Finite Group Databasebeta

← #10697909

Subgroups — #10697909

Order 768.

504 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C3×#3230C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic
3#83252C211yesmaximal, characteristic
4#71839C211yesmaximal, characteristic
5#71775C211yesmaximal, characteristic
6#70780C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic
8#10295—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#500—21no—
13C3×#487—21no—
14C3×#484—22no—
15C3×#477—22no—
16C3×#431C2211yescharacteristic
17#5229—22no—
18#5079—21no—
19#5078—21no—
20#5044—22no—
21#5021—22no—
22#4977—22no—
23#4973—22no—
24C42×Dic3C2211yescharacteristic
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27#4547—22no—
28#4532—22no—
29#4482—22no—
30#3230—31no—
31#3230D311yescharacteristic
32#2862—31no—
33#1994—31no—
34#1549—31no—
35#1549—31no—
36#1276—31no—
37C22×C3×Q8—21no—
38C22×Dic6—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C42C2311yescharacteristic
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3D412yes—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3D412yes—
59C2×#208—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64C3×#107—24no—
65C3×C4⋊C8D412yes—
66C3×#103—24no—
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69#835—24no—
70C3⋊#95—28no—
71#830—21no—
72#807—24no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77#533C2×D311yescharacteristic
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#500—21no—
81#500—61no—
82#487—21no—
83#487—61no—
84#484—22no—
85#484—62no—
86#477—22no—
87#477—62no—
88#431—31no—
89#431C2×D311yescharacteristic
90#423—31no—
91#391—62no—
92#391—62no—
93#382—62no—
94C43—31no—
95#339—62no—
96#332—62no—
97#332—62no—
98C2×C3×Q8—21no—
99C2×C3×Q8—21no—
100C2×C3×Q8—44no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×Dic3—22no—
108C22×Dic3—22no—
109C2×Dic6—44no—
110C2×Dic6—41no—
111C2×Dic6—41no—
112C2×C3×C8—22no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—21no—
118C3×C4⋊C4—21no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C42—41no—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129C4⋊Dic3—22no—
130C4⋊Dic3—22no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C4×Dic3—22no—
134C4×Dic3—42no—
135C4×Dic3—42no—
136C4×Dic3Q1614yes—
137#208—42no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—41no—
140C22×Q8—21no—
141C22×Q8—61no—
142C2×M16—61no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—61no—
148C4⋊Q8—61no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8#6711yescharacteristic
152C4⋊Q8#6711yescharacteristic
153#125—21no—
154C22⋊Q8—21no—
155C22⋊Q8—21no—
156C4×Q8—21no—
157C4×Q8—21no—
158#117—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—32no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4#6712yes—
163C2×C42—32no—
164C2×C42—61no—
165C2×C42—61no—
166C2×C42C22×D311yescharacteristic
167#107—24no—
168#107—64no—
169C4⋊C8—32no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172C4⋊C8#6712yes—
173#103—24no—
174#103—64no—
175C22⋊C8—21no—
176C22⋊C8—61no—
177C4×C8—21no—
178C4×C8—61no—
179#95—68no—
180C23×C3#12011yescharacteristic
181C3×Q8—42no—
182C3×Q8—42no—
183C3×Q8—82no—
184C3×Q8—82no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—44no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4#12011yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12111yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12711yescharacteristic
197C2×C3×C4C2×Q1612yes—
198C2×C3×C4#13712yes—
199C2×Dic3—24no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—44no—
203C2×Dic3—42no—
204Dic6—82no—
205Dic6—82no—
206C3×C8—42no—
207C3⋊C8—42no—
208C3⋊C8—81no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—44no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—124no—
215C2×Q8—121no—
216C2×Q8—121no—
217C22×C4—22no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4D3×D411yescharacteristic
221C22×C4D3×D411yescharacteristic
222C22×C4C2×#6711yescharacteristic
223M16—122no—
224C2×C8—22no—
225C2×C8—62no—
226C2×C8—121no—
227C4⋊C4—21no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—22no—
232C4⋊C4—21no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—62no—
238C4⋊C4—62no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C22⋊C4—41no—
242C42—34no—
243C42—41no—
244C42—62no—
245C42—122no—
246C42—122no—
247C42D3×D411yescharacteristic
248C42#23711yescharacteristic
249C42C2×#6711yescharacteristic
250C42C2×#6711yescharacteristic
251C22×C3—21no—
252C22×C3—21no—
253C22×C3#39211yescharacteristic
254C22×C3#45112yes—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—82no—
258C3×C4—81no—
259C3×C4—81no—
260C3×C4#45412yes—
261C3×C4#45412yes—
262C3×C4#46212yes—
263C3×C4#49411yescharacteristic
264C3×C4#49711yescharacteristic
265Dic3—44no—
266Dic3—82no—
267C23#93711yescharacteristic
268Q8—42no—
269Q8—42no—
270Q8—82no—
271Q8—82no—
272Q8—242no—
273Q8—242no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—64no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—124no—
284C2×C4—122no—
285C2×C4D3×Q1612yes—
286C2×C4#91812yes—
287C2×C4#93711yescharacteristic
288C2×C4#93811yescharacteristic
289C2×C4#93911yescharacteristic
290C2×C4#93911yescharacteristic
291C2×C4#94011yescharacteristic
292C2×C4#95311yescharacteristic
293C8—42no—
294C8—122no—
295C8—241no—
296C2×C3—22no—
297C2×C3#200311yescharacteristic
298C2×C3#200311yescharacteristic
299C2×C3#201611yescharacteristic
300C22—21no—
301C22—21no—
302C22#518612yes—
303C22#522311yescharacteristic
304C4—21no—
305C4—21no—
306C4—82no—
307C4—81no—
308C4—81no—
309C4—124no—
310C4—242no—
311C4#518212yes—
312C4#518311yescharacteristic
313C4#519011yescharacteristic
314C4#520112yes—
315C4#524512yes—
316C3#1029511yescharacteristic, complemented, Sylow
317C2—22no—
318C2#7186711yescharacteristic
319C2#7207311yescharacteristic
320C2#7209611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3230C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3230D3
hypocenterC3#10295
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
316. C3#10295semidirect

Maximal subgroups

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

subgroupindex
C3×#32302
#833242
#832522
#718392
#717752
#707802
#677362
#102953