Finite Group Databasebeta

← #10690259

Subgroups — #10690259

Order 768.

484 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 103 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#82467C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3C3×C4×#101C211yesmaximal, characteristic
4#69857C212yesmaximal
5#67665C212yesmaximal
6#9030—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C4×Dic6C2211yescharacteristic
9C2×C3×#101—21no—
10C3×C4×M16C2212yes—
11#5230—21no—
12#5024—24no—
13#4983—21no—
14#4981C411yescharacteristic
15#4981C411yescharacteristic
16#4979C411yescharacteristic
17#4979C411yescharacteristic
18#4932C2211yescharacteristic
19C4×Dic3⋊C4—21no—
20C4×Dic3⋊C4—21no—
21#4926—21no—
22#4926C2212yes—
23#4925—21no—
24C3×#343—21no—
25#4456—24no—
26#2983—31no—
27#2399—31no—
28C4×#101D311yescharacteristic
29#1491—32no—
30#1300—32no—
31C22×C3×Q8D411yescharacteristic
32C22×Dic6C2×C411yescharacteristic
33C2×C3×M16—22no—
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×C4×Dic3—21no—
50C4×Dic6—22no—
51C4×Dic6—22no—
52C3×#101—24no—
53C3×#97—22no—
54C3×C4×C8—22no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95C2×C411yescharacteristic
61C3⋊#95C2×C411yescharacteristic
62#830—22no—
63C4⋊(C3⋊C8)—22no—
64C4⋊(C3⋊C8)—22no—
65C4⋊(C3⋊C8)D414yes—
66#803—22no—
67#803—22no—
68C4×C3⋊C8—22no—
69C2×C4×Q8—31no—
70C2×C4×Q8C2×D311yescharacteristic
71#446—64no—
72#432—32no—
73#432—61no—
74#431—61no—
75C2×#101—21no—
76C4×M16C2×D312yes—
77#398—31no—
78#398—31no—
79#398—61no—
80#395—31no—
81#395—31no—
82#395—61no—
83C4×C4⋊C4—61no—
84C4×C4⋊C4—61no—
85#376—31no—
86#343—21no—
87Q8⋊C8—64no—
88C2×C3×Q8—22no—
89C2×C3×Q8—22no—
90C2×C3×Q8—41no—
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C2×Dic6—22no—
103C2×Dic6—42no—
104C3×M16—24no—
105C3×M16—42no—
106C2×C3×C8—42no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C42—21no—
111C3×C42—21no—
112C3×C42—21no—
113C3×C42—21no—
114C3×C42C2×D412yes—
115C3×C42#4112yes—
116C4⋊Dic3—22no—
117Dic3⋊C4—42no—
118Dic3⋊C4—42no—
119Dic3⋊C4—42no—
120Dic3⋊C4—42no—
121Dic3⋊C4—42no—
122C4×Dic3—24no—
123C4×Dic3—41no—
124C4×Dic3—41no—
125C2×C3⋊C8—24no—
126C22×Q8—31no—
127C22×Q8#6711yescharacteristic
128C2×M16—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—62no—
132C4×Q8—62no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4—61no—
139C2×C4⋊C4C4×D311yescharacteristic
140C2×C4⋊C4C4×D311yescharacteristic
141C2×C4⋊C4#6711yescharacteristic
142C2×C42—61no—
143C2×C42—61no—
144C2×C42D1211yescharacteristic
145C2×C42D1211yescharacteristic
146C2×C42C22×D311yescharacteristic
147C4⋊C8—34no—
148C4⋊C8—62no—
149C4⋊C8—62no—
150#101—24no—
151C22⋊C8—62no—
152#97—22no—
153#97—62no—
154#97—62no—
155C4×C8—22no—
156C4×C8—62no—
157#95—31no—
158#95—31no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164C23×C3C2×C22⋊C411yescharacteristic
165C3×Q8—44no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—22no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4#10412yes—
180C2×C3×C4C4×D412yes—
181C2×C3×C4#12011yescharacteristic
182C2×C3×C4#12112yes—
183C2×C3×C4#12311yescharacteristic
184C2×Dic3—22no—
185C2×Dic3—22no—
186C2×Dic3—22no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196Dic6—24no—
197Dic6—42no—
198C3×C8—44no—
199C3⋊C8—44no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—41no—
203C2×Q8—62no—
204C2×Q8—122no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4(C2×D3)⋊C411yescharacteristic
210C22×C4(C2×D3)⋊C411yescharacteristic
211C22×C4(C2×D3)⋊C411yescharacteristic
212C22×C4(C2×D3)⋊C411yescharacteristic
213C22×C4C2×C4×D311yescharacteristic
214C22×C4C2×D1211yescharacteristic
215C22×C4C2×#6711yescharacteristic
216M16—24no—
217M16—42no—
218C2×C8—42no—
219C2×C8—64no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—62no—
224C4⋊C4—122no—
225C4⋊C4—122no—
226C4⋊C4—122no—
227C4⋊C4—122no—
228C4⋊C4—122no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—64no—
234C42—121no—
235C42—121no—
236C42D3×D412yes—
237C42#23712yes—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3#38611yescharacteristic
241C22×C3C2×#10411yescharacteristic
242C22×C3#42111yescharacteristic
243C3×C4—22no—
244C3×C4—22no—
245C3×C4—21no—
246C3×C4—22no—
247C3×C4—41no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4#45612yes—
251Dic3—42no—
252Dic3—42no—
253Dic3—42no—
254Dic3—81no—
255C23C2×(C2×D3)⋊C411yescharacteristic
256Q8—44no—
257Q8—64no—
258Q8—122no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4#80512yes—
285C2×C4#67⋊C412yes—
286C2×C4#88211yescharacteristic
287C2×C4#88611yescharacteristic
288C2×C4#93912yes—
289C8—44no—
290C8—124no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3#187111yescharacteristic
294C2×C3#188611yescharacteristic
295C2×C3#189111yescharacteristic
296C22—21no—
297C22—21no—
298C22C2×#80511yescharacteristic
299C22#495811yescharacteristic
300C22#500511yescharacteristic
301C4—22no—
302C4—22no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—122no—
309C4—122no—
310C4—122no—
311C4—241no—
312C4#475212yes—
313C3#903011yescharacteristic, complemented, Sylow
314C2—21no—
315C2—21no—
316C2#6945311yescharacteristic
317C2#7010311yescharacteristic
318C2#7013611yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#101C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenterC4×#101D3
hypocenterC3#9030
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
313. C3#9030semidirect

Maximal subgroups

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

subgroupindex
#824672
#824362
C3×C4×#1012
#698572
#676652
#90303