Finite Group Databasebeta

← #11235192

Subgroups — #11235192

Order 768.

1514 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2327C211yesmaximal, characteristic
2#83239C212yesmaximal
3#82475C212yesmaximal
4#82463C212yesmaximal
5#76545C214yesmaximal
6#75213C218yesmaximal
7#75134C218yesmaximal
8#74159C212yesmaximal
9#73397C211yesmaximal, characteristic
10#73388C211yesmaximal, characteristic
11#45436—31nomaximal, Sylow
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4×Q8C2212yes—
14C2×#944C2214yes—
15C2×#869C2211yescharacteristic
16C2×C4×Dic6C2212yes—
17C2×C4×Dic6C2212yes—
18C3×#391C2212yes—
19C3×#387C2218yes—
20C3×#382C2211yescharacteristic
21C3×#376C2211yescharacteristic
22#5229C2214yes—
23#4983C2214yes—
24#4983C2214yes—
25#4981C2214yes—
26#4981C2214yes—
27#4980C2218yes—
28Dic3×C4⋊C4C2214yes—
29#4973C2214yes—
30#4972C2218yes—
31#4937C2214yes—
32#4936C2211yescharacteristic
33C4×C4⋊Dic3C2212yes—
34C4×C4⋊Dic3C2212yes—
35#4932C2214yes—
36#4929C2212yes—
37#4928C2218yes—
38#4661C2214yes—
39#4660C2218yes—
40#4658C2218yes—
41#4656C2214yes—
42#4655C2218yes—
43#4653C2218yes—
44#4651C2214yes—
45#4650C2218yes—
46#4648C2214yes—
47#4646C2218yes—
48#2769—38no—
49#2612—38no—
50#2568—34no—
51#2546—32no—
52#2546—32no—
53#2399—31no—
54#2356—32no—
55#2356—32no—
56#2327—31no—
57#2327D311yescharacteristic
58C22×C3×Q8C2312yes—
59C22×Dic6C2312yes—
60C22×Dic6C2312yes—
61C3×C4⋊Q8—24no—
62C3×C4×Q8—28no—
63C2×C3×C4⋊C4C2314yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2314yes—
66C2×C3×C42C2314yes—
67C2×C3×C42C2311yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70#944—48no—
71C2×C4⋊Dic3C2314yes—
72C2×C4⋊Dic3C2314yes—
73C2×C4⋊Dic3C2314yes—
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4C2314yes—
77C2×Dic3⋊C4C2318yes—
78C2×Dic3⋊C4C2314yes—
79C2×Dic3⋊C4C2318yes—
80C2×C4×Dic3C2318yes—
81C2×C4×Dic3C2314yes—
82C2×C4×Dic3C2314yes—
83#869—24no—
84C4×Dic6—28no—
85C4×Dic6—44no—
86C3×#95C2318yes—
87C3×#95C2314yes—
88C3×#95C2314yes—
89C3⋊#95C2318yes—
90C3⋊#95C2318yes—
91C3⋊#95C2318yes—
92C3⋊#95C2312yes—
93C3⋊#95C2318yes—
94C3⋊#95C2318yes—
95C3⋊#95C2312yes—
96C3⋊#95C2318yes—
97C3⋊#95C2314yes—
98C3⋊#95C2312yes—
99C3⋊#95C2312yes—
100C2×C4⋊Q8—34no—
101C2×C4⋊Q8—31no—
102C2×C4⋊Q8C2×D311yescharacteristic
103C2×C4×Q8—32no—
104C2×C4×Q8—32no—
105C2×C4×Q8C2×D312yes—
106#400—38no—
107#400—38no—
108#398—34no—
109#398—34no—
110#398—34no—
111#398—34no—
112#398—34no—
113#395—34no—
114#391—34no—
115#391—34no—
116#391—32no—
117#391—38no—
118#391C2×D312yes—
119#389—38no—
120#389—38no—
121#387—38no—
122#387—38no—
123#387—38no—
124#387C2×D318yes—
125#383—34no—
126#383—34no—
127#382—31no—
128#382—38no—
129#382C2×D311yescharacteristic
130C4×C4⋊C4—34no—
131C4×C4⋊C4—32no—
132C4×C4⋊C4—32no—
133#376—34no—
134#376—34no—
135#376C2×D311yescharacteristic
136C2×C3×Q8—24no—
137C2×C3×Q8—24no—
138C2×C3×Q8—24no—
139C22×C3×C4C2414yes—
140C22×C3×C4C2412yes—
141C22×C3×C4C2414yes—
142C22×C3×C4C2412yes—
143C22×C3×C4C2412yes—
144C22×C3×C4C2411yescharacteristic
145C22×Dic3C2414yes—
146C22×Dic3C2414yes—
147C22×Dic3C2414yes—
148C22×Dic3C2414yes—
149C2×Dic6—24no—
150C2×Dic6—24no—
151C2×Dic6—24no—
152C2×Dic6—28no—
153C2×Dic6—42no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—28no—
156C3×C4⋊C4—28no—
157C3×C42—28no—
158C3×C42—22no—
159C3×C42#4114yes—
160C3×C42#4114yes—
161C4⋊Dic3—24no—
162C4⋊Dic3—28no—
163C4⋊Dic3—28no—
164C4⋊Dic3—28no—
165C4⋊Dic3—42no—
166Dic3⋊C4—28no—
167Dic3⋊C4—44no—
168Dic3⋊C4—48no—
169Dic3⋊C4—48no—
170C4×Dic3—28no—
171C4×Dic3—28no—
172C4×Dic3—216no—
173C22×Q8—32no—
174C22×Q8—32no—
175C22×Q8C22×D312yes—
176C4⋊Q8—24no—
