Finite Group Databasebeta

← #10698859

Subgroups — #10698859

Order 768.

376 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 99 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×#5021C211yesmaximal, characteristic
2C3×#1479C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic
5#70779C211yesmaximal, characteristic
6#70750C211yesmaximal, characteristic
7#67745C211yesmaximal, characteristic
8#10455—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×#836—21no—
14C2×#835—21no—
15#5021—22no—
16#5021—22no—
17C2×#807—21no—
18#4924C2211yescharacteristic
19#4924C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C3×Q8⋊C8—22no—
22#4530—21no—
23#4482—22no—
24#4481—22no—
25#4458—22no—
26C2×#477—31no—
27#1801—31no—
28#1571—31no—
29#1549—31no—
30#1548—31no—
31#1479D311yescharacteristic
32#1301—31no—
33C22×C3×Q8D411yescharacteristic
34C22×Dic6—21no—
35C3×C4×Q8—21no—
36C3×C4×Q8—21no—
37C3×C4×Q8—21no—
38C3×C4×Q8—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×#208—21no—
46C2×#208—21no—
47C22×C3⋊C8—21no—
48#869—21no—
49#869—21no—
50#869—21no—
51#869D412yes—
52C3×C4⋊C8—21no—
53C3×C4⋊C8D412yes—
54C3×C22⋊C8—21no—
55C3×C22⋊C8—21no—
56C3×C4×C8—21no—
57C3×C4×C8—21no—
58#836—42no—
59#835—24no—
60#807—24no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)D412yes—
67C2×C4⋊Q8—31no—
68C2×C4×Q8C2×D311yescharacteristic
69#477—62no—
70#477—62no—
71#431C2×D311yescharacteristic
72#431C2×D311yescharacteristic
73C2×#107—61no—
74#423—31no—
75#423—31no—
76C2×C4⋊C8—31no—
77C2×#103—61no—
78C2×#101—61no—
79#343—61no—
80#335—62no—
81#333—62no—
82#332—62no—
83Q8⋊C8—22no—
84C2×C3×Q8—21no—
85C2×C3×Q8—21no—
86C2×C3×Q8—21no—
87C2×C3×Q8—21no—
88C2×C3×Q8—21no—
89C2×C3×Q8Q1612yes—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4111yescharacteristic
94C22×C3×C4#4111yescharacteristic
95C22×Dic3—41no—
96C2×Dic6—22no—
97C2×Dic6—41no—
98C2×Dic6—82no—
99C2×C3×C8—22no—
100C2×C3×C8—41no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C42—41no—
106C3×C42—41no—
107C3×C42C2×D411yescharacteristic
108C3×C42C2×D411yescharacteristic
109C3×C42C2×D411yescharacteristic
110C3×C42#4111yescharacteristic
111C4⋊Dic3—22no—
112C4⋊Dic3—22no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115#208—41no—
116#208—41no—
117#208—41no—
118#208—41no—
119C2×C3⋊C8—22no—
120C2×C3⋊C8—41no—
121C2×C3⋊C8—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8—41no—
124C22×Q8—61no—
125C22×Q8D1211yescharacteristic
126C2×M16—61no—
127C2×M16—61no—
128C22×C8—61no—
129C4⋊Q8—32no—
130C4⋊Q8—61no—
131C4⋊Q8—61no—
132C4⋊Q8—61no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4D1211yescharacteristic
141C2×C42—21no—
142C2×C42C22×D311yescharacteristic
143#107—64no—
144C4⋊C8—21no—
145C4⋊C8—32no—
146C4⋊C8—61no—
147C4⋊C8—61no—
148C4⋊C8—61no—
149C4⋊C8—61no—
150C4⋊C8—61no—
151C4⋊C8#6712yes—
152#103—64no—
153#101—122no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156C4×C8—21no—
157C4×C8—21no—
158C23×C3#12311yescharacteristic
159C3×Q8—22no—
160C3×Q8—22no—
161C3×Q8—41no—
162C3×Q8—41no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—81no—
170C2×C3×C4#12011yescharacteristic
171C2×C3×C4#12111yescharacteristic
172C2×C3×C4#12111yescharacteristic
173C2×C3×C4#12111yescharacteristic
174C2×C3×C4#12311yescharacteristic
175C2×C3×C4#12411yescharacteristic
176C2×C3×C4C2×Q1611yescharacteristic
177C2×C3×C4#13511yescharacteristic
178C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
179C2×C3×C4#13711yescharacteristic
180C2×Dic3—42no—
181C2×Dic3—82no—
182Dic6—82no—
183Dic6—82no—
184C3×C8—42no—
185C3×C8—81no—
186C3⋊C8—42no—
187C3⋊C8—81no—
188C3⋊C8—81no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8—62no—
195C2×Q8—121no—
196C2×Q8—242no—
197C2×Q8Dic1212yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—121no—
201C22×C4C2×D1211yescharacteristic
202C22×C4#23711yescharacteristic
203C22×C4#23711yescharacteristic
204M16—121no—
205M16—121no—
206M16—121no—
207M16—121no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—62no—
211C2×C8—121no—
212C2×C8—121no—
213C2×C8—121no—
214C2×C8—121no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—62no—
220C4⋊C4—62no—
221C4⋊C4—121no—
222C4⋊C4—121no—
223C42—41no—
224C42—41no—
225C42#23511yescharacteristic
226C42D3×D411yescharacteristic
227C42D3×D411yescharacteristic
228C42C2×#6711yescharacteristic
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—42no—
238C3×C4—81no—
239C3×C4#44911yescharacteristic
240C3×C4#45111yescharacteristic
241C3×C4#46711yescharacteristic
242C3×C4#46811yescharacteristic
243C3×C4#47312yes—
244Dic3—82no—
245C23#88611yescharacteristic
246Q8—22no—
247Q8—22no—
248Q8—41no—
249Q8—41no—
250Q8—242no—
251Q8—242no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—81no—
259C2×C4—122no—
260C2×C4—242no—
261C2×C4#88211yescharacteristic
262C2×C4#88311yescharacteristic
263C2×C4#88511yescharacteristic
264C2×C4#90411yescharacteristic
265C2×C4C2×Dic1211yescharacteristic
266C2×C4#90811yescharacteristic
267C2×C4#90911yescharacteristic
268C2×C4#93011yescharacteristic
269C2×C4#93811yescharacteristic
270C2×C4#93911yescharacteristic
271C8—42no—
272C8—81no—
273C8—122no—
274C8—241no—
275C8—241no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3#203611yescharacteristic
279C2×C3#204711yescharacteristic
280C2×C3#204811yescharacteristic
281C22—21no—
282C22—21no—
283C22#472211yescharacteristic
284C22#472311yescharacteristic
285C22#496111yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—42no—
290C4—81no—
291C4—242no—
292C4#473411yescharacteristic
293C4#473911yescharacteristic
294C4#511611yescharacteristic
295C4#513411yescharacteristic
296C4#520412yes—
297C3#1045511yescharacteristic, complemented, Sylow
298C2—21no—
299C2—21no—
300C2#6993711yescharacteristic
301C2#7034011yescharacteristic
302C2#7040211yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenter#1479D3
hypocenterC3#10455
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

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×#50212
C3×#14792
#716372
#707802
#707792
#707502
#677452
#104553