Finite Group Databasebeta

← #10695094

Subgroups — #10695094

Order 768.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#83344C211yesmaximal, characteristic
2C3×#1994C211yesmaximal, characteristic
3#71826C211yesmaximal, characteristic
4#70501C211yesmaximal, characteristic
5#70490C211yesmaximal, characteristic
6#69396C211yesmaximal, characteristic
7C4×#803C211yesmaximal, characteristic
8#9850—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C3×C43C2211yescharacteristic
13#5093C411yescharacteristic, complemented
14#5092C411yescharacteristic, complemented
15#5079C411yescharacteristic, complemented
16#5078C411yescharacteristic, complemented
17#5051—22no—
18C4×#216—22no—
19#4924C2211yescharacteristic
20C2×#803—22no—
21C2×C4×C3⋊C8C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×#339—22no—
24#4535—22no—
25#4523—22no—
26#4484—22no—
27#4480—22no—
28#3170—31no—
29#1994—31no—
30#1994D311yescharacteristic
31C4×#97—31no—
32#1566—31no—
33#1493—31no—
34#1493—31no—
35C2×C3×M16—21no—
36C3×C4⋊Q8C2×C411yescharacteristic
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×#117—21no—
44C2×C3×C42—21no—
45C2×C3×C42—21no—
46C2×C3×C42D412yes—
47C2×C3×C42C2311yescharacteristic
48C2×#216—22no—
49#926—21no—
50#924—21no—
51#924—21no—
52C2×#208—21no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8D412yes—
55#870—21no—
56#869C2×C411yescharacteristic
57#869C2×C411yescharacteristic
58C4×Dic6—21no—
59C4×Dic6—21no—
60C3×C4⋊C8—21no—
61C3×C4⋊C8—21no—
62#835—22no—
63#810—22no—
64#807—22no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)—21no—
67#803—28no—
68C4×C3⋊C8—21no—
69C4×C3⋊C8C2×C411yescharacteristic
70C4×C3⋊C8C2×C411yescharacteristic
71C4×C3⋊C8D412yes—
72C4×C3⋊C8D412yes—
73#533—31no—
74#533C2×D311yescharacteristic
75#500—31no—
76#494—31no—
77#487—31no—
78#487—31no—
79#467—62no—
80C4×Q16—62no—
81#423—31no—
82#423C2×D311yescharacteristic
83C2×#97—62no—
84C2×C4×C8—31no—
85C2×C4×C8—31no—
86C43C2×D311yescharacteristic
87#339—22no—
88#339—62no—
89#334—62no—
90Q8⋊C8—62no—
91Q8⋊C8—62no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C22×C3×C4—22no—
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4111yescharacteristic
98C2×Dic6—21no—
99C2×Dic6—21no—
100C3×M16—42no—
101C2×C3×C8—41no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C22⋊C4—41no—
109C3×C42—24no—
110C3×C42—22no—
111C3×C42—22no—
112C3×C42—42no—
113C3×C42—41no—
114C3×C42C22×C411yescharacteristic
115C3×C42C2×D411yescharacteristic
116C3×C42C2×D411yescharacteristic
117C3×C42#4111yescharacteristic
118#217—41no—
119#216—44no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124Dic3⋊C4—41no—
125Dic3⋊C4—41no—
126C4×Dic3—41no—
127#208—42no—
128C2×C3⋊C8—24no—
129C2×C3⋊C8—22no—
130C2×C3⋊C8—24no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8C22⋊C412yes—
133C2×C3⋊C8C22⋊C412yes—
134C2×Q16—62no—
135C2×M16—21no—
136C2×M16—61no—
137C22×C8—32no—
138C22×C8—61no—
139C4⋊Q8—31no—
140C4⋊Q8—31no—
141C4⋊Q8C4×D311yescharacteristic
142C4⋊Q8C4×D311yescharacteristic
143#125—21no—
144#125—61no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8—61no—
148C22⋊Q8—61no—
149C4×Q8—21no—
150C4×Q8—21no—
151C4×Q8—61no—
152C4×Q8—61no—
153#117—21no—
154#117—61no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42#6712yes—
