Finite Group Databasebeta

← #10697835

Subgroups — #10697835

Order 768.

626 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 111 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#83317C211yesmaximal, characteristic
2#83271C211yesmaximal, characteristic
3#73311C211yesmaximal, characteristic, complemented
4C3×#1962C211yesmaximal, characteristic
5#70508C211yesmaximal, characteristic, complemented
6#70507C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#10283—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#869—22no—
11C2×#869C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×#104—22no—
15C3×C43C2211yescharacteristic
16#5093—21no—
17#5080—21no—
18#5076—21no—
19#5070—21no—
20#5040—22no—
21#5034—22no—
22#5019—22no—
23#5008—22no—
24#4935—22no—
25#4935—22no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4486—22no—
29#4481—22no—
30#3227—31no—
31#3206—31no—
32#2868—31no—
33#1962D311yescharacteristic
34#1549—31no—
35#1548—31no—
36#1276—31no—
37C2×C3×#41—21no—
38C22×Dic6—21no—
39C22×Dic6D412yes—
40C2×C3×M16—21no—
41C3×C4⋊Q8D411yescharacteristic
42C3×#127D411yescharacteristic
43C3×#124—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C2×C3×C42—21no—
49C2×C3×C42—21no—
50C2×C3×C42D412yes—
51C2×C3×C42C2311yescharacteristic
52C2×#214—24no—
53C2×C4⋊Dic3—28no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3D412yes—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—48no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×#104—44no—
70#832—24no—
71#830—21no—
72#830—21no—
73#810—24no—
74#808—24no—
75C4⋊(C3⋊C8)D412yes—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79#532C2×D311yescharacteristic
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8—31no—
82C2×C4⋊Q8—62no—
83#498—61no—
84#492—61no—
85#487—61no—
86#486—61no—
87#481—62no—
88#476—62no—
89#461—62no—
90#448—62no—
91#431—31no—
92#431—31no—
93#423C2×D311yescharacteristic
94C2×#104—22no—
95#384—62no—
96#384—62no—
97C43C2×D311yescharacteristic
98#333—62no—
99#332—62no—
100C3×#41—42no—
101C3×#41—42no—
102C2×C3×Q8—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—41no—
107C22×C3×C4—22no—
108C22×C3×C4—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—22no—
113C22×Dic3—22no—
114C2×Dic6—22no—
115C2×Dic6—22no—
116C2×Dic6—44no—
117C2×Dic6—44no—
118C2×Dic6—41no—
119C2×Dic6—41no—
120C3×M16—42no—
121C2×C3×C8—41no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C42—24no—
127C3×C42—24no—
128C3×C42—22no—
129C3×C42SD1614yescomplemented
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134#214—84no—
135C4⋊Dic3—22no—
136C4⋊Dic3—22no—
137C4⋊Dic3—416no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C2×#41—21no—
147C22×Q8—32no—
148C22×Q8—61no—
149C2×SD16—64no—
150C2×M16—21no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—128no—
160C4⋊Q8D1211yescharacteristic
161#127D1211yescharacteristic
162#124—21no—
163#121—21no—
164#121—21no—
165C4×D4—21no—
166C4×D4—21no—
167C2×C4⋊C4—32no—
168C2×C4⋊C4—68no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4—61no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42#6712yes—
175C2×C42C22×D311yescharacteristic
176#106—64no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—32no—
180C4⋊C8—32no—
181#104—44no—
182#103—64no—
183#102—64no—
184C22⋊C8—61no—
185C22⋊C8—61no—
186C4×C8—61no—
187C4×C8—61no—
188C23×C3—41no—
189C23×C3#12011yescharacteristic
190C3×Q8—42no—
191C3×D4—42no—
192C3×D4—42no—
193C3×D4—42no—
194C3×D4—82no—
195C2×C3×C4—24no—
196C2×C3×C4—24no—
197C2×C3×C4—24no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12711yescharacteristic
209C2×C3×C4C2×SD1612yes—
210C2×C3×C4#13712yes—
211C2×Dic3—44no—
212C2×Dic3—44no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215Dic6—44no—
216Dic6—82no—
217Dic6—82no—
218Dic6—82no—
219C3×C8—81no—
220C3⋊C8—42no—
221C3⋊C8—42no—
222#41—42no—
223#41—42no—
224C2×Q8—21no—
225C2×Q8—62no—
226C2×Q8—62no—
227C2×Q8—124no—
228C2×Q8—124no—
229C2×Q8—121no—
230C2×Q8—121no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—41no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4D3×D411yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4C2×#6711yescharacteristic
242SD16—244no—
243M16—42no—
244C2×C8—41no—
245C2×C8—62no—
246C2×C8—62no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—62no—
250C4⋊C4—62no—
251C4⋊C4—1216no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C42—24no—
261C42—24no—
262C42—22no—
263C42#21414yescomplemented
264C42C2×D1211yescharacteristic
265C42#23511yescharacteristic
266C42D3×D411yescharacteristic
267C42D3×D411yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3#39211yescharacteristic
274C22×C3#44812yes—
275C3×C4—28no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—81no—
279C3×C4#45312yes—
280C3×C4#45612yes—
281C3×C4#46112yes—
282C3×C4#49211yescharacteristic
283C3×C4#49711yescharacteristic
284Dic3—82no—
285Dic3—82no—
286C23—41no—
287C23#93711yescharacteristic
288Q8—42no—
289Q8—124no—
290Q8—242no—
291Q8—242no—
292Q8—242no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—82no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—124no—
306C2×C4—124no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4#88211yescharacteristic
310C2×C4#88211yescharacteristic
311C2×C4#88411yescharacteristic
312C2×C4#88411yescharacteristic
313C2×C4#93011yescharacteristic
314C2×C4C2×#21412yes—
315C2×C4#94011yescharacteristic
316C2×C4#95112yes—
317C8—81no—
318C8—122no—
319C8—122no—
320C2×C3—22no—
321C2×C3—161no—
322C2×C3#200211yescharacteristic
323C2×C3#200511yescharacteristic
324C2×C3#201311yescharacteristic
325C22—21no—
326C22—21no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22#496011yescharacteristic
331C22#504012yes—
332C4—28no—
333C4—21no—
334C4—21no—
335C4—81no—
336C4—242no—
337C4—242no—
338C4#482212yes—
339C4#482612yes—
340C4#501912yes—
341C4#506011yescharacteristic
342C4#507611yescharacteristic
343C3#1028311yescharacteristic, complemented, Sylow
344C2—22no—
345C2—161no—
346C2#6966111yescharacteristic
347C2#7038511yescharacteristic
348C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1962D3
hypocenterC3#10283
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
3. #73311C2semidirect
5. #70508C2semidirect
6. #70507C2semidirect
7. #67611C2semidirect
129. C3×C42SD16semidirect
263. C42#214semidirect
343. C3#10283semidirect

Maximal subgroups

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

subgroupindex
#833172
#832712
#733112
C3×#19622
#705082
#705072
#676112
#102833