Finite Group Databasebeta

← #10692886

Subgroups — #10692886

Order 768.

554 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#2973C211yesmaximal, characteristic
2#82379C211yesmaximal, characteristic
3#71603C211yesmaximal, characteristic
4#70823C212yesmaximal, complemented
5#67750C212yesmaximal, complemented
6#9455—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#924—21no—
9C2×#924—21no—
10C2×#869C2211yescharacteristic
11C3×#435—24no—
12C3×#432C2212yes—
13C3×#407—21no—
14C2×C3×C22⋊C8—21no—
15#5221C412yes—
16#4975C412yes—
17#4949—21no—
18#4949—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4926C2212yes—
21#4525—21no—
22#4525—21no—
23#4482—24no—
24#2973D311yescharacteristic
25#2542—31no—
26#1556—32no—
27#1466—31no—
28#1285—32no—
29C22×C3×D4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×Dic6—21no—
33C2×C3×M16—22no—
34C22×C3×C8—22no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×#217—22no—
42C2×#217—22no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3C2×C412yes—
46#924—44no—
47#924—44no—
48C2×Dic3⋊C4—22no—
49C2×Dic3⋊C4—22no—
50C2×C4×Dic3C2×C412yes—
51#869—22no—
52#869—22no—
53C3×C4⋊C8D414yes—
54C3×C22⋊C8—22no—
55C3×C22⋊C8—22no—
56C3×C22⋊C8—22no—
57C3×#97—22no—
58C3⋊#95—21no—
59#830—22no—
60C4⋊(C3⋊C8)—22no—
61#803—22no—
62C2×C4⋊Q8—31no—
63C2×C22⋊Q8—61no—
64C2×C22⋊Q8—61no—
65C2×C4×D4C2×D311yescharacteristic
66#435—24no—
67#432—32no—
68#432C2×D312yes—
69#407—21no—
70C2×C22⋊C8—21no—
71#399—61no—
72#399—61no—
73#390—32no—
74#384—32no—
75C4×C4⋊C4—31no—
76#332—64no—
77#323—61no—
78#323—61no—
79C24×C3C22⋊C411yescharacteristic
80C24×C3C22⋊C411yescharacteristic
81C2×C3×D4—22no—
82C2×C3×D4—22no—
83C2×C3×D4—41no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—42no—
89C22×C3×C4—42no—
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×Dic3—22no—
96C22×Dic3—22no—
97C2×Dic6—42no—
98C2×Dic6—42no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C3×M16—82no—
102C2×C3×C8—24no—
103C2×C3×C8—82no—
104C3×C4⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C22⋊C4—22no—
107C3×C42C2×D412yes—
108C3×C42#4112yes—
109#217—44no—
110#217—44no—
111C4⋊Dic3—24no—
112C4⋊Dic3—24no—
113C4⋊Dic3—42no—
114Dic3⋊C4—44no—
115Dic3⋊C4—44no—
116C4×Dic3—42no—
117C2×C3⋊C8—42no—
118C22×Q8—61no—
119C22×D4C4×D311yescharacteristic
120C23×C4#6711yescharacteristic
121C23×C4#6711yescharacteristic
122C2×M16—22no—
123C22×C8—22no—
124C4⋊Q8—62no—
125C4⋊Q8—62no—
126C22⋊Q8—124no—
127C22⋊Q8—124no—
128C4×D4—22no—
129C4×D4—22no—
130C2×C4⋊C4—32no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4C4×D311yescharacteristic
136C2×C22⋊C4—62no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4D1211yescharacteristic
139C2×C22⋊C4D1211yescharacteristic
140C2×C42—32no—
141C2×C42C22×D311yescharacteristic
142C4⋊C8—62no—
143C4⋊C8#6714yes—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—62no—
148#97—22no—
149#97—62no—
150#95—61no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3#10112yes—
161C23×C3C2×C22⋊C411yescharacteristic
162C3×D4—44no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—22no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—41no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—81no—
174C2×C3×C4—81no—
175C2×C3×C4C4×D412yes—
176C2×C3×C4#12011yescharacteristic
177C2×C3×C4#12112yes—
178C2×C3×C4#12311yescharacteristic
179C2×Dic3—24no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184C2×Dic3—42no—
185Dic6—82no—
186Dic6—82no—
187C3×C8—44no—
188C3⋊C8—82no—
189C24(C2×D3)⋊C411yescharacteristic
190C24(C2×D3)⋊C411yescharacteristic
191C2×Q8—122no—
192C2×Q8—122no—
193C2×Q8—121no—
194C2×Q8—121no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—41no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4(C2×D3)⋊C411yescharacteristic
207C22×C4(C2×D3)⋊C411yescharacteristic
208C22×C4C2×C4×D311yescharacteristic
209C22×C4C2×D1211yescharacteristic
210C22×C4C2×#6711yescharacteristic
211M16—82no—
212C2×C8—24no—
213C2×C8—82no—
214C2×C8—122no—
215C4⋊C4—22no—
216C4⋊C4—64no—
217C4⋊C4—64no—
218C4⋊C4—124no—
219C4⋊C4—124no—
220C4⋊C4—122no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—124no—
224C22⋊C4—124no—
225C42—122no—
226C42#23512yes—
227C42C2×#6712yes—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—22no—
231C22×C3—42no—
232C22×C3—42no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3—81no—
239C22×C3—81no—
240C22×C3#38611yescharacteristic
241C22×C3C2×#10111yescharacteristic
242C22×C3#41311yescharacteristic
243C3×C4—21no—
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—41no—
247C3×C4—41no—
248C3×C4#45312yes—
249Dic3—44no—
250Dic3—82no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#82412yes—
261C23C2×(C2×D3)⋊C411yescharacteristic
262Q8—242no—
263Q8—242no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—64no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4C4×#6712yes—
284C2×C4#92911yescharacteristic
285C2×C4#93012yes—
286C2×C4#95311yescharacteristic
287C8—44no—
288C8—242no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—42no—
292C2×C3—81no—
293C2×C3#188711yescharacteristic
294C2×C3#190111yescharacteristic
295C2×C3#190911yescharacteristic
296C22—21no—
297C22—21no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22—81no—
308C22#513211yescharacteristic
309C22C2×#82411yescharacteristic
310C22#521311yescharacteristic
311C4—22no—
312C4—22no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—124no—
317C4—242no—
318C4#524112yes—
319C3#945511yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#7165011yescharacteristic
325C2#7200311yescharacteristic
326C2#7202411yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2973D3
hypocenterC3#9455
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #70823C2semidirect
5. #67750C2semidirect
319. C3#9455semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823792
#716032
#708232
#677502
#94553