Finite Group Databasebeta

← #10697912

Subgroups — #10697912

Order 768.

504 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 107 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#83252C211yesmaximal, characteristic
2#83224C211yesmaximal, characteristic
3#79272C211yesmaximal, characteristic
4C3×#1549C211yesmaximal, characteristic
5#70514C211yesmaximal, characteristic
6#70508C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic
8#10295—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5229—22no—
15#5093—21no—
16#5091—22no—
17#5078—21no—
18#4977—22no—
19#4973—22no—
20#4970—22no—
21C42×Dic3C2211yescharacteristic
22#4925C2211yescharacteristic
23#4850—22no—
24#4722—22no—
25#4682—21no—
26#4681—21no—
27C3×#332—22no—
28#4555—22no—
29#4486—22no—
30#3230—31no—
31#3230—31no—
32#2862—31no—
33#1994—31no—
34#1549—31no—
35#1549D311yescharacteristic
36#1276—31no—
37C22×C3×Q8—21no—
38C22×Dic6—21no—
39C2×C3×M16—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C42C2311yescharacteristic
47#926—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D412yes—
50#924—21no—
51#924—21no—
52C2×C4×Dic3—21no—
53C2×C4×Dic3—21no—
54C2×C4×Dic3D412yes—
55#870—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D412yes—
67C3×C22⋊C8—21no—
68C3×C4×C8—21no—
69C3⋊#95—28no—
70#830—21no—
71#818—24no—
72#816—24no—
73#810—24no—
74#807—24no—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77#533—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80#500—61no—
81#500—61no—
82#487—61no—
83#487—61no—
84#484—62no—
85#484—62no—
86#477—62no—
87#477—62no—
88#431—31no—
89#431C2×D311yescharacteristic
90#423C2×D311yescharacteristic
91#391—62no—
92#391—62no—
93#382—62no—
94C43—31no—
95#339—62no—
96#332—22no—
97#332—62no—
98C2×C3×Q8—44no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C22×C3×C4—22no—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×Dic3—22no—
106C22×Dic3—22no—
107C2×Dic6—21no—
108C2×Dic6—21no—
109C2×Dic6—44no—
110C2×Dic6—41no—
111C2×Dic6—41no—
112C3×M16—42no—
113C2×C3×C8—22no—
114C2×C3×C8—41no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—41no—
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#217—41no—
124C4⋊Dic3—21no—
125C4⋊Dic3—22no—
126C4⋊Dic3—21no—
127C4⋊Dic3—22no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134C4×Dic3—22no—
135C4×Dic3—42no—
136C4×Dic3—42no—
137C4×Dic3—41no—
138C4×Dic3Q1614yes—
139C2×C3⋊C8—22no—
140C22×Q8—21no—
141C22×Q8—61no—
142C2×M16—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—31no—
148C4⋊Q8—31no—
149C4⋊Q8—61no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153#125—61no—
154C22⋊Q8—61no—
155C22⋊Q8—61no—
156C4×Q8—61no—
157C4×Q8—61no—
158#117—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—32no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4D1212yes—
163C2×C42—32no—
164C2×C42—61no—
165C2×C42—61no—
166C2×C42C22×D311yescharacteristic
167#107—64no—
168#107—64no—
169C4⋊C8—21no—
170C4⋊C8—21no—
171C4⋊C8—32no—
172C4⋊C8#6712yes—
173#103—64no—
174#103—64no—
175C22⋊C8—21no—
176C22⋊C8—61no—
177C4×C8—21no—
178C4×C8—61no—
179#95—68no—
180C23×C3#12011yescharacteristic
181C3×Q8—82no—
182C3×Q8—82no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—44no—
186C2×C3×C4—42no—
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4#12711yescharacteristic
193C2×C3×C4C2×Q1612yes—
194C2×C3×C4#13712yes—
195C2×Dic3—24no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—44no—
199C2×Dic3—41no—
200C2×Dic3—41no—
201C2×Dic3—42no—
202Dic6—42no—
203Dic6—42no—
204Dic6—82no—
205Dic6—82no—
206C3×C8—42no—
207C3×C8—81no—
208C3⋊C8—42no—
209C2×Q8—44no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8—124no—
215C2×Q8—121no—
216C2×Q8—121no—
217C22×C4—22no—
218C22×C4—62no—
219C22×C4—62no—
220C22×C4C2×D1211yescharacteristic
221C22×C4D3×D411yescharacteristic
222C22×C4D3×D411yescharacteristic
223M16—42no—
224C2×C8—22no—
225C2×C8—41no—
226C2×C8—62no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—61no—
232C4⋊C4—62no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—62no—
236C4⋊C4—61no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—121no—
241C22⋊C4—121no—
242C42—34no—
243C42—62no—
244C42—122no—
245C42—122no—
246C42—121no—
247C42#23511yescharacteristic
248C42D3×D411yescharacteristic
249C42D3×D411yescharacteristic
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#45412yes—
259C3×C4#45412yes—
260C3×C4#46212yes—
261C3×C4#49411yescharacteristic
262C3×C4#49711yescharacteristic
263Dic3—44no—
264Dic3—81no—
265Dic3—81no—
266Dic3—82no—
267C23#88211yescharacteristic
268Q8—82no—
269Q8—82no—
270Q8—122no—
271Q8—122no—
272Q8—242no—
273Q8—242no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—64no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—124no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—122no—
285C2×C4#88211yescharacteristic
286C2×C4#88411yescharacteristic
287C2×C4#88411yescharacteristic
288C2×C4#91512yes—
289C2×C4D3×Q1612yes—
290C2×C4#93011yescharacteristic
291C2×C4#93711yescharacteristic
292C2×C4#94011yescharacteristic
293C8—42no—
294C8—81no—
295C8—122no—
296C2×C3—22no—
297C2×C3#200311yescharacteristic
298C2×C3#200311yescharacteristic
299C2×C3#201611yescharacteristic
300C22—21no—
301C22—21no—
302C22#480912yes—
303C22#496011yescharacteristic
304C4—21no—
305C4—21no—
306C4—82no—
307C4—124no—
308C4—241no—
309C4—242no—
310C4—241no—
311C4#475312yes—
312C4#479512yes—
313C4#516711yescharacteristic
314C4#518311yescharacteristic
315C4#520112yes—
316C3#1029511yescharacteristic, complemented, Sylow
317C2—22no—
318C2#7011611yescharacteristic
319C2#7028311yescharacteristic
320C2#7032611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#1549D3
hypocenterC3#10295
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

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
#832522
#832242
#792722
C3×#15492
#705142
#705082
#677362
#102953