Finite Group Databasebeta

← #10697121

Subgroups — #10697121

Order 768.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×C3×#498C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic
3#83198C211yesmaximal, characteristic
4#71790C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic
7#67656C211yesmaximal, characteristic
8#10173—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#498—22no—
13C3×#498—22no—
14C2×C3×#106—22no—
15C2×C3×C4×C8C2211yescharacteristic
16#5225—22no—
17#5093—21no—
18#5091—22no—
19#5082—22no—
20#5079—21no—
21#5079—21no—
22#5078—21no—
23#4970—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C4×#208C2211yescharacteristic
27#4547—22no—
28#4530—22no—
29#4456—22no—
30#3230—31no—
31C2×#498D311yescharacteristic
32#3139—31no—
33#1997—31no—
34#1546—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D412yes—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D412yes—
52C2×C3×C42C2311yescharacteristic
53#926—21no—
54#924—21no—
55#924—21no—
56C2×#208—21no—
57C2×#208D412yes—
58#870—21no—
59#869D411yescharacteristic
60#869D411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×#106—28no—
64C3×C4×C8—21no—
65C3×C4×C8D412yes—
66#835—24no—
67#835—24no—
68#830—21no—
69#810—24no—
70#807—24no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)D412yes—
74#803—21no—
75C4×C3⋊C8—21no—
76C4×C3⋊C8—21no—
77#533—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4⋊Q8C2×D311yescharacteristic
80#500—61no—
81#498—22no—
82#498—22no—
83#491—62no—
84#487—61no—
85#487—61no—
86#487—61no—
87#484—62no—
88#477—62no—
89#431—31no—
90C2×#106—22no—
91#423—31no—
92#419—62no—
93C4×M16—31no—
94C2×C4×C8C2×D311yescharacteristic
95#343—62no—
96#332—62no—
97Q8⋊C8—62no—
98C2×C3×Q8—22no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—42no—
103C2×C3×Q8—42no—
104C2×C3×Q8—82no—
105C22×C3×C4—22no—
106C22×C3×C4—41no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×Dic6—21no—
111C2×Dic6—21no—
112C2×C3×C8—24no—
113C2×C3×C8—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42C2×D411yescharacteristic
125C3×C42C2×D411yescharacteristic
126#217—41no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—21no—
131Dic3⋊C4—41no—
132Dic3⋊C4—41no—
133C4×Dic3—41no—
134#208—22no—
135#208—22no—
136#208—42no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—41no—
140C22×Q8—21no—
141C22×Q8—21no—
142C2×M16—32no—
143C2×M16—61no—
144C22×C8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—31no—
153C4⋊Q8—31no—
154C4⋊Q8#6712yes—
155#125—61no—
156C22⋊Q8—61no—
157C22⋊Q8—61no—
158C4×Q8—61no—
159C4×Q8—61no—
160#117—61no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4#6712yes—
165C2×C42C22×D311yescharacteristic
166#107—64no—
167#106—28no—
168C4⋊C8—32no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171#103—64no—
172#103—64no—
173#103—64no—
174C22⋊C8—61no—
175#97—61no—
176C4×C8—21no—
177C4×C8—61no—
178C4×C8—61no—
179C4×C8#6712yes—
180C23×C3#12111yescharacteristic
181C3×Q8—44no—
182C3×Q8—82no—
183C3×Q8—82no—
184C3×Q8—81no—
185C3×Q8—81no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—82no—
191C2×C3×C4—82no—
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12011yescharacteristic
194C2×C3×C4#12011yescharacteristic
195C2×C3×C4#12111yescharacteristic
196C2×C3×C4#12111yescharacteristic
197C2×C3×C4#12711yescharacteristic
198C2×C3×C4#13512yes—
199C2×C3×C4#13712yes—
200C2×Dic3—41no—
201C2×Dic3—41no—
202Dic6—42no—
203Dic6—42no—
204C3×C8—24no—
205C3⋊C8—42no—
206C3⋊C8—42no—
207C3⋊C8—81no—
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—61no—
215C2×Q8—61no—
216C2×Q8—82no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4D3×D411yescharacteristic
220C22×C4#23711yescharacteristic
221C22×C4C2×#6711yescharacteristic
222M16—62no—
223M16—62no—
224M16—122no—
225C2×C8—24no—
226C2×C8—22no—
227C2×C8—62no—
228C2×C8—62no—
229C2×C8—121no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—121no—
243C4⋊C4—121no—
244C22⋊C4—121no—
245C42—121no—
246C42D3×D411yescharacteristic
247C42D3×D411yescharacteristic
248C42C2×#6711yescharacteristic
249C42C2×#6711yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3#39211yescharacteristic
253C22×C3#46412yes—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—82no—
257C3×C4—82no—
258C3×C4#45212yes—
259C3×C4#45612yes—
260C3×C4#47112yes—
261C3×C4#49511yescharacteristic
262C3×C4#49711yescharacteristic
263Dic3—81no—
264Dic3—81no—
265C23#93911yescharacteristic
266Q8—44no—
267Q8—82no—
268Q8—81no—
269Q8—81no—
270Q8—82no—
271Q8—122no—
272Q8—122no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4#91512yes—
282C2×C4#91612yes—
283C2×C4#93711yescharacteristic
284C2×C4#93711yescharacteristic
285C2×C4#93811yescharacteristic
286C2×C4#93911yescharacteristic
287C2×C4#94011yescharacteristic
288C2×C4#95311yescharacteristic
289C8—24no—
290C8—122no—
291C8—122no—
292C8—241no—
293C2×C3—22no—
294C2×C3#200511yescharacteristic
295C2×C3#201511yescharacteristic
296C2×C3#201911yescharacteristic
297C22—21no—
298C22—21no—
299C22#516612yes—
300C22#522311yescharacteristic
301C4—21no—
302C4—21no—
303C4—82no—
304C4—82no—
305C4—241no—
306C4—241no—
307C4#509512yes—
308C4#516711yescharacteristic
309C4#516811yescharacteristic
310C4#517312yes—
311C4#517712yes—
312C3#1017311yescharacteristic, complemented, Sylow
313C2—22no—
314C2#7179511yescharacteristic
315C2#7205811yescharacteristic
316C2#7205911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#498C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenterC2×#498D3
hypocenterC3#10173
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
312. C3#10173semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4982
#832242
#831982
#717902
#717732
#697382
#676562
#101733