158C2×C42C22×D311yescharacteristic
159#107—62no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—61no—
163C4⋊C8—61no—
164#103—62no—
165#103—62no—
166#97—68no—
167C4×C8—31no—
168C4×C8—31no—
169C4×C8—32no—
170C4×C8—32no—
171C4×C8—61no—
172C23×C3#12111yescharacteristic
173C3×Q8—42no—
174C3×Q8—42no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—24no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4C2×C22⋊C411yescharacteristic
183C2×C3×C4C4×D411yescharacteristic
184C2×C3×C4C4×D411yescharacteristic
185C2×C3×C4#12111yescharacteristic
186C2×C3×C4#12411yescharacteristic
187C2×C3×C4#12711yescharacteristic
188C2×C3×C4#13712yes—
189C2×C3×C4#13712yes—
190C2×Dic3—41no—
191C2×Dic3—41no—
192Dic6—42no—
193Dic6—42no—
194C3×C8—81no—
195C3⋊C8—24no—
196C3⋊C8—22no—
197C3⋊C8—81no—
198C3⋊C8#10414yes—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—61no—
202C2×Q8—61no—
203C22×C4—22no—
204C22×C4D3×D411yescharacteristic
205C22×C4#23711yescharacteristic
206C22×C4C2×#6711yescharacteristic
207Q16—124no—
208M16—42no—
209M16—122no—
210C2×C8—32no—
211C2×C8—32no—
212C2×C8—41no—
213C2×C8—64no—
214C2×C8—64no—
215C2×C8—62no—
216C2×C8—121no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—61no—
224C4⋊C4—61no—
225C4⋊C4—61no—
226C4⋊C4—61no—
227C4⋊C4—121no—
228C4⋊C4—121no—
229C22⋊C4—41no—
230C22⋊C4—121no—
231C42—24no—
232C42—22no—
233C42—22no—
234C42—42no—
235C42—41no—
236C42—121no—
237C42C2×C4×D311yescharacteristic
238C42#23511yescharacteristic
239C42D3×D411yescharacteristic
240C42D3×D411yescharacteristic
241C22×C3—21no—
242C22×C3—21no—
243C22×C3#39011yescharacteristic
244C22×C3#47012yes—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—44no—
248C3×C4—81no—
249C3×C4—81no—
250C3×C4C2×#10412yes—
251C3×C4#44112yes—
252C3×C4#44412yes—
253C3×C4#49711yescharacteristic
254C3×C4#49711yescharacteristic
255Dic3—81no—
256Dic3—81no—
257C23#93911yescharacteristic
258Q8—42no—
259Q8—42no—
260Q8—122no—
261Q8—122no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—121no—
270C2×C4—121no—
271C2×C4D3×C22⋊C411yescharacteristic
272C2×C4#67⋊C411yescharacteristic
273C2×C4#88411yescharacteristic
274C2×C4#88511yescharacteristic
275C2×C4C4×#6711yescharacteristic
276C2×C4#94011yescharacteristic
277C2×C4#94212yes—
278C2×C4#95112yes—
279C8—34no—
280C8—64no—
281C8—62no—
282C8—81no—
283C8—241no—
284C2×C3—22no—
285C2×C3#196211yescharacteristic
286C2×C3#197111yescharacteristic
287C2×C3#197511yescharacteristic
288C22—21no—
289C22—21no—
290C22#495711yescharacteristic
291C22#505212yes—
292C4—21no—
293C4—21no—
294C4—44no—
295C4—81no—
296C4—81no—
297C4—241no—
298C4—241no—
299C4D3×#10412yes—
300C4#482312yes—
301C4#503012yes—
302C4#506011yescharacteristic
303C4#508611yescharacteristic
304C3#985011yescharacteristic, complemented, Sylow
305C2—22no—
306C2#6967811yescharacteristic
307C2#7035611yescharacteristic
308C2#7037411yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1994D3
hypocenterC3#9850
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
13. #5093C4semidirect
14. #5092C4semidirect
15. #5079C4semidirect
16. #5078C4semidirect
304. C3#9850semidirect

Maximal subgroups

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

subgroupindex
#833442
C3×#19942
#718262
#705012
#704902
#693962
C4×#8032
#98503