Finite Group Databasebeta

← #87011

Subgroups — #87011

Order 384.

910 conjugacy classes of subgroups.

The subgroup lattice has 414 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#5712C214yesmaximal, complemented
5#5701C214yesmaximal, complemented
6#5682C212yesmaximal
7#5671C212yesmaximal, complemented
8#5642C212yesmaximal, complemented
9#5625C212yesmaximal, complemented
10#5583C212yesmaximal, complemented
11#5578C211yesmaximal, characteristic, complemented
12#5570C212yesmaximal, complemented
13Q8×Dic6C211yesmaximal, characteristic, complemented
14#5557C211yesmaximal, characteristic
15#5543C211yesmaximal, characteristic, complemented
16#5533C212yesmaximal, complemented
17#5519C212yesmaximal, complemented
18#3555—31nomaximal, Sylow
19C3×C4⋊Q8C2211yescharacteristic
20C3×#126C2214yes—
21C3×#124C2212yes—
22C3×C22⋊Q8C2212yes—
23C3×C4×Q8C2211yescharacteristic
24C3×C4×Q8C2212yes—
25C3×C4×D4C2211yescharacteristic
26C3×#117C2212yes—
27#946C2211yescharacteristic
28#946C2212yes—
29Q8×Dic3C2212yes—
30Q8×Dic3C2212yes—
31Q8×Dic3C2212yes—
32#944C2212yes—
33#944C2211yescharacteristic
34#938C2211yescharacteristic
35#936C2211yescharacteristic
36D4×Dic3C2212yes—
37C4×#67C2212yes—
38#926C2212yes—
39C2×C4⋊Dic3C2212yes—
40#924C2212yes—
41#898C2214yes—
42#897C2212yes—
43#897C2211yescharacteristic
44#897C2211yescharacteristic
45#896C2212yes—
46#892C2212yes—
47#890C2214yes—
48#890C2212yes—
49#890C2214yes—
50#890C2212yes—
51#890C2212yes—
52#890C2211yescharacteristic
53#889C2212yes—
54#889C2214yes—
55#889C2214yes—
56#888C2214yes—
57#888C2211yescharacteristic
58#888C2212yes—
59#887C2212yes—
60#887C2212yes—
61#886C2214yes—
62#885C2214yes—
63#883C2212yes—
64#879C2214yes—
65#879C2214yes—
66#879C2214yes—
67#879C2212yes—
68#878C2214yes—
69#878C2214yes—
70#878C2214yes—
71#877C2214yes—
72#876C2214yes—
73#872C2212yes—
74C42×D3C2211yescharacteristic
75#870C2214yes—
76#869C2212yes—
77#869C2214yes—
78#869C2211yescharacteristic
79C4×Dic6C2212yes—
80C4×Dic6C2212yes—
81C4×Dic6C2214yes—
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2211yescharacteristic
84#563—34no—
85Q82—31no—
86#557—31no—
87#557—32no—
88#556—32no—
89#556—32no—
90#556D311yescharacteristic, complemented
91#554—31no—
92#552—32no—
93#544—32no—
94#542—32no—
95#541—34no—
96#541—32no—
97#541—32no—
98#533—31no—
99#533—31no—
100#529—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×Q8C2312yes—
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2312yes—
105C2×#67C2312yes—
106C22×Dic3C2314yes—
107C2×C4×D3C2311yescharacteristic
108C2×C4×D3C2311yescharacteristic
109C2×C4×D3C2311yescharacteristic
110C2×Dic6C2311yescharacteristic
111C2×Dic6C2311yescharacteristic
112C2×Dic6C2312yes—
113C2×Dic6C2314yes—
114C2×Dic6C2312yes—
115C3×C4⋊C4C2312yes—
116C3×C4⋊C4C2312yes—
117C3×C4⋊C4C2311yescharacteristic
118C3×C4⋊C4C2312yes—
119C3×C4⋊C4C2314yes—
120C3×C4⋊C4C2311yescharacteristic
121C3×C22⋊C4C2314yes—
122C3×C22⋊C4C2312yes—
123C3×C22⋊C4C2314yes—
