Finite Group Databasebeta

← #10698445

Subgroups — #10698445

Order 768.

746 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73845C211yesmaximal, characteristic, complemented
2C3×#1781C211yesmaximal, characteristic
3#72130C212yesmaximal
4C3⋊#1304C211yesmaximal, characteristic, complemented
5C3⋊#1304C212yesmaximal, complemented
6#10392—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C2×C3×#99—22no—
9C3×#379C2211yescharacteristic
10C3×#379C2211yescharacteristic
11#5247C2212yes—
12#5247C2212yes—
13C2×#834—24no—
14#4949—22no—
15#4948—24no—
16#2842—31no—
17#1781—32no—
18#1781D311yescharacteristic
19#1304—32no—
20#1304—31no—
21C25×C3Q811yescharacteristic
22C25×C3C2311yescharacteristic
23C22×C3×D4—22no—
24C22×C3×D4—21no—
25C22×C3×D4—21no—
26C22×C3×D4—22no—
27C22×C3×D4—44no—
28C23×C3×C4Q811yescharacteristic
29C2×C3×C22⋊C4—21no—
30C2×C3×C22⋊C4—22no—
31C2×C3×C22⋊C4—22no—
32C2×C3×C22⋊C4D412yes—
33C2×#217—22no—
34C2×#217—22no—
35C2×#217—22no—
36C2×#217—22no—
37C2×#217—22no—
38C2×#217D414yes—
39C2×C4⋊Dic3—22no—
40C2×Dic3⋊C4—24no—
41C3×#99—44no—
42#834—48no—
43C3⋊#95—22no—
44C23×D4C2×D311yescharacteristic
45C2×#99—22no—
46C2×#99—64no—
47#399—62no—
48#396—64no—
49#379—32no—
50#379—32no—
51#379C2×D311yescharacteristic
52#379C2×D311yescharacteristic
53C24×C3—22no—
54C24×C3—21no—
55C24×C3—42no—
56C24×C3—42no—
57C24×C3—82no—
58C24×C3—82no—
59C24×C3C2×D412yes—
60C24×C3C2×D411yescharacteristic
61C24×C3C2×Q811yescharacteristic
62C24×C3#4111yescharacteristic
63C24×C3#4112yes—
64C2×C3×D4—28no—
65C2×C3×D4—24no—
66C2×C3×D4—42no—
67C2×C3×D4—44no—
68C2×C3×D4—42no—
69C2×C3×D4—44no—
70C2×C3×D4—44no—
71C2×C3×D4—42no—
72C2×C3×D4—41no—
73C2×C3×D4—42no—
74C2×C3×D4—41no—
75C22×C3×C4—22no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—42no—
79C22×C3×C4—41no—
80C22×Dic3—24no—
81C22×Dic3—42no—
82C3×C22⋊C4—42no—
83C3×C22⋊C4—42no—
84C3×C22⋊C4—82no—
85C3×C22⋊C4—82no—
86#217—44no—
87#217—44no—
88#217—82no—
89#217—82no—
90#217—82no—
91#217—82no—
92C4⋊Dic3—82no—
93Dic3⋊C4—84no—
94C25Dic611yescharacteristic
95C25C22×D311yescharacteristic
96C22×D4—22no—
97C22×D4—21no—
98C22×D4—21no—
99C22×D4—22no—
100C22×D4—44no—
101C23×C4Dic611yescharacteristic
102C2×C4⋊C4—64no—
103C2×C4⋊C4—62no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—34no—
108C2×C22⋊C4—62no—
109C2×C22⋊C4—62no—
110C2×C22⋊C4—62no—
111C2×C22⋊C4—62no—
112C2×C22⋊C4—62no—
113C2×C22⋊C4#6712yes—
114#99—44no—
115#99—128no—
116#95—62no—
117C23×C3—24no—
118C23×C3—22no—
119C23×C3—24no—
120C23×C3—22no—
121C23×C3—41no—
122C23×C3—42no—
123C23×C3—42no—
124C23×C3—42no—
125C23×C3—42no—
126C23×C3—42no—
127C23×C3—42no—
128C23×C3—42no—
129C23×C3—41no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3—41no—
133C23×C3—41no—
134C23×C3—82no—
135C23×C3—82no—
136C23×C3—82no—
137C23×C3—88no—
138C23×C3—88no—
139C23×C3#12011yescharacteristic
140C23×C3C22⋊Q812yes—
141C23×C3C22⋊Q811yescharacteristic
142C23×C3#12412yes—
143C23×C3#12411yescharacteristic