177C4⋊Q8—64no—
178C4⋊Q8—128no—
179C4×Q8—28no—
180C4×Q8—68no—
181C4×Q8—124no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—34no—
184C2×C4⋊C4—34no—
185C2×C4⋊C4—34no—
186C2×C4⋊C4—34no—
187C2×C4⋊C4—34no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—38no—
190C2×C4⋊C4—38no—
191C2×C4⋊C4C22×D312yes—
192C2×C4⋊C4C22×D314yes—
193C2×C4⋊C4C22×D314yes—
194C2×C42—34no—
195C2×C42—34no—
196C2×C42—38no—
197C2×C42C22×D311yescharacteristic
198C2×C42C22×D314yes—
199C2×C42C22×D311yescharacteristic
200C2×C42C22×D311yescharacteristic
201#95—32no—
202#95—32no—
203#95—34no—
204#95—38no—
205#95—38no—
206#95—38no—
207#95—32no—
208#95—32no—
209#95—38no—
210#95—38no—
211#95—38no—
212#95C22×D314yes—
213#95C22×D314yes—
214#95C22×D318yes—
215C23×C3C2511yescharacteristic
216C3×Q8—48no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—24no—
220C2×C3×C4—24no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—22no—
224C2×C3×C4—28no—
225C2×C3×C4—24no—
226C2×C3×C4C2×#4114yes—
227C2×C3×C4C2×#4112yes—
228C2×C3×C4C2×#4114yes—
229C2×C3×C4C2×#4112yes—
230C2×C3×C4H(5,𝔽2)12yes—
231C2×Dic3—24no—
232C2×Dic3—24no—
233C2×Dic3—24no—
234C2×Dic3—24no—
235C2×Dic3—24no—
236C2×Dic3—24no—
237C2×Dic3—24no—
238C2×Dic3—28no—
239C2×Dic3—24no—
240C2×Dic3—24no—
241C2×Dic32−1+418yes—
242Dic6—48no—
243Dic6—48no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—68no—
248C2×Q8—64no—
249C2×Q8—64no—
250C2×Q8—64no—
251C2×Q8—122no—
252C22×C4—34no—
253C22×C4—34no—
254C22×C4—34no—
255C22×C4—34no—
256C22×C4C23×D314yes—
257C22×C4C23×D312yes—
258C22×C4C23×D314yes—
259C22×C4C23×D312yes—
260C22×C4C23×D312yes—
261C22×C4C23×D311yescharacteristic
262C4⋊C4—28no—
263C4⋊C4—24no—
264C4⋊C4—28no—
265C4⋊C4—68no—
266C4⋊C4—68no—
267C4⋊C4—68no—
268C4⋊C4—64no—
269C4⋊C4—68no—
270C4⋊C4—124no—
271C4⋊C4—122no—
272C4⋊C4—128no—
273C4⋊C4—128no—
274C42—28no—
275C42—22no—
276C42—616no—
277C42—68no—
278C42—68no—
279C42#23514yes—
280C42#23714yes—
281C22×C3C22×#4111yescharacteristic
282C22×C3C22×#4111yescharacteristic
283C22×C3C2×H(5,𝔽2)11yescharacteristic
284C22×C3C2×2−1+414yes—
285C3×C4—24no—
286C3×C4—24no—
287C3×C4—44no—
288C3×C4—44no—
289C3×C4—42no—
290C3×C4#55514yes—
291Dic3—28no—
292Dic3—44no—
293Dic3—44no—
294Dic3—44no—
295C23C24×D311yescharacteristic
296Q8—48no—
297Q8—128no—
298Q8—128no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—28no—
307C2×C4—24no—
308C2×C4—38no—
309C2×C4—64no—
310C2×C4—64no—
311C2×C4—64no—
312C2×C4—64no—
313C2×C4—64no—
314C2×C4—64no—
315C2×C4—64no—
316C2×C4—68no—
317C2×C4—64no—
318C2×C4—64no—
319C2×C4C2×#23514yes—
320C2×C4C2×#23512yes—
321C2×C4C2×#23714yes—
322C2×C4C2×#23712yes—
323C2×C4#100912yes—
324C2×C3C2×#55511yescharacteristic
325C2×C3#350912yes—
326C2×C3#350912yes—
327C2×C3#353812yes—
328C22C22×#23511yescharacteristic
329C22C22×#23711yescharacteristic
330C22C2×#100911yescharacteristic
331C22D3×2−1+414yes—
332C4—24no—
333C4—24no—
334C4—44no—
335C4—44no—
336C4—42no—
337C4—68no—
338C4—124no—
339C4—124no—
340C4—124no—
341C4#556314yes—
342C3#4543611yescharacteristic, complemented, Sylow
343C2C2×#556311yescharacteristic
344C2#8587612yes—
345C2#8634312yes—
346C2#8651512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, prime radical, cosocleC23×C3C25
FittingC3×#2327C2
hypercenter#2327D3
hypocenterC3#45436
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2327

Complements

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

NH ≅ G/Ntype
342. C3#45436semidirect

Maximal subgroups

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

subgroupindex
C3×#23272
#832392
#824752
#824632
#765452
#752132
#751342
#741592
#733972
#733882
#454363