Finite Group Databasebeta

← #11233289

Subgroups — #11233289

Order 768.

1922 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5721C212yesmaximal
2C3×#2327C211yesmaximal, characteristic
3#82475C212yesmaximal
4#82460C212yesmaximal
5#76503C214yesmaximal
6#75166C218yesmaximal
7#74995C218yesmaximal
8#74191C212yesmaximal
9#73397C211yesmaximal, characteristic
10#73388C211yesmaximal, characteristic
11#45334—31nomaximal, Sylow
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4×Q8C2212yes—
14C2×Q8×Dic3C2214yes—
15C2×#944C2214yes—
16#5721—216no—
17C2×#890C2214yes—
18C2×#889C2218yes—
19C2×#887C2214yes—
20C2×#869C2211yescharacteristic
21C2×C4×Dic6C2212yes—
22C2×C4×Dic6C2212yes—
23C3×#391C2212yes—
24C3×#387C2218yes—
25C3×#382C2211yescharacteristic
26C3×#376C2211yescharacteristic
27#4984C2218yes—
28#4984C2214yes—
29#4981C2214yes—
30#4981C2214yes—
31#4973C2214yes—
32#4936C2211yescharacteristic
33#4936C2214yes—
34C4×C4⋊Dic3C2212yes—
35C4×C4⋊Dic3C2214yes—
36C4×C4⋊Dic3C2212yes—
37#4929C2212yes—
38#4928C2218yes—
39#4661C2214yes—
40#4661C2218yes—
41#4659C2214yes—
42#4659C2218yes—
43#4656C2214yes—
44#4656C2214yes—
45#4653C2218yes—
46#4653C2218yes—
47#4653C2218yes—
48#4653C2218yes—
49C2×#557—32no—
50#2719—38no—
51#2581—32no—
52#2546—32no—
53#2462—38no—
54#2446—34no—
55#2399—31no—
56#2351—32no—
57#2327—31no—
58#2327D311yescharacteristic
59C22×C3×Q8C2312yes—
60C22×Dic6C2312yes—
61C22×Dic6C2312yes—
62C3×C4⋊Q8D418yes—
63C3×C4×Q8—28no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2314yes—
66C2×C3×C4⋊C4C2314yes—
67C2×C3×C42C2314yes—
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71Q8×Dic3—216no—
72#944—216no—
73C2×C4⋊Dic3C2314yes—
74C2×C4⋊Dic3C2318yes—
75C2×C4⋊Dic3C2314yes—
76C2×C4⋊Dic3C2314yes—
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2318yes—
79C2×C4⋊Dic3C2312yes—
80C2×Dic3⋊C4C2314yes—
81C2×Dic3⋊C4C2314yes—
82C2×Dic3⋊C4C2318yes—
83C2×Dic3⋊C4C2318yes—
84C2×C4×Dic3C2314yes—
85C2×C4×Dic3C2318yes—
86C2×C4×Dic3C2314yes—
87#890—216no—
88#889—232no—
89#887—216no—
90#869—24no—
91C4×Dic6—28no—
92C4×Dic6—28no—
93C3×#95C2314yes—
94C3×#95C2314yes—
95C3×#95C2318yes—
96C3⋊#95C2312yes—
97C3⋊#95C2318yes—
98C3⋊#95C2318yes—
99C3⋊#95C2318yes—
100C3⋊#95C2314yes—
101C3⋊#95C2318yes—
102C3⋊#95C2312yes—
103C3⋊#95C2312yes—
104C3⋊#95C2312yes—
105#557—616no—
106C2×C4⋊Q8—34no—
107C2×C4⋊Q8—31no—
108C2×C4⋊Q8C2×D311yescharacteristic
109C2×#125—38no—
110C2×#125—34no—
111C2×C4×Q8—34no—
112C2×C4×Q8—34no—
113C2×C4×Q8—32no—
114C2×C4×Q8—32no—
115C2×C4×Q8C2×D312yes—
116#400—34no—
117#400—38no—
118#398—34no—
119#398—34no—
120#398—34no—
121#398—38no—
122#395—34no—
123#395—34no—
124#391—34no—
125#391—32no—
126#391C2×D312yes—
127#389—34no—
128#389—38no—
129#389—38no—
130#387—38no—
131#387—38no—
132#387—38no—
133#387—38no—
134#387C2×D318yes—
135#382—31no—
136#382—34no—
137#382C2×D311yescharacteristic
138C4×C4⋊C4—32no—
139C4×C4⋊C4—34no—
140C4×C4⋊C4—32no—
141#376C2×D311yescharacteristic
142C2×C3×Q8—24no—
143C2×C3×Q8—24no—
144C2×C3×Q8C2×D418yes—
145C22×C3×C4C2412yes—
146C22×C3×C4C2414yes—
147C22×C3×C4C2412yes—
148C22×C3×C4C2414yes—
149C22×C3×C4C2412yes—
150C22×C3×C4C2411yescharacteristic
151C22×Dic3C2414yes—
152C22×Dic3C2414yes—
153C22×Dic3C2414yes—
154C22×Dic3C2414yes—
155C2×Dic6—24no—
156C2×Dic6—28no—
157C2×Dic6—24no—
158C2×Dic6—24no—
159C2×Dic6—24no—
160C3×C4⋊C4—28no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4C2×D4116yes—
163C3×C42—28no—
164C3×C42—22no—
165C3×C42C2×D414yes—
166C3×C42#4114yes—
167C4⋊Dic3—24no—
168C4⋊Dic3—24no—
169C4⋊Dic3—216no—
170C4⋊Dic3—28no—
171C4⋊Dic3—28no—
172C4⋊Dic3—28no—
173C4⋊Dic3—216no—
174Dic3⋊C4—28no—
175Dic3⋊C4—28no—
176Dic3⋊C4—216no—
177Dic3⋊C4—216no—
178C4×Dic3—28no—
179C4×Dic3—28no—
180C4×Dic3—216no—
181C22×Q8—32no—
182C22×Q8—32no—