124C3×C42C2311yescharacteristic
125C3×C42C2312yes—
126C3×C42C2314yes—
127#217C2312yes—
128#217C2312yes—
129#217C2312yes—
130#217C2314yes—
131(C2×D3)⋊C4C2312yes—
132(C2×D3)⋊C4C2312yes—
133(C2×D3)⋊C4C2312yes—
134(C2×D3)⋊C4C2314yes—
135C4⋊Dic3C2312yes—
136C4⋊Dic3C2312yes—
137C4⋊Dic3C2311yescharacteristic
138C4⋊Dic3C2311yescharacteristic
139C4⋊Dic3C2312yes—
140C4⋊Dic3C2314yes—
141C4⋊Dic3C2314yes—
142C4⋊Dic3C2314yes—
143C4⋊Dic3C2314yes—
144C4⋊Dic3C2312yes—
145C4⋊Dic3C2312yes—
146Dic3⋊C4C2312yes—
147Dic3⋊C4C2314yes—
148Dic3⋊C4C2314yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2314yes—
158C4×Dic3C2311yescharacteristic
159C4×Dic3C2312yes—
160C4×Dic3C2314yes—
161C4×Dic3C2314yes—
162C4×Dic3C2314yes—
163C4×Dic3C2312yes—
164C4×Dic3C2311yescharacteristic
165C4×Dic3C2311yescharacteristic
166C4⋊Q8—31no—
167C4⋊Q8—32no—
168C4⋊Q8—31no—
169C4⋊Q8—34no—
170C4⋊Q8—32no—
171C4⋊Q8—31no—
172C4⋊Q8—32no—
173C4⋊Q8—34no—
174C4⋊Q8C2×D311yescharacteristic
175#126—34no—
176#126—32no—
177#126—34no—
178#126—34no—
179#126—34no—
180#126—34no—
181#126C2×D314yes—
182#125—31no—
183#125—32no—
184#125—32no—
185#125—34no—
186#125—34no—
187#125—32no—
188#125—34no—
189#125—34no—
190#125—32no—
191#125—34no—
192#124—31no—
193#124—34no—
194#124C2×D312yes—
195#123—32no—
196#123—34no—
197C22⋊Q8—32no—
198C22⋊Q8—31no—
199C22⋊Q8—32no—
200C22⋊Q8—31no—
201C22⋊Q8—31no—
202C22⋊Q8—32no—
203C22⋊Q8—32no—
204C22⋊Q8—34no—
205C22⋊Q8—34no—
206C22⋊Q8—34no—
207C22⋊Q8C2×D312yes—
208#121—31no—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—32no—
212C4×Q8—32no—
213C4×Q8—32no—
214C4×Q8—31no—
215C4×Q8—31no—
216C4×Q8—32no—
217C4×Q8—34no—
218C4×Q8—32no—
219C4×Q8C2×D311yescharacteristic
220C4×Q8C2×D312yes—
221C4×D4—32no—
222C4×D4—32no—
223C4×D4C2×D311yescharacteristic
224#117—32no—
225#117—32no—
226#117—34no—
227#117—32no—
228#117C2×D312yes—
229C2×C4⋊C4—32no—
230C2×C42—31no—
231C23×C3C2412yes—
232C22×D3C2411yescharacteristic
233C3×Q8—22no—
234C3×Q8—24no—
235C3×D4—22no—
236C2×C3×C4—24no—
237C2×C3×C4C2414yes—
238C2×C3×C4C2412yes—
239C2×C3×C4C2411yescharacteristic
240C2×C3×C4C2412yes—
241C2×C3×C4C2412yes—
242C2×C3×C4C2411yescharacteristic
243C2×C3×C4C2411yescharacteristic
244#67—42no—
245C2×Dic3—44no—
246C2×Dic3C2411yescharacteristic
247C2×Dic3C2412yes—
248C2×Dic3C2412yes—
249C2×Dic3C2412yes—
250C2×Dic3C2411yescharacteristic
251C2×Dic3C2414yes—
252C2×Dic3C2411yescharacteristic
253C2×Dic3C2412yes—
254C4×D3—22no—
255C4×D3—22no—
256C4×D3—41no—
257Dic6—22no—
258Dic6—22no—
259Dic6—42no—
260Dic6—42no—
261Dic6—44no—
262C2×Q8—31no—
263C2×Q8—31no—
264C2×Q8—32no—
265C2×Q8—32no—
266C2×Q8—34no—
267C2×Q8C22×D311yescharacteristic
268C2×Q8C22×D312yes—
269C2×D4—32no—
270C2×D4C22×D311yescharacteristic
271C22×C4—34no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4C22×D312yes—
276C4⋊C4—31no—
277C4⋊C4—32no—
278C4⋊C4—34no—
279C4⋊C4—32no—
280C4⋊C4—32no—
281C4⋊C4—32no—
282C4⋊C4—34no—
283C4⋊C4—32no—
284C4⋊C4—32no—
285C4⋊C4—32no—