144C3×D4—48no—
145C3×D4—44no—
146C3×D4—44no—
147C2×C3×C4—24no—
148C2×C3×C4—22no—
149C2×C3×C4—42no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
152C2×C3×C4—44no—
153C2×C3×C4—42no—
154C2×C3×C4—81no—
155C2×C3×C4—82no—
156C2×Dic3—44no—
157C2×Dic3—44no—
158C2×Dic3—44no—
159C2×Dic3—82no—
160C2×Dic3—82no—
161C2×Dic3—82no—
162C24—22no—
163C24—21no—
164C24—42no—
165C24—42no—
166C24—82no—
167C24—82no—
168C24C2×Dic611yescharacteristic
169C24#23511yescharacteristic
170C24D3×D412yes—
171C24#23712yes—
172C24C2×#6711yescharacteristic
173C2×D4—24no—
174C2×D4—28no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—42no—
180C2×D4—44no—
181C2×D4—42no—
182C2×D4—41no—
183C2×D4—41no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—64no—
190C22×C4—122no—
191C4⋊C4—244no—
192C4⋊C4—242no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C22⋊C4—124no—
198C22⋊C4—124no—
199C22⋊C4—242no—
200C22⋊C4—242no—
201C22⋊C4—242no—
202C22⋊C4—242no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—44no—
215C22×C3—42no—
216C22×C3—84no—
217C22×C3—82no—
218C22×C3—82no—
219C22×C3—84no—
220C22×C3—84no—
221C22×C3—82no—
222C22×C3—84no—
223C22×C3—82no—
224C22×C3—82no—
225C22×C3#39311yescharacteristic
226C22×C3B(4,𝔽2)12yes—
227C3×C4—42no—
228C3×C4—82no—
229C3×C4—161no—
230Dic3—84no—
231Dic3—162no—
232C23—22no—
233C23—24no—
234C23—24no—
235C23—22no—
236C23—41no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23—41no—
248C23—41no—
249C23—82no—
250C23—82no—
251C23—82no—
252C23—88no—
253C23—88no—
254C23#87812yes—
255C23#88512yes—
256C23#92411yescharacteristic
257C23#93611yescharacteristic
258C23#93711yescharacteristic
259D4—44no—
260D4—48no—
261D4—44no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—124no—
272C2×C4—124no—
273C2×C4—124no—
274C2×C4—242no—
275C2×C4—242no—
276C2×C4—242no—
277C2×C3—22no—
278C2×C3—42no—
279C2×C3—41no—
280C2×C3—42no—
281C2×C3—41no—
282C2×C3—82no—
283C2×C3#202911yescharacteristic
284C2×C3#203212yes—
285C22—22no—
286C22—22no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—44no—
297C22—42no—
298C22—82no—
299C22—84no—
300C22—84no—
301C22—82no—
302C22—84no—
303C22—82no—
304C22—84no—
305C22—82no—
306C22—82no—
307C22#494711yescharacteristic
308C22#503212yes—
309C4—42no—
310C4—82no—
311C4—161no—
312C4—244no—
313C4—482no—
314C3#1039211yescharacteristic, complemented, Sylow
315C2—22no—
316C2—41no—
317C2—42no—
318C2—42no—
319C2—41no—
320C2—82no—
321C2#6949711yescharacteristic
322C2#7002512yes—
subgroupG/H
centerC22#4947
commutator, cosocleC25×C3C23
FrattiniC25C22×D3
FittingC3×#1781C2
socleC22×C3#393
prime radicalC23×C3×D4C22
hypercenter#1781D3
hypocenterC3#10392
LCSG ▷ C25×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
1. #73845C2semidirect
4. C3⋊#1304C2semidirect
5. C3⋊#1304C2semidirect
314. C3#10392semidirect

Maximal subgroups

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

subgroupindex
#738452
C3×#17812
#721302
C3⋊#13042
#103923