Finite Group Databasebeta

← #10697859

Subgroups — #10697859

Order 768.

566 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 109 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×#3206C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic
3#83252C211yesmaximal, characteristic, complemented
4#71815C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic, complemented
6#70779C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#10287—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#492—21no—
13C3×#487—21no—
14C3×#461—22no—
15C3×#448—22no—
16C3×#431C2211yescharacteristic
17C2×#837—22no—
18#5229—22no—
19#5093—21no—
20#5076—21no—
21#5040—22no—
22#5019—22no—
23#4977—22no—
24#4973—22no—
25C42×Dic3C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4546—22no—
29#4481—22no—
30#3206—31no—
31#3206D311yescharacteristic
32#2862—31no—
33#1962—31no—
34#1548—31no—
35#1548—31no—
36#1276—31no—
37C2×C3×#41—21no—
38C22×C3×Q8D412yes—
39C22×Dic6D412yes—
40C2×C3×SD16—24no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×#127D411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#214—24no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3D412yes—
61C2×#208—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66C3×C4⋊C8D412yes—
67C3×#103—24no—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70#837—44no—
71C3⋊#95—28no—
72#830—21no—
73#810—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78#532C2×D311yescharacteristic
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8C2×D311yescharacteristic
81#492—21no—
82#492—61no—
83#487—21no—
84#487—61no—
85#461—22no—
86#461—62no—
87#448—22no—
88#448—62no—
89#431—31no—
90#431C2×D311yescharacteristic
91#423—31no—
92C2×#104—62no—
93#391—62no—
94#391—62no—
95#382—62no—
96C43—31no—
97#333—62no—
98#333—62no—
99C3×#41—42no—
100C3×#41—42no—
101C2×C3×Q8—22no—
102C2×C3×Q8—21no—
103C2×C3×Q8—22no—
104C2×C3×Q8—44no—
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—41no—
109C22×C3×C4—22no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—22no—
115C22×Dic3—22no—
116C2×Dic6—22no—
117C2×Dic6—22no—
118C2×Dic6—44no—
119C3×SD16—84no—
120C2×C3×C8—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133#214—84no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138C4×Dic3—22no—
139C4×Dic3—42no—
140C4×Dic3—42no—
141C4×Dic3SD1614yescomplemented
142#208—42no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C2×#41—21no—
146C22×Q8—32no—
147C22×Q8#6712yes—
148C2×SD16—24no—
149C2×SD16—64no—
150C2×M16—61no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8#6711yescharacteristic
160#127#6711yescharacteristic
161#124—21no—
162#121—21no—
163#121—21no—
164C4×D4—21no—
165C4×D4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—61no—
169C2×C4⋊C4—61no—
170C2×C42—32no—
171C2×C42—61no—
172C2×C42—61no—
173C2×C42C22×D311yescharacteristic
174C4⋊C8—32no—
175C4⋊C8—61no—
176C4⋊C8—61no—
177C4⋊C8#6712yes—
178#104—124no—
179#103—24no—
180#103—64no—
181C22⋊C8—21no—
182C22⋊C8—61no—
183C4×C8—21no—
184C4×C8—61no—
185#95—68no—
186C23×C3—41no—
187C23×C3#12011yescharacteristic
188C3×Q8—44no—
189C3×Q8—42no—
190C3×Q8—82no—
191C3×D4—42no—
192C3×D4—42no—
193C3×D4—42no—
194C3×D4—82no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—44no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—81no—
201C2×C3×C4—81no—
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12711yescharacteristic
208C2×C3×C4C2×SD1612yes—
209C2×C3×C4#13712yes—
210C2×Dic3—24no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—44no—
214C2×Dic3—42no—
215Dic6—44no—
216Dic6—82no—
217C3×C8—42no—
218C3⋊C8—42no—
219C3⋊C8—81no—
220#41—42no—
221#41—42no—
222C2×Q8—21no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—44no—
226C2×Q8—62no—
227C2×Q8—62no—
228C2×Q8—124no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—41no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4D3×D411yescharacteristic
238C22×C4D3×D411yescharacteristic
239C22×C4C2×#6711yescharacteristic
240SD16—84no—
241SD16—244no—
242M16—122no—
243C2×C8—22no—
244C2×C8—62no—
245C2×C8—121no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42—34no—
259C42—62no—
260C42—122no—
261C42—122no—
262C42D3×D411yescharacteristic
263C42#23711yescharacteristic
264C42C2×#6711yescharacteristic
265C42C2×#6711yescharacteristic
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—81no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3#39211yescharacteristic
272C22×C3#44812yes—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—82no—
276C3×C4—81no—
277C3×C4#45512yes—
278C3×C4#45512yes—
279C3×C4#46112yes—
280C3×C4#49211yescharacteristic
281C3×C4#49711yescharacteristic
282Dic3—44no—
283Dic3—82no—
284C23—41no—
285C23#93711yescharacteristic
286Q8—42no—
287Q8—44no—
288Q8—82no—
289Q8—124no—
290Q8—242no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—82no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—64no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—124no—
306C2×C4—122no—
307C2×C4D3×SD1612yes—
308C2×C4#91512yes—
309C2×C4#93711yescharacteristic
310C2×C4#93811yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#93911yescharacteristic
313C2×C4#94011yescharacteristic
314C2×C4#95311yescharacteristic
315C8—42no—
316C8—122no—
317C8—241no—
318C2×C3—22no—
319C2×C3—161no—
320C2×C3#200411yescharacteristic
321C2×C3#200411yescharacteristic
322C2×C3#201311yescharacteristic
323C22—21no—
324C22—21no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22#517012yes—
329C22#522311yescharacteristic
330C4—21no—
331C4—21no—
332C4—81no—
333C4—82no—
334C4—124no—
335C4—242no—
336C4#516212yes—
337C4#516711yescharacteristic
338C4#517511yescharacteristic
339C4#520212yes—
340C4#524212yes—
341C3#1028711yescharacteristic, complemented, Sylow
342C2—22no—
343C2—161no—
344C2#7185711yescharacteristic
345C2#7205711yescharacteristic
346C2#7210611yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3206C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3206D3
hypocenterC3#10287
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
3. #83252C2semidirect
5. #71774C2semidirect
6. #70779C2semidirect
7. #67736C2semidirect
141. C4×Dic3SD16semidirect
341. C3#10287semidirect

Maximal subgroups

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

subgroupindex
C3×#32062
#833172
#832522
#718152
#717742
#707792
#677362
#102873