Finite Group Databasebeta

← #11233291

Subgroups — #11233291

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×#5570C212yesmaximal
2C3×#2399C211yesmaximal, characteristic
3#82480C212yesmaximal
4#77860C218yesmaximal
5#77178C218yesmaximal
6#73711C212yesmaximal
7#73672C214yesmaximal
8#73637C212yesmaximal
9#73388C211yesmaximal, characteristic
10#73388C211yesmaximal, characteristic
11#45334—31nomaximal, Sylow
12C2×C3×C4×Q8C2212yes—
13C2×Q8×Dic3C2214yes—
14#5570—216no—
15C2×#890C2214yes—
16C2×#890C2218yes—
17C2×#869C2214yes—
18C2×#869C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C4×Dic6C2212yes—
21C2×C4×Dic6C2214yes—
22C2×C4×Dic6C2212yes—
23C3×#398C2214yes—
24C3×#395C2214yes—
25C3×C4×C4⋊C4C2212yes—
26C3×C4×C4⋊C4C2212yes—
27C3×#376C2211yescharacteristic
28#5230C2214yes—
29#4985C2214yes—
30#4984C2218yes—
31#4982C2218yes—
32#4980C2218yes—
33#4978C2218yes—
34#4976C2218yes—
35#4936C2211yescharacteristic
36#4936C2211yescharacteristic
37#4929C2214yes—
38#4929C2212yes—
39#4929C2212yes—
40#4928C2214yes—
41#4661C2214yes—
42#4661C2214yes—
43#4660C2218yes—
44#4655C2218yes—
45#4653C2218yes—
46#4653C2218yes—
47#4650C2214yes—
48#4649C2214yes—
49C2×#557—32no—
50#2719—38no—
51#2581—32no—
52#2546—32no—
53#2462—38no—
54#2446—34no—
55#2399D311yescharacteristic
56#2351—32no—
57#2327—31no—
58#2327—31no—
59C22×C3×Q8C2312yes—
60C22×Dic6C2312yes—
61C22×Dic6C2312yes—
62C3×C4×Q8—28no—
63C2×C3×C4⋊C4C2314yes—
64C2×C3×C4⋊C4C2314yes—
65C2×C3×C4⋊C4C2314yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C42C2314yes—
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71Q8×Dic3—216no—
72C2×C4⋊Dic3C2314yes—
73C2×C4⋊Dic3C2318yes—
74C2×C4⋊Dic3C2318yes—
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2314yes—
78C2×Dic3⋊C4C2318yes—
79C2×Dic3⋊C4C2318yes—
80C2×Dic3⋊C4C2314yes—
81C2×Dic3⋊C4C2314yes—
82C2×C4×Dic3C2314yes—
83C2×C4×Dic3C2318yes—
84C2×C4×Dic3C2314yes—
85#890—216no—
86#890—232no—
87#869—216no—
88#869—24no—
89#869D418yes—
90C4×Dic6—216no—
91C4×Dic6—28no—
92C4×Dic6—28no—
93C3×#95C2312yes—
94C3×#95C2312yes—
95C3×#95C2314yes—
96C3×#95C2312yes—
97C3×#95C2312yes—
98C3⋊#95C2314yes—
99C3⋊#95C2318yes—
100C3⋊#95C2318yes—
101C3⋊#95C2318yes—
102C3⋊#95C2314yes—
103C3⋊#95C2318yes—
104C3⋊#95C2318yes—
105#557—616no—
106C2×C4⋊Q8—31no—
107C2×C4⋊Q8—31no—
108C2×C4⋊Q8—34no—
109C2×#125—38no—
110C2×#125—34no—
111C2×C4×Q8—34no—
112C2×C4×Q8—32no—
113C2×C4×Q8—32no—
114C2×C4×Q8—34no—
115C2×C4×Q8C2×D312yes—
116#400—34no—
117#400—38no—
118#398—38no—
119#398—34no—
120#398—34no—
121#398C2×D314yes—
122#395—34no—
123#395C2×D314yes—
124#391—34no—
125#391—32no—
126#391—32no—
127#389—38no—
128#389—38no—
129#389—34no—
130#387—38no—
131#387—38no—
132#387—38no—
133#387—38no—
134#387—38no—
135#382—31no—
136#382—31no—
137#382—34no—
138C4×C4⋊C4—34no—
139C4×C4⋊C4C2×D312yes—
140C4×C4⋊C4C2×D312yes—
141#376C2×D311yescharacteristic
142C2×C3×Q8—24no—
143C2×C3×Q8—28no—
144C22×C3×C4C2412yes—
145C22×C3×C4C2414yes—
146C22×C3×C4C2412yes—
147C22×C3×C4C2412yes—
148C22×C3×C4C2414yes—
149C22×C3×C4C2411yescharacteristic
150C22×Dic3C2414yes—
151C22×Dic3C2414yes—
152C22×Dic3C2414yes—
153C22×Dic3C2414yes—
154C2×Dic6—24no—
155C2×Dic6—24no—
156C2×Dic6—24no—
157C2×Dic6—24no—
158C2×Dic6—24no—
159C2×Dic6C2×D418yes—
160C3×C4⋊C4—28no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—28no—
163C3×C4⋊C4—28no—
