Finite Group Databasebeta

← #10698967

Subgroups — #10698967

Order 768.

366 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3078C211yesmaximal, characteristic
2#71838C211yesmaximal, characteristic
3#71828C211yesmaximal, characteristic
4#71627C211yesmaximal, characteristic
5#71624C211yesmaximal, characteristic
6#70494C211yesmaximal, characteristic
7#67713C211yesmaximal, characteristic
8#10471—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C3×#479—22no—
12C3×#477—21no—
13C3×#475—21no—
14C3×#428—21no—
15C3×#423C2211yescharacteristic
16C3×#419—21no—
17C2×#835—21no—
18#4936C2211yescharacteristic
19#4925C2211yescharacteristic
20#4925C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22#4547—21no—
23#4546—21no—
24#4538—22no—
25#4529—21no—
26#4528—22no—
27#4479—22no—
28#3078D311yescharacteristic
29#1805—31no—
30#1571—31no—
31#1542—31no—
32#1518—31no—
33#1461—31no—
34#1286—31no—
35C22×C3×Q8D411yescharacteristic
36C2×C3×M16—21no—
37C3×C4⋊Q8D411yescharacteristic
38C3×C4⋊Q8D411yescharacteristic
39C3×#125—21no—
40C3×C22⋊Q8—21no—
41C3×C22⋊Q8—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×C4⋊Dic3—21no—
54C22×C3⋊C8—21no—
55C3×#107—22no—
56C3×#106—22no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8D412yes—
59C3×#103—22no—
60C3×#103—22no—
61#835—42no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64#830—21no—
65#830D412yes—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4×C3⋊C8—21no—
71C4×C3⋊C8—21no—
72#533C2×D311yescharacteristic
73C2×C4×Q8C2×D311yescharacteristic
74#479—22no—
75#477—21no—
76#475—21no—
77#431—31no—
78#431—31no—
79#428—21no—
80#423C2×D311yescharacteristic
81C2×C4⋊C8—31no—
82#419—21no—
83C2×#103—61no—
84#382—31no—
85#340—61no—
86#335—62no—
87#333—61no—
88#332—61no—
89#328—62no—
90#324—62no—
91C2×C3×Q8—21no—
92C2×C3×Q8—21no—
93C2×C3×Q8—21no—
94C2×C3×Q8—21no—
95C2×C3×Q8—21no—
96C2×C3×Q8—21no—
97C2×C3×Q8—21no—
98C2×C3×Q8Q1612yes—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×Dic3—41no—
105C3×M16—41no—
106C3×M16—41no—
107C2×C3×C8—22no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4—21no—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C42—41no—
120C3×C42—41no—
121C3×C42—41no—
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42#4111yescharacteristic
125C3×C42#4111yescharacteristic
126C4⋊Dic3—41no—
127C4⋊Dic3—41no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C22×Q8#6711yescharacteristic
134C2×M16—21no—
135C22×C8—61no—
136C4⋊Q8#6711yescharacteristic
137C4⋊Q8#6711yescharacteristic
138#125—21no—
139C22⋊Q8—21no—
140C22⋊Q8—21no—
141C4×Q8—21no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—21no—
147#117—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4#6711yescharacteristic
151C2×C42—21no—
152C2×C42C22×D311yescharacteristic
153#107—22no—
154#106—22no—
155C4⋊C8—21no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8#6712yes—
161#103—22no—
162#103—22no—
163#103—122no—
164C22⋊C8—32no—
165C22⋊C8—61no—
166C4×C8—61no—
167C4×C8—61no—
168#95—61no—
169#95—61no—
170C23×C3#12111yescharacteristic
171C3×Q8—22no—
172C3×Q8—22no—
173C3×Q8—41no—
174C3×Q8—41no—
175C3×Q8—81no—
176C3×Q8—81no—
177C2×C3×C4—22no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—81no—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12311yescharacteristic
190C2×C3×C4#12311yescharacteristic
191C2×C3×C4#12411yescharacteristic
192C2×C3×C4C2×Q1611yescharacteristic
193C2×C3×C4#13511yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C3×C4#13711yescharacteristic
196C2×Dic3—81no—
197C2×Dic3—81no—
198C2×Dic3—81no—
199C3×C8—42no—
200C3⋊C8—42no—
201C3⋊C8—81no—
202C3⋊C8—81no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8#21612yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—121no—
214C22×C4D3×D411yescharacteristic
215C22×C4#23711yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—41no—
218M16—41no—
219C2×C8—22no—
220C2×C8—62no—
221C2×C8—121no—
222C2×C8—121no—
223C2×C8—121no—
224C2×C8—121no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C22⋊C4—41no—
238C42—41no—
239C42—41no—
240C42—41no—
241C42#23711yescharacteristic
242C42#23711yescharacteristic
243C42C2×#6711yescharacteristic
244C42C2×#6711yescharacteristic
245C22×C3—21no—
246C22×C3—21no—
247C22×C3#39411yescharacteristic
248C22×C3#46711yescharacteristic
249C22×C3#46811yescharacteristic
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—42no—
255C3×C4—81no—
256C3×C4—81no—
257C3×C4—81no—
258C3×C4#44911yescharacteristic
259C3×C4#45111yescharacteristic
260C3×C4#48411yescharacteristic
261C3×C4#48511yescharacteristic
262Dic3—161no—
263C23#93911yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—41no—
267Q8—41no—
268Q8—81no—
269Q8—81no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—81no—
279C2×C4—241no—
280C2×C4—241no—
281C2×C4—241no—
282C2×C4#93511yescharacteristic
283C2×C4#93511yescharacteristic
284C2×C4#93611yescharacteristic
285C2×C4#93811yescharacteristic
286C2×C4#93911yescharacteristic
287C2×C4#94211yescharacteristic
288C2×C4C2×#21611yescharacteristic
289C2×C4#94911yescharacteristic
290C2×C4#95011yescharacteristic
291C2×C4#95311yescharacteristic
292C8—42no—
293C8—122no—
294C8—241no—
295C8—241no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3#203611yescharacteristic
299C2×C3#204911yescharacteristic
300C2×C3#205311yescharacteristic
301C22—21no—
302C22—21no—
303C22#504911yescharacteristic
304C22#505111yescharacteristic
305C22#522211yescharacteristic
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—42no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—481no—
315C4#504411yescharacteristic
316C4#504511yescharacteristic
317C4#522811yescharacteristic
318C4#523811yescharacteristic
319C3#1047111yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2#7169511yescharacteristic
323C2#7210411yescharacteristic
324C2#7211511yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3078C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#3078D3
hypocenterC3#10471
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
319. C3#10471semidirect

Maximal subgroups

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

subgroupindex
C3×#30782
#718382
#718282
#716272
#716242
#704942
#677132
#104713