Finite Group Databasebeta

← #10699222

Subgroups — #10699222

Order 768.

390 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 95 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#83223C211yesmaximal, characteristic
2C3×#1988C211yesmaximal, characteristic
3#71789C211yesmaximal, characteristic
4#70438C211yesmaximal, characteristic
5#70420C211yesmaximal, characteristic
6#69493C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic
8#10507—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13C2×#836—21no—
14C2×#836—21no—
15#5094—21no—
16#5094—21no—
17#5093—21no—
18#5080—21no—
19#4971—22no—
20#4925C2211yescharacteristic
21#4924C2211yescharacteristic
22C4×#208C2211yescharacteristic
23C3×#340—22no—
24#4523—21no—
25#4523—21no—
26#4482—21no—
27#4481—21no—
28#4480—21no—
29#4479—21no—
30#3229—31no—
31#1988D311yescharacteristic
32#1986—31no—
33#1966—31no—
34#1565—31no—
35#1550—31no—
36#1303—31no—
37C22×C3×Q8—21no—
38C22×C3×C8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48#926—21no—
49#924—21no—
50#924—21no—
51C2×#208—21no—
52C2×#208D411yescharacteristic
53C2×#208D411yescharacteristic
54#870—21no—
55#869D411yescharacteristic
56#869D411yescharacteristic
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×#95—21no—
62#836—42no—
63#836—42no—
64#830—21no—
65#810—22no—
66#810—22no—
67#808—22no—
68#808—22no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)D411yescharacteristic
72C4⋊(C3⋊C8)D411yescharacteristic
73#803—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8—21no—
76#533—31no—
77C2×C4⋊Q8C2×D311yescharacteristic
78#498—61no—
79#487—61no—
80#483—62no—
81#475—61no—
82#475—61no—
83#431—31no—
84#423—31no—
85C2×C4⋊C8C2×D311yescharacteristic
86C2×#101—61no—
87C2×#101—61no—
88C4×M16—31no—
89C4×C4⋊C4C2×D311yescharacteristic
90#340—22no—
91#339—61no—
92#339—61no—
93#335—61no—
94#334—61no—
95#333—61no—
96#332—61no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C22×C3×C4—22no—
104C22×C3×C4—41no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C2×Dic6—21no—
109C2×Dic6—21no—
110C2×C3×C8—42no—
111C2×C3×C8—41no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C42—41no—
119C3×C42—41no—
120C3×C42C2×D411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123C3×C42#4111yescharacteristic
124#217—41no—
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131C4×Dic3—41no—
132#208—21no—
133#208—21no—
134#208—21no—
135#208—41no—
136#208—41no—
137#208SD1612yes—
138C2×C3⋊C8—21no—
139C2×C3⋊C8—21no—
140C2×C3⋊C8—21no—
141C2×C3⋊C8—21no—
142C2×C3⋊C8—41no—
143C22×Q8—21no—
144C2×M16—31no—
145C2×M16—31no—
146C2×M16—61no—
147C22×C8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—31no—
153C4⋊Q8—31no—
154#125—61no—
155C22⋊Q8—61no—
156C22⋊Q8—61no—
157C4×Q8—61no—
158C4×Q8—61no—
159#117—61no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4#6712yes—
163C2×C42—21no—
164C2×C42C22×D311yescharacteristic
165#106—62no—
166#106—62no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—31no—
170C4⋊C8—31no—
171C4⋊C8—61no—
172C4⋊C8—61no—
173#103—62no—
174#103—62no—
175#101—122no—
176#101—122no—
177C22⋊C8—61no—
178#97—61no—
179C4×C8—61no—
180C4×C8—61no—
181#95—21no—
182C23×C3#12111yescharacteristic
183C3×Q8—81no—
184C3×Q8—81no—
185C3×Q8—81no—
186C3×Q8—81no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—41no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4#12011yescharacteristic
197C2×C3×C4#12111yescharacteristic
198C2×C3×C4#12111yescharacteristic
199C2×C3×C4#12311yescharacteristic
200C2×C3×C4#12311yescharacteristic
201C2×C3×C4#12411yescharacteristic
202C2×C3×C4C2×SD1611yescharacteristic
203C2×C3×C4#13511yescharacteristic
204C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
205C2×C3×C4#13711yescharacteristic
206C2×Dic3—41no—
207C2×Dic3—41no—
208Dic6—41no—
209Dic6—41no—
210Dic6—81no—
211C3×C8—81no—
212C3⋊C8—22no—
213C3⋊C8—41no—
214C3⋊C8—41no—
215C3⋊C8—41no—
216C3⋊C8—81no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—61no—
224C2×Q8—61no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4D3×D411yescharacteristic
228C22×C4#23711yescharacteristic
229C22×C4C2×#6711yescharacteristic
230M16—32no—
231M16—61no—
232M16—61no—
233M16—61no—
234M16—121no—
235M16—121no—
236C2×C8—42no—
237C2×C8—41no—
238C2×C8—61no—
239C2×C8—61no—
240C2×C8—61no—
241C2×C8—61no—
242C2×C8—121no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C22⋊C4—121no—
256C42—41no—
257C42—41no—
258C42—121no—
259C42#23511yescharacteristic
260C42#23511yescharacteristic
261C42D3×D411yescharacteristic
262C42D3×D411yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#39411yescharacteristic
266C22×C3#46111yescharacteristic
267C22×C3#46311yescharacteristic
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—82no—
272C3×C4—82no—
273C3×C4#45411yescharacteristic
274C3×C4#45611yescharacteristic
275C3×C4#46011yescharacteristic
276C3×C4#46411yescharacteristic
277C3×C4#46511yescharacteristic
278C3×C4#46811yescharacteristic
279Dic3—81no—
280Dic3—81no—
281C23#93911yescharacteristic
282Q8—81no—
283Q8—81no—
284Q8—81no—
285Q8—81no—
286Q8—121no—
287Q8—121no—
288Q8—241no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4#88311yescharacteristic
301C2×C4#88411yescharacteristic
302C2×C4#88411yescharacteristic
303C2×C4#88511yescharacteristic
304C2×C4#91111yescharacteristic
305C2×C4#91211yescharacteristic
306C2×C4D3×SD1611yescharacteristic
307C2×C4#91511yescharacteristic
308C2×C4#92911yescharacteristic
309C2×C4#93711yescharacteristic
310C8—62no—
311C8—81no—
312C8—121no—
313C8—121no—
314C8—121no—
315C8—241no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3#203811yescharacteristic
319C2×C3#204911yescharacteristic
320C2×C3#205511yescharacteristic
321C22—21no—
322C22—21no—
323C22#495911yescharacteristic
324C22#515311yescharacteristic
325C22#516211yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—82no—
330C4—82no—
331C4—241no—
332C4—241no—
333C4#476211yescharacteristic
334C4#476711yescharacteristic
335C4#477811yescharacteristic
336C4#478011yescharacteristic
337C4#506211yescharacteristic
338C4#509511yescharacteristic
339C3#1050711yescharacteristic, complemented, Sylow
340C2—21no—
341C2—21no—
342C2#6969511yescharacteristic
343C2#7021711yescharacteristic
344C2#7035211yescharacteristic
subgroupG/H
centerC22#4959
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1988C2
socleC22×C3#394
prime radicalC23×C3#121
hypercenter#1988D3
hypocenterC3#10507
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
339. C3#10507semidirect

Maximal subgroups

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

subgroupindex
#832232
C3×#19882
#717892
#704382
#704202
#694932
#676202
#105073