Finite Group Databasebeta

← #10695096

Subgroups — #10695096

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×#1493C211yesmaximal, characteristic
3#71839C211yesmaximal, characteristic
4#70514C211yesmaximal, characteristic
5#70490C211yesmaximal, characteristic
6C4×#815C211yesmaximal, characteristic
7#69738C211yesmaximal, characteristic
8#9850—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#815—22no—
14#5093C411yescharacteristic, complemented
15#5092C411yescharacteristic, complemented
16#5079C411yescharacteristic, complemented
17#5078C411yescharacteristic, complemented
18#5051—22no—
19C4×#216—22no—
20C42×Dic3C2211yescharacteristic
21#4924C2211yescharacteristic
22C2×C4×C3⋊C8C2211yescharacteristic
23C3×Q8⋊C8—22no—
24#4555—22no—
25#4532—22no—
26#4480—22no—
27#4456—22no—
28#3170—31no—
29#1994—31no—
30#1994—31no—
31C4×#97—31no—
32#1566—31no—
33#1493—31no—
34#1493D311yescharacteristic
35C2×C3×M16—21no—
36C22×C3×C8—21no—
37C3×C4⋊Q8C2×C411yescharacteristic
38C3×C4⋊Q8C2×C411yescharacteristic
39C3×#125—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×#117—21no—
45C2×C3×C42C2311yescharacteristic
46C2×#216—22no—
47#926—21no—
48#924—21no—
49#924—21no—
50C2×C4×Dic3—21no—
51C2×C4×Dic3—21no—
52C2×C4×Dic3D412yes—
53C2×#208—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—
62C3×C4×C8—21no—
63C3×C4×C8D412yes—
64#835—22no—
65#815—28no—
66#810—22no—
67#807—22no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4×C3⋊C8C2×C411yescharacteristic
71C4×C3⋊C8C2×C411yescharacteristic
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×C8C2×D311yescharacteristic
86C43—31no—
87#339—62no—
88#339—62no—
89#334—62no—
90Q8⋊C8—22no—
91Q8⋊C8—62no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C22×Dic3—22no—
98C2×Dic6—21no—
99C2×Dic6—21no—
100C3×M16—42no—
101C2×C3×C8—24no—
102C2×C3×C8—22no—
103C2×C3×C8—41no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—41no—
109C3×C4⋊C4—41no—
110C3×C22⋊C4—41no—
111C3×C42—41no—
112C3×C42C22×C411yescharacteristic
113C3×C42C2×D411yescharacteristic
114C3×C42C2×D411yescharacteristic
115C3×C42#4111yescharacteristic
116#217—41no—
117#216—44no—
118C4⋊Dic3—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122Dic3⋊C4—41no—
123Dic3⋊C4—41no—
124C4×Dic3—24no—
125C4×Dic3—22no—
126C4×Dic3—22no—
127C4×Dic3—42no—
128C4×Dic3—41no—
129#208—42no—
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—21no—
138C22×C8—32no—
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—32no—
156C2×C42—61no—
157C2×C42—61no—
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—21no—
168C4×C8—31no—
169C4×C8—31no—
170C4×C8—32no—
171C4×C8#6712yes—
172C23×C3#12111yescharacteristic
173C3×Q8—42no—
174C3×Q8—42no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4C2×C22⋊C411yescharacteristic
180C2×C3×C4C4×D411yescharacteristic
181C2×C3×C4C4×D411yescharacteristic
182C2×C3×C4#12111yescharacteristic
183C2×C3×C4#12411yescharacteristic
184C2×C3×C4#12711yescharacteristic
185C2×C3×C4#13712yes—
186C2×C3×C4#13712yes—
187C2×Dic3—24no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192Dic6—42no—
193Dic6—42no—
194C3×C8—24no—
195C3×C8—81no—
196C3⋊C8—22no—
197C3⋊C8—81no—
198C3⋊C8#10414yes—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—61no—
202C2×Q8—61no—
203C22×C4—62no—
204C22×C4#23511yescharacteristic
205C22×C4D3×D411yescharacteristic
206C22×C4D3×D411yescharacteristic
207Q16—124no—
208M16—42no—
209M16—122no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—32no—
213C2×C8—32no—
214C2×C8—41no—
215C2×C8—64no—
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—41no—
232C42—64no—
233C42—62no—
234C42—62no—
235C42—122no—
236C42—121no—
237C42C2×C4×D311yescharacteristic
238C42D3×D411yescharacteristic
239C42#23711yescharacteristic
240C42C2×#6711yescharacteristic
241C22×C3—21no—
242C22×C3—21no—
243C22×C3#39011yescharacteristic
244C22×C3#47012yes—
245C3×C4—21no—
246C3×C4—21no—
247C3×C4—81no—
248C3×C4—81no—
249C3×C4C2×#10412yes—
250C3×C4#44112yes—
251C3×C4#44412yes—
252C3×C4#49711yescharacteristic
253C3×C4#49711yescharacteristic
254Dic3—44no—
255Dic3—81no—
256Dic3—81no—
257C23#88411yescharacteristic
258Q8—42no—
259Q8—42no—
260Q8—122no—
261Q8—122no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—64no—
267C2×C4—122no—
268C2×C4—122no—
269C2×C4—121no—
270C2×C4—121no—
271C2×C4D3×C22⋊C411yescharacteristic
272C2×C4#67⋊C411yescharacteristic
273C2×C4#88511yescharacteristic
274C2×C4#91512yes—
275C2×C4#91812yes—
276C2×C4C4×#6711yescharacteristic
277C2×C4#93911yescharacteristic
278C2×C4#94011yescharacteristic
279C8—24no—
280C8—34no—
281C8—62no—
282C8—81no—
283C8—241no—
284C2×C3—22no—
285C2×C3#196211yescharacteristic
286C2×C3#197111yescharacteristic
287C2×C3#197511yescharacteristic
288C22—21no—
289C22—21no—
290C22#481612yes—
291C22#495711yescharacteristic
292C4—21no—
293C4—21no—
294C4—81no—
295C4—81no—
296C4—124no—
297C4—241no—
298C4—241no—
299C4#478912yes—
300C4D3×#10412yes—
301C4#514512yes—
302C4#516711yescharacteristic
303C4#519011yescharacteristic
304C3#985011yescharacteristic, complemented, Sylow
305C2—22no—
306C2#7000011yescharacteristic
307C2#7025911yescharacteristic
308C2#7035611yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1493C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#1493D3
hypocenterC3#9850
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
14. #5093C4semidirect
15. #5092C4semidirect
16. #5079C4semidirect
17. #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×#14932
#718392
#705142
#704902
C4×#8152
#697382
#98503