Finite Group Databasebeta

← #10698861

Subgroups — #10698861

Order 768.

376 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 95 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×#4850C211yesmaximal, characteristic
2#70780C211yesmaximal, characteristic
3#70767C211yesmaximal, characteristic
4#70528C211yesmaximal, characteristic
5#70507C211yesmaximal, characteristic
6#70422C211yesmaximal, characteristic
7C3×#1301C211yesmaximal, characteristic
8#10455—31nomaximal, Sylow
9C2×#869C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×#824—21no—
15C2×#818—21no—
16C2×#816—21no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19#4850—22no—
20#4850—22no—
21#4554—21no—
22#4482—22no—
23#4481—22no—
24#4479—22no—
25#4456—22no—
26C2×#477—31no—
27#1801—31no—
28#1571—31no—
29#1549—31no—
30#1548—31no—
31#1479—31no—
32#1301D311yescharacteristic
33C22×Dic6—21no—
34C22×Dic6D411yescharacteristic
35C2×C3×M16—21no—
36C22×C3×C8—21no—
37C2×C3×C42C2311yescharacteristic
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3D411yescharacteristic
41C2×Dic3⋊C4—21no—
42C2×C4×Dic3—21no—
43C2×#208—21no—
44#869—21no—
45#869—21no—
46#869—21no—
47#869D412yes—
48C4×Dic6—21no—
49C4×Dic6—21no—
50C4×Dic6—21no—
51C4×Dic6—21no—
52C3×C4⋊C8—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8D412yes—
57C3×C22⋊C8—21no—
58C3×C4×C8—21no—
59#830—21no—
60#824—42no—
61#818—24no—
62#816—24no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)D412yes—
66C4×C3⋊C8—21no—
67C2×C4⋊Q8—31no—
68C2×C4×Q8—31no—
69#477—62no—
70#477—62no—
71#431—31no—
72#431C2×D311yescharacteristic
73C2×#107—61no—
74#423—31no—
75#423C2×D311yescharacteristic
76C2×C4⋊C8C2×D311yescharacteristic
77C2×#103—61no—
78C2×#101—61no—
79#343—61no—
80#335—62no—
81#333—62no—
82#332—62no—
83Q8⋊C8—62no—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4#4111yescharacteristic
86C22×C3×C4#4111yescharacteristic
87C22×Dic3—21no—
88C22×Dic3—21no—
89C22×Dic3—41no—
90C2×Dic6—21no—
91C2×Dic6—21no—
92C2×Dic6—21no—
93C2×Dic6—21no—
94C2×Dic6—21no—
95C2×Dic6—22no—
96C2×Dic6—41no—
97C2×Dic6—82no—
98C2×Dic6Q1612yes—
99C3×M16—41no—
100C3×M16—41no—
101C2×C3×C8—22no—
102C2×C3×C8—41no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C3×C42C2×D411yescharacteristic
107C3×C42C2×D411yescharacteristic
108C3×C42C2×D411yescharacteristic
109C3×C42#4111yescharacteristic
110C4⋊Dic3—22no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—22no—
114C4⋊Dic3—41no—
115C4⋊Dic3—41no—
116Dic3⋊C4—41no—
117Dic3⋊C4—41no—
118C4×Dic3—41no—
119C4×Dic3—41no—
120#208—41no—
121#208—41no—
122C2×C3⋊C8—22no—
123C2×C3⋊C8—41no—
124C22×Q8—31no—
125C22×Q8—61no—
126C2×M16—21no—
127C2×M16—61no—
128C22×C8—21no—
129C4⋊Q8—32no—
130C4⋊Q8—61no—
131C4⋊Q8—61no—
132C4⋊Q8—61no—
133C4×Q8—61no—
134C4×Q8—61no—
135C4×Q8—61no—
136C4×Q8—61no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4—61no—
141C2×C42—61no—
142C2×C42C22×D311yescharacteristic
143#107—64no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—21no—
147C4⋊C8—21no—
148C4⋊C8—32no—
149C4⋊C8—61no—
150C4⋊C8—61no—
151C4⋊C8D1212yes—
152#103—64no—
153#101—122no—
154C22⋊C8—21no—
155C22⋊C8—61no—
156C4×C8—21no—
157C4×C8—61no—
158C23×C3#12311yescharacteristic
159C2×C3×C4—22no—
160C2×C3×C4—41no—
161C2×C3×C4#12011yescharacteristic
162C2×C3×C4#12111yescharacteristic
163C2×C3×C4#12111yescharacteristic