164C3×C42—28no—
165C3×C42—22no—
166C3×C42C2×D414yes—
167C3×C42#4114yes—
168C4⋊Dic3—24no—
169C4⋊Dic3—24no—
170C4⋊Dic3—28no—
171C4⋊Dic3—216no—
172C4⋊Dic3—216no—
173C4⋊Dic3C2×D4116yes—
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—38no—
185C4⋊Q8—616no—
186C4⋊Q8—64no—
187#125—632no—
188#125—616no—
189C4×Q8—28no—
190C4×Q8—616no—
191C4×Q8—616no—
192C4×Q8—68no—
193C4×Q8—68no—
194C2×C4⋊C4—34no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—38no—
197C2×C4⋊C4—34no—
198C2×C4⋊C4—38no—
199C2×C4⋊C4—32no—
200C2×C4⋊C4—34no—
201C2×C4⋊C4—38no—
202C2×C4⋊C4—34no—
203C2×C4⋊C4—38no—
204C2×C4⋊C4C22×D314yes—
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—34no—
216#95—34no—
217#95—38no—
218#95—38no—
219#95—38no—
220#95—38no—
221#95—38no—
222#95C22×D314yes—
223#95C22×D312yes—
224#95C22×D312yes—
225#95C22×D312yes—
226#95C22×D312yes—
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—22no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4C22×D414yes—
239C2×C3×C4C22×D412yes—
240C2×C3×C4C2×#4114yes—
241C2×C3×C4C2×#4112yes—
242C2×C3×C42−1+412yes—
243C2×C3×C42−1+418yes—
244C2×Dic3—24no—
245C2×Dic3—24no—
246C2×Dic3—24no—
247C2×Dic3—24no—
248C2×Dic3—24no—
249C2×Dic3—24no—
250C2×Dic3—28no—
251C2×Dic3—28no—
252C2×Dic3—24no—
253C2×Dic3C22×D418yes—
254Dic6—216no—
255Dic6—48no—
256C2×Q8—24no—
257C2×Q8—28no—
258C2×Q8—38no—
259C2×Q8—64no—
260C2×Q8—64no—
261C2×Q8—64no—
262C2×Q8—64no—
263C2×Q8—64no—
264C22×C4—34no—
265C22×C4—34no—
266C22×C4—34no—
267C22×C4—34no—
268C22×C4C23×D312yes—
269C22×C4C23×D314yes—
270C22×C4C23×D312yes—
271C22×C4C23×D312yes—
272C22×C4C23×D314yes—
273C22×C4C23×D311yescharacteristic
274C4⋊C4—28no—
275C4⋊C4—24no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—316no—
279C4⋊C4—616no—
280C4⋊C4—616no—
281C4⋊C4—616no—
282C4⋊C4—64no—
283C4⋊C4—68no—
284C4⋊C4—68no—
285C4⋊C4—64no—
286C4⋊C4—68no—
287C4⋊C4—616no—
288C42—28no—
289C42—22no—
290C42—616no—
291C42—68no—
292C42—68no—
293C42D3×D414yes—
294C42#23914yes—
295C22×C3C23×D411yescharacteristic
296C22×C3C22×#4111yescharacteristic
297C22×C3C2×2−1+411yescharacteristic
298C22×C3C2×2−1+414yes—
299C3×C4—24no—
300C3×C4—28no—
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—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—38no—
323C2×C4—64no—
324C2×C4—64no—
325C2×C4—64no—
326C2×C4—64no—
327C2×C4—64no—
328C2×C4—64no—
329C2×C4—64no—
330C2×C4—68no—
331C2×C4—68no—
332C2×C4C2×D3×D414yes—
333C2×C4C2×D3×D412yes—
334C2×C4C2×#23914yes—
335C2×C4C2×#23912yes—
336C2×C4#100712yes—
337C2×C4#101018yes—
338C2×C3C2×#54711yescharacteristic
339C2×C3#348612yes—
340C2×C3#350912yes—
341C2×C3#355612yes—
342C22C22×D3×D411yescharacteristic
343C22C22×#23911yescharacteristic
344C22C2×#100711yescharacteristic
345C22C2×#101014yes—
346C4—24no—
347C4—28no—
348C4—24no—
349C4—44no—
350C4—42no—
351C4—68no—
352C4—124no—
353C4—124no—
354C4—124no—
355C4#572614yes—
356C3#4533411yescharacteristic, complemented, Sylow
357C2#8602212yes—
358C2C2×#572611yescharacteristic
359C2#8626212yes—
360C2#8627012yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, prime radical, cosocleC23×C3C25
FittingC3×#2399C2
hypercenter#2399D3
hypocenterC3#45334
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2399

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×#55702
C3×#23992
#824802
#778602
#771782
#737112
#736722
#736372
#733882
#453343