Finite Group Databasebeta

← #10698969

Subgroups — #10698969

Order 768.

366 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 93 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#83367C211yesmaximal, characteristic
2C3×#1518C211yesmaximal, characteristic
3#71828C211yesmaximal, characteristic
4#70786C211yesmaximal, characteristic
5#70750C211yesmaximal, characteristic
6#69559C211yesmaximal, characteristic
7#67631C211yesmaximal, characteristic
8#10471—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#5094—21no—
14#5091—21no—
15#5063—22no—
16#5004—21no—
17#4995—21no—
18C2×#810—21no—
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21C2×C4⋊(C3⋊C8)C2211yescharacteristic
22C3×#328—22no—
23#4547—21no—
24#4546—21no—
25#4529—21no—
26#4469—22no—
27#4458—22no—
28#3078—31no—
29#1805—31no—
30#1571—31no—
31#1542—31no—
32#1518D311yescharacteristic
33#1461—31no—
34#1286—31no—
35C22×Dic6D411yescharacteristic
36C3×C4⋊Q8D411yescharacteristic
37C3×C4⋊Q8D411yescharacteristic
38C3×#125—21no—
39C3×C22⋊Q8—21no—
40C3×C22⋊Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8—21no—
43C3×#117—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46C2×C4⋊Dic3D411yescharacteristic
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C2×#208—21no—
50C22×C3⋊C8—21no—
51C4×Dic6—21no—
52C4×Dic6—21no—
53C4×Dic6—21no—
54C4×Dic6—21no—
55C3×C4⋊C8—21no—
56C3×C22⋊C8D412yes—
57C3×C4×C8—21no—
58C3×#95—21no—
59C3×#95—21no—
60#830—21no—
61#810—22no—
62#810—22no—
63#810—42no—
64#808—22no—
65#807—22no—
66C4⋊(C3⋊C8)—21no—
67C4⋊(C3⋊C8)—21no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)D412yes—
71C4×C3⋊C8—21no—
72#533C2×D311yescharacteristic
73C2×C4×Q8—31no—
74#479—62no—
75#477—61no—
76#475—61no—
77#431—31no—
78#431C2×D311yescharacteristic
79#428—61no—
80#423—31no—
81C2×C4⋊C8—31no—
82#419—61no—
83C2×#103—61no—
84#382C2×D311yescharacteristic
85#340—61no—
86#335—62no—
87#333—61no—
88#332—61no—
89#328—22no—
90#324—62no—
91C2×C3×Q8—21no—
92C2×C3×Q8—21no—
93C22×C3×C4—41no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C2×Dic6—21no—
100C2×Dic6—21no—
101C2×Dic6—21no—
102C2×Dic6—21no—
103C2×Dic6—21no—
104C2×Dic6Q1612yes—
105C2×C3×C8—22no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C42—41no—
116C3×C42C2×D411yescharacteristic
117C3×C42C2×D411yescharacteristic
118C3×C42#4111yescharacteristic
119C3×C42#4111yescharacteristic
120C4⋊Dic3—21no—
121C4⋊Dic3—21no—
122Dic3⋊C4—41no—
123Dic3⋊C4—41no—
124C4×Dic3—41no—
125C4×Dic3—41no—
126#208—41no—
127#208—41no—
128C2×C3⋊C8—22no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8—41no—
131C2×C3⋊C8—41no—
132C2×C3⋊C8—41no—
133C22×Q8—31no—
134C2×M16—61no—
135C22×C8—61no—
136C4⋊Q8D1211yescharacteristic
137C4⋊Q8D1211yescharacteristic
138#125—21no—
139C22⋊Q8—21no—
140C22⋊Q8—21no—
141C4×Q8—21no—
142C4×Q8—21no—
143C4×Q8—61no—
144C4×Q8—61no—
145C4×Q8—61no—
146C4×Q8—61no—
147#117—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—61no—
151C2×C42—61no—
152C2×C42C22×D311yescharacteristic
153#107—62no—
154#106—62no—
155C4⋊C8—21no—
156C4⋊C8—32no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161#103—62no—
162#103—62no—
163#103—122no—
164C22⋊C8—61no—
165C22⋊C8#6712yes—
166C4×C8—21no—
167C4×C8—61no—
168#95—21no—
169#95—21no—
170C23×C3#12111yescharacteristic
171C3×Q8—81no—
172C3×Q8—81no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—81no—
178C2×C3×C4—81no—
179C2×C3×C4—81no—
180C2×C3×C4#12011yescharacteristic
181C2×C3×C4#12111yescharacteristic
182C2×C3×C4#12111yescharacteristic
183C2×C3×C4#12311yescharacteristic
184C2×C3×C4#12311yescharacteristic
185C2×C3×C4#12411yescharacteristic
186C2×C3×C4C2×Q1611yescharacteristic
187C2×C3×C4#13511yescharacteristic
188C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
189C2×C3×C4#13711yescharacteristic
190C2×Dic3—22no—
191C2×Dic3—41no—
192C2×Dic3—41no—
193C2×Dic3—41no—
194C2×Dic3—81no—
195Dic6—22no—
196Dic6—22no—
197Dic6—41no—
198Dic6—41no—
199C3×C8—42no—
200C3⋊C8—42no—
201C3⋊C8—81no—
202C3⋊C8—81no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—32no—
206C2×Q8—61no—
207C2×Q8—61no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—61no—
211C22×C4—41no—
212C22×C4—61no—
213C22×C4—61no—
214C22×C4D3×D411yescharacteristic
215C22×C4#23711yescharacteristic
216C22×C4C2×#6711yescharacteristic
217M16—121no—
218M16—121no—
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—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—121no—
236C4⋊C4—121no—
237C22⋊C4—41no—
238C42—41no—
239C42—121no—
240C42—121no—
241C42C2×D1211yescharacteristic
242C42#23511yescharacteristic
243C42D3×D411yescharacteristic
244C42#23711yescharacteristic
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—81no—
255C3×C4—81no—
256C3×C4—161no—
257C3×C4#44911yescharacteristic
258C3×C4#45111yescharacteristic
259C3×C4#48411yescharacteristic
260C3×C4#48511yescharacteristic
261Dic3—42no—
262Dic3—81no—
263C23#93811yescharacteristic
264Q8—62no—
265Q8—62no—
266Q8—81no—
267Q8—81no—
268Q8—121no—
269Q8—121no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—62no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—241no—
282C2×C4#88211yescharacteristic
283C2×C4#88311yescharacteristic
284C2×C4#88511yescharacteristic
285C2×C4#88611yescharacteristic
286C2×C4#91411yescharacteristic
287C2×C4#91511yescharacteristic
288C2×C4D3×Q1611yescharacteristic
289C2×C4#91911yescharacteristic
290C2×C4#93011yescharacteristic
291C2×C4#93911yescharacteristic
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#496111yescharacteristic
304C22#517411yescharacteristic
305C22#518811yescharacteristic
306C4—21no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—81no—
311C4—81no—
312C4—122no—
313C4—161no—
314C4—241no—
315C4#480911yescharacteristic
316C4#481011yescharacteristic
317C4#481111yescharacteristic
318C4#481411yescharacteristic
319C3#1047111yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2#7004311yescharacteristic
323C2#7024411yescharacteristic
324C2#7040311yescharacteristic
subgroupG/H
centerC22#4961
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1518C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1518D3
hypocenterC3#10471
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

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
#833672
C3×#15182
#718282
#707862
#707502
#695592
#676312
#104713