Finite Group Databasebeta

← #10684249

Subgroups — #10684249

Order 768.

374 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 83 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#76129C211yesmaximal, characteristic
2C3×#2083C211yesmaximal, characteristic
3#69571C211yesmaximal, characteristic
4#69569C211yesmaximal, characteristic
5#67638C211yesmaximal, characteristic
6#67478C211yesmaximal, characteristic
7#67477C211yesmaximal, characteristic
8#7792—31nomaximal, Sylow
9C2×#889—21no—
10C2×C3×#107—21no—
11C3×#389C2211yescharacteristic
12C3×#384C2211yescharacteristic
13#4983—21no—
14#4982—21no—
15#4976—21no—
16C2×#807—21no—
17#4928C2211yescharacteristic
18#4928C2211yescharacteristic
19#4657—21no—
20#4654—21no—
21C3×#340—21no—
22C3×#336C2211yescharacteristic
23#4529—21no—
24#4529—21no—
25#4529—21no—
26#4524C2211yescharacteristic
27#4524C2211yescharacteristic
28#2726—31no—
29#2099—31no—
30#2083—31no—
31#2083D311yescharacteristic
32#1271—31no—
33#1271—31no—
34#1271—31no—
35C22×C3×C8—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4D411yescharacteristic
42C2×C3×C42C2311yescharacteristic
43C2×C4⋊Dic3D411yescharacteristic
44C2×C4⋊Dic3D411yescharacteristic
45C2×C4⋊Dic3D411yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—21no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×C4×Dic3—21no—
54C22×C3⋊C8—21no—
55C22×C3⋊C8—21no—
56#889—42no—
57#889—42no—
58C3×#107—42no—
59C3×#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#807—42no—
66C2×#125—61no—
67C2×#107—21no—
68C2×#107—61no—
69#398—61no—
70#398—61no—
71#398—61no—
72#389—31no—
73#389—31no—
74#389—61no—
75#389C2×D311yescharacteristic
76#387—61no—
77#384C2×D311yescharacteristic
78#340—21no—
79#340—61no—
80#340—61no—
81#340—61no—
82#336—31no—
83#336—31no—
84#336C2×D311yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—41no—
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×Dic3—21no—
92C22×Dic3—21no—
93C22×Dic3—21no—
94C22×Dic3—21no—
95C2×C3×C8—22no—
96C2×C3×C8—41no—
97C2×C3×C8—41no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—42no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C42—41no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3—41no—
112C4⋊Dic3—41no—
113C4⋊Dic3D812yes—
114C4⋊Dic3Q1612yes—
115Dic3⋊C4—42no—
116Dic3⋊C4—42no—
117Dic3⋊C4—42no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120Dic3⋊C4—41no—
121Dic3⋊C4—41no—
122Dic3⋊C4—81no—
123C4×Dic3—42no—
124C2×C3⋊C8—22no—
125C2×C3⋊C8—41no—
126C2×C3⋊C8—41no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C22×C8—21no—
131C22×C8—61no—
132C22×C8—61no—
133#125—122no—
134#125—122no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4#6711yescharacteristic
150C2×C4⋊C4#6711yescharacteristic
151C2×C42—61no—
152C2×C42C22×D311yescharacteristic
153#107—42no—
154#107—122no—
155#95—21no—
156#95—61no—
157#95—61no—
158#95—61no—
159#95—61no—
160#95—61no—
161C23×C3#12011yescharacteristic
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—81no—
175C2×C3×C4—81no—
176C2×C3×C4—81no—
177C2×C3×C4C2×D811yescharacteristic
178C2×C3×C4C2×Q1611yescharacteristic
179C2×C3×C4#13511yescharacteristic
180C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
181C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
182C2×C3×C4#13711yescharacteristic
183C2×Dic3—22no—
184C2×Dic3—22no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C3×C8—42no—
196C3⋊C8—42no—
197C3⋊C8—81no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4—61no—
202C22×C4—61no—
203C22×C4—61no—
204C22×C4—61no—
205C22×C4D3×D411yescharacteristic
206C22×C4D3×D411yescharacteristic
207C22×C4C2×#6711yescharacteristic
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—62no—
212C2×C8—121no—
213C2×C8—121no—
214C2×C8—121no—
215C2×C8—121no—
216C2×C8—121no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—32no—
220C4⋊C4—32no—
221C4⋊C4—41no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C4⋊C4—121no—
232C4⋊C4—121no—
233C4⋊C4—122no—
234C4⋊C4—122no—
235C4⋊C4—122no—
236C4⋊C4—121no—
237C4⋊C4—121no—
238C4⋊C4—121no—
239C4⋊C4—121no—
240C4⋊C4—241no—
241C42—41no—
242C42—122no—
243C22×C3#44711yescharacteristic
244C22×C3#44911yescharacteristic
245C22×C3#45111yescharacteristic
246C22×C3#45411yescharacteristic
247C22×C3#45611yescharacteristic
248C22×C3#45611yescharacteristic
249C22×C3#45811yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—81no—
257C3×C4—81no—
258C3×C4—161no—
259Dic3—42no—
260Dic3—42no—
261Dic3—81no—
262Dic3—81no—
263C23#93711yescharacteristic
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4D3×D811yescharacteristic
292C2×C4#91111yescharacteristic
293C2×C4D3×Q1611yescharacteristic
294C2×C4#91811yescharacteristic
295C2×C4#93211yescharacteristic
296C2×C4#95011yescharacteristic
297C8—42no—
298C8—122no—
299C8—241no—
300C2×C3#160911yescharacteristic
301C2×C3#161011yescharacteristic
302C2×C3#161711yescharacteristic
303C2×C3#162511yescharacteristic
304C2×C3#164811yescharacteristic
305C2×C3#164911yescharacteristic
306C2×C3#166311yescharacteristic
307C22#503711yescharacteristic
308C22#505311yescharacteristic
309C22#509511yescharacteristic
310C22#515611yescharacteristic
311C22#518611yescharacteristic
312C22#520111yescharacteristic
313C22#520511yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—81no—
321C4—81no—
322C4—122no—
323C4—122no—
324C4—161no—
325C4—241no—
326C4—241no—
327C3#779211yescharacteristic, complemented, Sylow
328C2#6971111yescharacteristic
329C2#7101211yescharacteristic
330C2#7106411yescharacteristic
331C2#7116711yescharacteristic
332C2#7117911yescharacteristic
333C2#7137611yescharacteristic
334C2#7139311yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2083C2
socle, prime radicalC23×C3#120
hypercenter#2083D3
hypocenterC3#7792
LCSG ▷ C2×C3×C42 ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2083

Complements

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

NH ≅ G/Ntype
327. C3#7792semidirect

Maximal subgroups

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

subgroupindex
#761292
C3×#20832
#695712
#695692
#676382
#674782
#674772
#77923