Finite Group Databasebeta

← #11235186

Subgroups — #11235186

Order 768.

1658 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2399C211yesmaximal, characteristic
2#82478C212yesmaximal
3#82475C212yesmaximal
4#77912C214yesmaximal
5#77872C214yesmaximal
6#77194C214yesmaximal
7#77186C214yesmaximal
8#73715C214yesmaximal
9#73659C214yesmaximal
10#73390C212yesmaximal
11#45435—31nomaximal, Sylow
12C2×C3×C4×Q8C2211yescharacteristic
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#944C2214yes—
15C2×#888C2218yes—
16C2×#870C2212yes—
17C2×#869C2212yes—
18C2×#869C2212yes—
19C3×#398C2212yes—
20C3×#398C2212yes—
21C3×#395C2212yes—
22C3×#395C2212yes—
23C3×C4×C4⋊C4C2214yes—
24C3×#376C2211yescharacteristic
25#5229C2214yes—
26#4984C2218yes—
27#4981C2214yes—
28#4981C2214yes—
29#4979C2214yes—
30#4979C2214yes—
31#4978C2218yes—
32#4976C2218yes—
33#4975C2214yes—
34#4973C2214yes—
35#4935C2212yes—
36C4×C4⋊Dic3C2212yes—
37C4×C4⋊Dic3C2212yes—
38C4×C4⋊Dic3C2214yes—
39#4928C2214yes—
40#4928C2214yes—
41#4928C2214yes—
42#4656C2218yes—
43#4656C2218yes—
44#4655C2218yes—
45#4652C2218yes—
46#4649C2218yes—
47#4647C2218yes—
48#2792—34no—
49#2734—34no—
50#2586—34no—
51#2567—32no—
52#2551—34no—
53#2551—34no—
54#2546—32no—
55#2399D311yescharacteristic
56#2370—34no—
57#2330—32no—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×Dic6C2314yes—
61C22×Dic6C2314yes—
62C3×C4×Q8—42no—
63C3×C4×Q8—42no—
64C2×C3×C4⋊C4C2314yes—
65C2×C3×C4⋊C4C2314yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×C3×C42C2312yes—
72C2×C3×C42C2312yes—
73C2×C3×C42C2312yes—
74#944—48no—
75C2×C4⋊Dic3C2314yes—
76C2×C4⋊Dic3C2318yes—
77C2×C4⋊Dic3C2314yes—
78C2×C4⋊Dic3C2314yes—
79C2×C4⋊Dic3C2314yes—
80C2×Dic3⋊C4C2318yes—
81C2×Dic3⋊C4C2318yes—
82C2×Dic3⋊C4C2318yes—
83C2×Dic3⋊C4C2318yes—
84C2×C4×Dic3C2314yes—
85C2×C4×Dic3C2318yes—
86C2×C4×Dic3C2314yes—
87#888—416no—
88#870Q8116yescomplemented
89#869—44no—
90#869—44no—
91C3×#95C2312yes—
92C3×#95C2314yes—
93C3×#95C2314yes—
94C3×#95C2312yes—
95C3⋊#95C2318yes—
96C3⋊#95C2318yes—
97C3⋊#95C2314yes—
98C3⋊#95C2314yes—
99C3⋊#95C2318yes—
100C3⋊#95C2318yes—
101C2×C4⋊Q8—34no—
102C2×C4⋊Q8—38no—
103C2×C4⋊Q8—32no—
104C2×C4⋊Q8—32no—
105C2×#125—32no—
106C2×C4×Q8C2×D311yescharacteristic
107C2×C4×Q8C2×D311yescharacteristic
108#398—34no—
109#398—34no—
110#398C2×D312yes—
111#398C2×D312yes—
112#395—34no—
113#395—34no—
114#395—38no—
115#395—38no—
116#395C2×D312yes—
117#395C2×D312yes—
118#391—34no—
119#391—34no—
120#389—38no—
121#389—38no—
122#389—34no—
123#389—34no—
124#389—34no—
125#389—38no—
126#387—38no—
127#387—38no—
128#387—38no—
129#384—34no—
130#384—32no—
131C4×C4⋊C4—32no—
132C4×C4⋊C4—34no—
133C4×C4⋊C4—32no—
134C4×C4⋊C4—38no—
135C4×C4⋊C4C2×D314yes—
136#376C2×D311yescharacteristic
137C2×C3×Q8—24no—
138C2×C3×Q8—24no—
139C2×C3×Q8—41no—
140C2×C3×Q8—41no—
141C22×C3×C4C2411yescharacteristic
142C22×C3×C4C2412yes—
143C22×C3×C4C2412yes—
144C22×C3×C4C2411yescharacteristic
145C22×C3×C4C2414yes—
146C22×C3×C4C2412yes—
147C22×C3×C4C2412yes—
148C22×C3×C4C2411yescharacteristic
149C22×Dic3C2418yes—
150C22×Dic3C2418yes—
151C2×Dic6—28no—
152C2×Dic6—44no—
153C2×Dic6—44no—
154C2×Dic6—48no—
155C2×Dic6—44no—
156C3×C4⋊C4—28no—
157C3×C4⋊C4—28no—
158C3×C4⋊C4—42no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—41no—
162C3×C42—24no—
163C3×C42—24no—
164C3×C42—22no—
165C3×C42C2×Q818yes—
166C4⋊Dic3—28no—
167C4⋊Dic3—44no—
168C4⋊Dic3—44no—
169C4⋊Dic3—48no—
170C4⋊Dic3C2×Q8116yes—
171Dic3⋊C4—216no—
172Dic3⋊C4—48no—
173Dic3⋊C4—48no—
174Dic3⋊C4C2×Q8132yes—
175C4×Dic3—28no—
176C4×Dic3—48no—
177C4×Dic3—44no—
178C22×Q8—34no—
179C22×Q8—34no—
