—
378C2×C4—68no—
379C2×C4—62no—
380C2×C4—62no—
381C2×C4C23×D312yes—
382C2×C4C23×D314yes—
383C2×C4C23×D311yescharacteristic
384C2×C4C23×D312yes—
385C2×C4C23×D312yes—
386C2×C4C23×D311yescharacteristic
387C2×C4C23×D311yescharacteristic
388C2×C3—42no—
389C2×C3C22×#4111yescharacteristic
390C2×C3C2×2−1+411yescharacteristic
391C2×C3C2×2−1+411yescharacteristic
392D3—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—32no—
397C22—61no—
398C22—61no—
399C22C24×D311yescharacteristic
400C4—22no—
401C4—22no—
402C4—24no—
403C4—22no—
404C4—21no—
405C4—34no—
406C4—64no—
407C4—61no—
408C4—61no—
409C4—62no—
410C4—61no—
411C4—62no—
412C4—62no—
413C4D3×#4112yes—
414C4#101012yes—
415C3#352311yescharacteristic, complemented, Sylow
416C2—42no—
417C2—62no—
418C2C2×D3×#4111yescharacteristic
419C2C2×#101011yescharacteristic
420C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
prime radical#938C22
hypercenter#556D3
hypocenterC3#3523
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. #5728C2semidirect
3. #5721C2semidirect
4. #5708C2semidirect
5. #5701C2semidirect
7. #5671C2semidirect
8. #5632C2semidirect
9. #5625C2semidirect
10. #5578C2semidirect
11. C4×D3×Q8C2semidirect
12. #5569C2semidirect
13. #5567C2semidirect
15. D4×Dic6C2semidirect
16. #5529C2semidirect
17. #5516C2semidirect
88. #556D3semidirect
415. C3#3523semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#57282
#57212
#57082
#57012
#56802
#56712
#56322
#56252
#55782
C4×D3×Q82
#55692
#55672
#55572
D4×Dic62
#55292
#55162
#35233