183C22×Q8C22×D312yes—
184C4⋊Q8—616no—
185C4⋊Q8—64no—
186C4⋊Q8D1218yes—
187#125—632no—
188#125—616no—
189C4×Q8—28no—
190C4×Q8—616no—
191C4×Q8—616no—
192C4×Q8—68no—
193C4×Q8—68no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—38no—
196C2×C4⋊C4—34no—
197C2×C4⋊C4—34no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—38no—
200C2×C4⋊C4—38no—
201C2×C4⋊C4—34no—
202C2×C4⋊C4—38no—
203C2×C4⋊C4—34no—
204C2×C4⋊C4—34no—
205C2×C4⋊C4C22×D312yes—
206C2×C4⋊C4C22×D314yes—
207C2×C4⋊C4C22×D314yes—
208C2×C42—38no—
209C2×C42—34no—
210C2×C42—34no—
211C2×C42C22×D314yes—
212C2×C42C22×D311yescharacteristic
213C2×C42C22×D311yescharacteristic
214C2×C42C22×D311yescharacteristic
215#95—32no—
216#95—34no—
217#95—32no—
218#95—32no—
219#95—32no—
220#95—38no—
221#95—38no—
222#95—38no—
223#95—38no—
224#95C22×D314yes—
225#95C22×D314yes—
226#95C22×D318yes—
227C23×C3C2511yescharacteristic
228C3×Q8—216no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—28no—
235C2×C3×C4—24no—
236C2×C3×C4—22no—
237C2×C3×C4C22×D418yes—
238C2×C3×C4C22×D414yes—
239C2×C3×C4C22×D412yes—
240C2×C3×C4C2×#4114yes—
241C2×C3×C4C2×#4112yes—
242C2×C3×C42−1+412yes—
243C2×Dic3—24no—
244C2×Dic3—24no—
245C2×Dic3—24no—
246C2×Dic3—24no—
247C2×Dic3—24no—
248C2×Dic3—24no—
249C2×Dic3—28no—
250C2×Dic3—24no—
251C2×Dic3—24no—
252C2×Dic3—24no—
253C2×Dic32−1+418yes—
254Dic6—216no—
255Dic6—48no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—68no—
259C2×Q8—64no—
260C2×Q8—64no—
261C2×Q8—64no—
262C2×Q8—64no—
263C2×Q8C2×D1218yes—
264C22×C4—34no—
265C22×C4—34no—
266C22×C4—34no—
267C22×C4—34no—
268C22×C4C23×D312yes—
269C22×C4C23×D314yes—
270C22×C4C23×D312yes—
271C22×C4C23×D314yes—
272C22×C4C23×D312yes—
273C22×C4C23×D311yescharacteristic
274C4⋊C4—28no—
275C4⋊C4—24no—
276C4⋊C4—68no—
277C4⋊C4—616no—
278C4⋊C4—68no—
279C4⋊C4—64no—
280C4⋊C4—68no—
281C4⋊C4—68no—
282C4⋊C4—616no—
283C4⋊C4—64no—
284C4⋊C4—616no—
285C4⋊C4—616no—
286C4⋊C4—68no—
287C4⋊C4C2×D12116yes—
288C42—28no—
289C42—22no—
290C42—616no—
291C42—68no—
292C42—68no—
293C42C2×D1214yes—
294C42#23714yes—
295C22×C3C23×D411yescharacteristic
296C22×C3C22×#4111yescharacteristic
297C22×C3C2×2−1+411yescharacteristic
298C22×C3C2×2−1+414yes—
299C3×C4—28no—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—44no—
303C3×C4—42no—
304C3×C4#54714yes—
305Dic3—28no—
306Dic3—44no—
307Dic3—44no—
308Dic3—44no—
309C23C24×D311yescharacteristic
310Q8—216no—
311Q8—616no—
312Q8—128no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—28no—
320C2×C4—24no—
321C2×C4—38no—
322C2×C4—64no—
323C2×C4—64no—
324C2×C4—64no—
325C2×C4—64no—
326C2×C4—64no—
327C2×C4—64no—
328C2×C4—64no—
329C2×C4—68no—
330C2×C4—64no—
331C2×C4—64no—
332C2×C4C22×D1214yes—
333C2×C4C22×D1218yes—
334C2×C4C22×D1212yes—
335C2×C4C2×#23714yes—
336C2×C4C2×#23712yes—
337C2×C4#101012yes—
338C2×C3C2×#54711yescharacteristic
339C2×C3#348612yes—
340C2×C3#350912yes—
341C2×C3#355612yes—
342C22C23×D1211yescharacteristic
343C22C22×#23711yescharacteristic
344C22C2×#101011yescharacteristic
345C22D3×2−1+414yes—
346C4—24no—
347C4—24no—
348C4—28no—
349C4—44no—
350C4—42no—
351C4—68no—
352C4—124no—
353C4—124no—
354C4—124no—
355C4#555514yes—
356C3#4533411yescharacteristic, complemented, Sylow
357C2C2×#555511yescharacteristic
358C2#8588112yes—
359C2#8634312yes—
360C2#8650212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, prime radical, cosocleC23×C3C25
FittingC3×#2327C2
hypercenter#2327D3
hypocenterC3#45334
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2327

Complements

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

NH ≅ G/Ntype
356. C3#45334semidirect

Maximal subgroups

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

subgroupindex
C2×#57212
C3×#23272
#824752
#824602
#765032
#751662
#749952
#741912
#733972
#733882
#453343