180C22×Q8C22×D311yescharacteristic
181C22×Q8C22×D311yescharacteristic
182C4⋊Q8—124no—
183C4⋊Q8—124no—
184C4⋊Q8—1216no—
185C4⋊Q8—128no—
186#125—316no—
187C4×Q8—42no—
188C4×Q8—42no—
189C2×C4⋊C4—34no—
190C2×C4⋊C4—38no—
191C2×C4⋊C4—34no—
192C2×C4⋊C4—38no—
193C2×C4⋊C4—38no—
194C2×C4⋊C4—38no—
195C2×C4⋊C4—38no—
196C2×C4⋊C4—34no—
197C2×C4⋊C4—34no—
198C2×C4⋊C4C22×D314yes—
199C2×C4⋊C4C22×D314yes—
200C2×C4⋊C4C22×D312yes—
201C2×C4⋊C4C22×D312yes—
202C2×C4⋊C4C22×D311yescharacteristic
203C2×C4⋊C4C22×D311yescharacteristic
204C2×C42—34no—
205C2×C42—34no—
206C2×C42—38no—
207C2×C42C22×D311yescharacteristic
208C2×C42C22×D312yes—
209C2×C42C22×D312yes—
210C2×C42C22×D312yes—
211#95—34no—
212#95—34no—
213#95—38no—
214#95—38no—
215#95—38no—
216#95—38no—
217#95C22×D312yes—
218#95C22×D312yes—
219#95C22×D314yes—
220#95C22×D314yes—
221C23×C3C2511yescharacteristic
222C3×Q8—44no—
223C3×Q8—44no—
224C2×C3×C4—24no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—24no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4C22×Q818yes—
239C2×C3×C4C22×Q814yes—
240C2×C3×C4H(5,𝔽2)12yes—
241C2×C3×C4H(5,𝔽2)14yes—
242C2×C3×C42−1+414yes—
243C2×Dic3—28no—
244C2×Dic3—28no—
245C2×Dic3—28no—
246C2×Dic3—28no—
247C2×Dic3—28no—
248C2×Dic3C22×Q8116yes—
249Dic6—88no—
250Dic6—88no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—68no—
256C2×Q8—124no—
257C2×Q8—124no—
258C2×Q8—124no—
259C2×Q8—128no—
260C22×C4—38no—
261C22×C4—38no—
262C22×C4C23×D311yescharacteristic
263C22×C4C23×D312yes—
264C22×C4C23×D312yes—
265C22×C4C23×D311yescharacteristic
266C22×C4C23×D314yes—
267C22×C4C23×D312yes—
268C22×C4C23×D312yes—
269C22×C4C23×D311yescharacteristic
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—332no—
273C4⋊C4—316no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—68no—
279C4⋊C4—616no—
280C4⋊C4—128no—
281C4⋊C4—128no—
282C4⋊C4—124no—
283C4⋊C4—128no—
284C4⋊C4—124no—
285C42—22no—
286C42—24no—
287C42—24no—
288C42—68no—
289C42—128no—
290C42—124no—
291C42D3×Q818yes—
292C22×C3C23×Q812yes—
293C22×C3C2×H(5,𝔽2)11yescharacteristic
294C22×C3C2×H(5,𝔽2)12yes—
295C22×C3C2×2−1+412yes—
296C3×C4—28no—
297C3×C4—24no—
298C3×C4—24no—
299C3×C4—42no—
300C3×C4—41no—
301C3×C4—42no—
302C3×C4—41no—
303C3×C4Q8214yes—
304Dic3—216no—
305Dic3—48no—
306C23C24×D311yescharacteristic
307Q8—44no—
308Q8—44no—
309Q8—248no—
310Q8—248no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—316no—
326C2×C4—68no—
327C2×C4—68no—
328C2×C4—68no—
329C2×C4—68no—
330C2×C4—68no—
331C2×C4#100412yes—
332C2×C4C2×D3×Q818yes—
333C2×C4C2×D3×Q814yes—
334C2×C4#100914yes—
335C2×C4#101014yes—
336C2×C3C2×Q8211yescharacteristic
337C2×C3#350314yes—
338C2×C3#353711yescharacteristic
339C2×C3#353811yescharacteristic
340C22C2×#100411yescharacteristic
341C22C22×D3×Q812yes—
342C22C2×#100912yes—
343C22C2×#101012yes—
344C4—28no—
345C4—24no—
346C4—24no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—616no—
352C4—128no—
353C4#572214yes—
354C3#4543511yescharacteristic, complemented, Sylow
355C2#8603111yescharacteristic
356C2#8605211yescharacteristic
357C2C2×#572211yescharacteristic
358C2#8628114yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, prime radical, cosocleC23×C3C25
FittingC3×#2399C2
hypercenter#2399D3
hypocenterC3#45435
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2399

Complements

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

NH ≅ G/Ntype
88. #870Q8semidirect
354. C3#45435semidirect

Maximal subgroups

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

subgroupindex
C3×#23992
#824782
#824752
#779122
#778722
#771942
#771862
#737152
#736592
#733902
#454353