164C2×C3×C4#12111yescharacteristic
165C2×C3×C4#12311yescharacteristic
166C2×C3×C4#12411yescharacteristic
167C2×C3×C4C2×Q1611yescharacteristic
168C2×C3×C4#13511yescharacteristic
169C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
170C2×C3×C4#13711yescharacteristic
171C2×Dic3—22no—
172C2×Dic3—41no—
173C2×Dic3—41no—
174C2×Dic3—41no—
175C2×Dic3—42no—
176C2×Dic3—82no—
177C2×Dic3—81no—
178Dic6—22no—
179Dic6—22no—
180Dic6—41no—
181Dic6—41no—
182Dic6—82no—
183Dic6—82no—
184C3×C8—42no—
185C3×C8—81no—
186C3×C8—81no—
187C3⋊C8—42no—
188C3⋊C8—81no—
189C2×Q8—32no—
190C2×Q8—61no—
191C2×Q8—61no—
192C2×Q8—61no—
193C2×Q8—62no—
194C2×Q8—61no—
195C2×Q8—61no—
196C2×Q8—121no—
197C2×Q8—242no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4—121no—
201C22×C4#23511yescharacteristic
202C22×C4D3×D411yescharacteristic
203C22×C4#23911yescharacteristic
204M16—41no—
205M16—41no—
206M16—121no—
207M16—121no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—62no—
214C2×C8—121no—
215C4⋊C4—62no—
216C4⋊C4—61no—
217C4⋊C4—61no—
218C4⋊C4—62no—
219C4⋊C4—121no—
220C4⋊C4—121no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C42—121no—
224C42—121no—
225C42C2×D1211yescharacteristic
226C42#23511yescharacteristic
227C42D3×D411yescharacteristic
228C42D3×D411yescharacteristic
229C22×C3—21no—
230C22×C3—21no—
231C22×C3#39411yescharacteristic
232C22×C3#48411yescharacteristic
233C22×C3#48511yescharacteristic
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4#44911yescharacteristic
238C3×C4#45111yescharacteristic
239C3×C4#46711yescharacteristic
240C3×C4#46811yescharacteristic
241C3×C4#47312yes—
242Dic3—42no—
243Dic3—82no—
244Dic3—81no—
245C23#89411yescharacteristic
246Q8—62no—
247Q8—62no—
248Q8—121no—
249Q8—121no—
250Q8—242no—
251Q8—242no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—62no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4—122no—
259C2×C4—242no—
260C2×C4—241no—
261C2×C4#87511yescharacteristic
262C2×C4#88211yescharacteristic
263C2×C4#88411yescharacteristic
264C2×C4#88411yescharacteristic
265C2×C4#89411yescharacteristic
266C2×C4#89511yescharacteristic
267C2×C4#91111yescharacteristic
268C2×C4#91511yescharacteristic
269C2×C4#91611yescharacteristic
270C2×C4D3×Q1611yescharacteristic
271C8—42no—
272C8—81no—
273C8—81no—
274C8—122no—
275C8—241no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3#203611yescharacteristic
279C2×C3#204711yescharacteristic
280C2×C3#204811yescharacteristic
281C22—21no—
282C22—21no—
283C22#467211yescharacteristic
284C22#486811yescharacteristic
285C22#488411yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—122no—
290C4—242no—
291C4—241no—
292C4#478011yescharacteristic
293C4#478111yescharacteristic
294C4#480911yescharacteristic
295C4#481311yescharacteristic
296C4#489612yes—
297C3#1045511yescharacteristic, complemented, Sylow
298C2—21no—
299C2—21no—
300C2#6882511yescharacteristic
301C2#6893411yescharacteristic
302C2#6901811yescharacteristic
subgroupG/H
centerC22#4672
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenter#1301D3
hypocenterC3#10455
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
297. C3#10455semidirect

Maximal subgroups

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

subgroupindex
C2×#48502
#707802
#707672
#705282
#705072
#704222
C3×#13012
#104553