Finite Group Databasebeta

← #10691198

Subgroups — #10691198

Order 768.

504 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 127 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#82264C211yesmaximal, characteristic
2#82204C211yesmaximal, characteristic
3C3×#1458C211yesmaximal, characteristic
4#70780C212yesmaximal, complemented
5#67738C212yesmaximal, complemented
6#9193—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×#208—21no—
9C2×#869C2211yescharacteristic
10C3×#431C2212yes—
11#5212—21no—
12#4999—24no—
13#4986C412yes—
14#4979—21no—
15#4949C412yes—
16#4947—21no—
17#4936C2211yescharacteristic
18#4928—21no—
19#4924—21no—
20#4924C2212yes—
21C3×#324—21no—
22C3×#323—21no—
23#4482—24no—
24#2905—31no—
25#2628—31no—
26#1549—32no—
27#1458D311yescharacteristic
28#1286—32no—
29C23×C3×C4D411yescharacteristic
30C22×Dic6—21no—
31C3×#117—22no—
32C3×#117—22no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C22⋊C4C2×C411yescharacteristic
36C2×C3×C22⋊C4D411yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×C3×C42C2311yescharacteristic
39C2×#217—22no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3C2×C412yes—
43C2×Dic3⋊C4—22no—
44C2×#208—22no—
45C2×#208—22no—
46C2×#208—42no—
47C2×#208—42no—
48C22×C3⋊C8—22no—
49#869—22no—
50#869—22no—
51C3×C4⋊C8—22no—
52C3×C22⋊C8—22no—
53C3×C4×C8—22no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95—21no—
57C3⋊#95C2×C412yes—
58C4⋊(C3⋊C8)—24no—
59C4⋊(C3⋊C8)—22no—
60C4⋊(C3⋊C8)D414yes—
61C22×M16—61no—
62C2×C4⋊Q8—31no—
63C2×#117C2×D311yescharacteristic
64#431C2×D312yes—
65#424—64no—
66#423—32no—
67#423—61no—
68#400—32no—
69#399—32no—
70#395—61no—
71#393—61no—
72#389—61no—
73#382—31no—
74#380—61no—
75#332—64no—
76#324—21no—
77#323—21no—
78C24×C3C22⋊C411yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4SD1612yes—
87C22×C3×C4Q1612yes—
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×Dic3—22no—
92C22×Dic3—22no—
93C2×Dic6—42no—
94C2×Dic6—82no—
95C2×C3×C8—42no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—22no—
98C3×C22⋊C4—22no—
99C3×C22⋊C4—22no—
100C3×C42—21no—
101C3×C42—21no—
102C3×C42C2×D412yes—
103C3×C42#4112yes—
104#217—82no—
105C4⋊Dic3—24no—
106C4⋊Dic3—41no—
107C4⋊Dic3—41no—
108C4⋊Dic3—41no—
109C4⋊Dic3—41no—
110Dic3⋊C4—82no—
111#208—44no—
112#208—42no—
113#208—42no—
114C2×C3⋊C8—24no—
115C2×C3⋊C8—44no—
116C22×Q8—61no—
117C23×C4D1211yescharacteristic
118C2×M16—62no—
119C2×M16—62no—
120C2×M16—122no—
121C2×M16—122no—
122C22×C8—62no—
123C4⋊Q8—62no—
124C4⋊Q8—62no—
125#117—22no—
126#117—22no—
127C2×C4⋊C4—32no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4C4×D311yescharacteristic
132C2×C4⋊C4#6711yescharacteristic
133C2×C22⋊C4—62no—
134C2×C22⋊C4C4×D311yescharacteristic
135C2×C22⋊C4#6711yescharacteristic
136C2×C42D1211yescharacteristic
137C2×C42C22×D311yescharacteristic
138C4⋊C8—22no—
139C4⋊C8—34no—
140C4⋊C8—64no—
141C4⋊C8—62no—
142C22⋊C8—22no—
143C4×C8—22no—
144#95—32no—
145#95—62no—
146#95—62no—
147#95—61no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3#10312yes—
153C23×C3C2×C22⋊C411yescharacteristic
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4#10312yes—
167C2×C3×C4C4×D412yes—
168C2×C3×C4#12011yescharacteristic
169C2×C3×C4#12112yes—
170C2×C3×C4#12311yescharacteristic
171C2×C3×C4C2×SD1611yescharacteristic
172C2×C3×C4C2×Q1611yescharacteristic
173C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
174C2×C3×C4#13711yescharacteristic
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181Dic6—84no—
182C3×C8—82no—
183C3⋊C8—44no—
184C24(C2×D3)⋊C411yescharacteristic
185C2×Q8—122no—
186C2×Q8—242no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4#20412yes—
194C22×C4Dic1212yes—
195C22×C4(C2×D3)⋊C411yescharacteristic
196C22×C4(C2×D3)⋊C411yescharacteristic
197C22×C4(C2×D3)⋊C411yescharacteristic
198C22×C4C2×C4×D311yescharacteristic
199C22×C4C2×D1211yescharacteristic
200C22×C4C2×#6711yescharacteristic
201M16—124no—
202M16—122no—
203M16—122no—
204C2×C8—42no—
205C2×C8—64no—
206C2×C8—124no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—64no—
210C4⋊C4—121no—
211C4⋊C4—121no—
212C4⋊C4—121no—
213C4⋊C4—121no—
214C4⋊C4—242no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—242no—
218C42—21no—
219C42—21no—
220C42D3×D412yes—
221C42#23712yes—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—22no—
225C22×C3—42no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—81no—
229C22×C3#38611yescharacteristic
230C22×C3C2×#10311yescharacteristic
231C22×C3#41711yescharacteristic
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4#45011yescharacteristic
238C3×C4#45111yescharacteristic
239C3×C4#48311yescharacteristic
240C3×C4#48411yescharacteristic
241Dic3—82no—
242Dic3—82no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23#81612yes—
248C23C2×(C2×D3)⋊C411yescharacteristic
249Q8—244no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4#81612yes—
269C2×C4#67⋊C412yes—
270C2×C4#88211yescharacteristic
271C2×C4#88611yescharacteristic
272C2×C4C2×#20411yescharacteristic
273C2×C4C2×Dic1211yescharacteristic
274C2×C4#90811yescharacteristic
275C2×C4#90911yescharacteristic
276C2×C4#93912yes—
277C8—82no—
278C8—124no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3#187211yescharacteristic
283C2×C3#191011yescharacteristic
284C2×C3#191511yescharacteristic
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22#495811yescharacteristic
293C22C2×#81611yescharacteristic
294C22#513311yescharacteristic
295C4—22no—
296C4—22no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—242no—
301C4—242no—
302C4#472111yescharacteristic
303C4#472211yescharacteristic
304C4#473111yescharacteristic
305C4#473911yescharacteristic
306C3#919311yescharacteristic, complemented, Sylow
307C2—21no—
308C2—21no—
309C2—42no—
310C2#6993011yescharacteristic
311C2#7004711yescharacteristic
312C2#7013811yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1458C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1458D3
hypocenterC3#9193
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
4. #70780C2semidirect
5. #67738C2semidirect
306. C3#9193semidirect

Maximal subgroups

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

subgroupindex
#822642
#822042
C3×#14582
#707802
#677382
#91933