286C4⋊C4—32no—
287C4⋊C4—31no—
288C4⋊C4—32no—
289C4⋊C4—32no—
290C4⋊C4—34no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4—34no—
294C4⋊C4—32no—
295C4⋊C4—34no—
296C4⋊C4—32no—
297C4⋊C4—34no—
298C4⋊C4—34no—
299C4⋊C4C22×D312yes—
300C4⋊C4C22×D311yescharacteristic
301C4⋊C4C22×D311yescharacteristic
302C4⋊C4C22×D312yes—
303C4⋊C4C22×D312yes—
304C4⋊C4C22×D314yes—
305C22⋊C4—32no—
306C22⋊C4—32no—
307C22⋊C4—32no—
308C22⋊C4—34no—
309C22⋊C4—32no—
310C22⋊C4—34no—
311C22⋊C4—32no—
312C22⋊C4—32no—
313C22⋊C4C22×D314yes—
314C22⋊C4C22×D314yes—
315C22⋊C4C22×D312yes—
316C42—31no—
317C42—31no—
318C42—31no—
319C42—34no—
320C42—32no—
321C42—32no—
322C42—34no—
323C42—34no—
324C42C22×D311yescharacteristic
325C42C22×D312yes—
326C42C22×D314yes—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3C2511yescharacteristic
331C2×D3—21no—
332C2×D3—21no—
333C2×D3—21no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—24no—
337C3×C4—22no—
338C3×C4—21no—
339C3×C42−1+412yes—
340C3×C42−1+412yes—
341Dic3—22no—
342Dic3—22no—
343Dic3—21no—
344Dic3—21no—
345Dic3—21no—
346Dic3—24no—
347Dic3—22no—
348Dic3—22no—
349C23—31no—
350C23C23×D312yes—
351Q8—22no—
352Q8—24no—
353Q8—62no—
354Q8—62no—
355Q8—122no—
356Q8—122no—
357Q8—124no—
358D4—22no—
359D4—122no—
360C2×C4—24no—
361C2×C4—31no—
362C2×C4—32no—
363C2×C4—32no—
364C2×C4—32no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—34no—
368C2×C4—32no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—121no—
372C2×C4—124no—
373C2×C4C23×D314yes—
374C2×C4C23×D312yes—
375C2×C4C23×D311yescharacteristic
376C2×C4C23×D312yes—
377C2×C4C23×D312yes—
378C2×C4C23×D311yescharacteristic
379C2×C4C23×D311yescharacteristic
380C2×C3—42no—
381C2×C3C2×2−1+411yescharacteristic
382C2×C3C2×2−1+411yescharacteristic
383C2×C3#58511yescharacteristic
384D3—41no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—61no—
389C22—61no—
390C22—61no—
391C22C24×D311yescharacteristic
392C4—22no—
393C4—22no—
394C4—24no—
395C4—21no—
396C4—22no—
397C4—61no—
398C4—61no—
399C4—62no—
400C4—62no—
401C4—62no—
402C4—61no—
403C4—64no—
404C4—62no—
405C4#101012yes—
406C4#101012yes—
407C3#355511yescharacteristic, complemented, Sylow
408C2—42no—
409C2—121no—
410C2C2×#101011yescharacteristic
411C2C2×#101011yescharacteristic
412C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#556C2
prime radical#938C22
hypercenter#556D3
hypocenterC3#3555
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. #5712C2semidirect
5. #5701C2semidirect
7. #5671C2semidirect
8. #5642C2semidirect
9. #5625C2semidirect
10. #5583C2semidirect
11. #5578C2semidirect
12. #5570C2semidirect
13. Q8×Dic6C2semidirect
15. #5543C2semidirect
16. #5533C2semidirect
17. #5519C2semidirect
90. #556D3semidirect
407. C3#3555semidirect

Maximal subgroups

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

subgroupindex
C3×#5562
#57282
#57212
#57122
#57012
#56822
#56712
#56422
#56252
#55832
#55782
#55702
Q8×Dic62
#55572
#55432
#55332
#55192
#35553