Finite Group Databasebeta

← #10697866

Subgroups — #10697866

Order 768.

554 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 103 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#83270C211yesmaximal, characteristic
3#73308C211yesmaximal, characteristic, complemented
4C3×#1962C211yesmaximal, characteristic
5#70507C211yesmaximal, characteristic, complemented
6#70506C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#10288—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×#104—22no—
14C3×C43C2211yescharacteristic
15#5093—21no—
16#5078—21no—
17#5076—21no—
18#5070—21no—
19#5040—22no—
20#5033—22no—
21#5019—22no—
22#5007—22no—
23#4936—22no—
24#4929—22no—
25#4929—22no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4482—22no—
29#4481—22no—
30#3226—31no—
31#3206—31no—
32#2862—31no—
33#1962D311yescharacteristic
34#1548—31no—
35#1546—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—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D412yes—
57#869—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×#104—44no—
68#832—24no—
69C3⋊#95—28no—
70#830—21no—
71#830—21no—
72#810—24no—
73#807—24no—
74C4⋊(C3⋊C8)D412yes—
75C4⋊(C3⋊C8)D412yes—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#532C2×D311yescharacteristic
79C2×C4⋊Q8—31no—
80C2×C4⋊Q8—31no—
81#500—61no—
82#492—61no—
83#487—61no—
84#486—61no—
85#480—62no—
86#474—62no—
87#461—62no—
88#448—62no—
89#431—31no—
90#431—31no—
91#423C2×D311yescharacteristic
92C2×#104—22no—
93#391—62no—
94#391—62no—
95#382—62no—
96C43C2×D311yescharacteristic
97#333—62no—
98#332—62no—
99C3×#41—42no—
100C3×#41—42no—
101C2×C3×Q8—21no—
102C2×C3×D4—21no—
103C2×C3×D4—21no—
104C2×C3×D4—21no—
105C2×C3×D4—41no—
106C22×C3×C4—22no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—22no—
112C22×Dic3—22no—
113C2×Dic6—22no—
114C2×Dic6—22no—
115C2×Dic6—44no—
116C2×Dic6—41no—
117C2×Dic6—41no—
118C2×Dic6—42no—
119C2×Dic6—42no—
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—22no—
128C3×C42—22no—
129C3×C42—42no—
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134#214—84no—
135C4⋊Dic3—22no—
136C4⋊Dic3—22no—
137C4⋊Dic3—41no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140C4⋊Dic3—41no—
141C4⋊Dic3—41no—
142C4⋊Dic3—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×#41—21no—
146C22×Q8—32no—
147C22×Q8—61no—
148C2×SD16—64no—
149C2×M16—21no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8D1211yescharacteristic
159#127D1211yescharacteristic
160#124—21no—
161#121—21no—
162#121—21no—
163C4×D4—21no—
164C4×D4—21no—
165C2×C4⋊C4—32no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4—61no—
168C2×C4⋊C4—61no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42#6712yes—
172C2×C42C22×D311yescharacteristic
173#107—64no—
174C4⋊C8—21no—
175C4⋊C8—21no—
176C4⋊C8—32no—
177C4⋊C8—32no—
178#104—44no—
179#103—64no—
180#102—64no—
181C22⋊C8—61no—
182C22⋊C8—61no—
183C4×C8—61no—
184C4×C8—61no—
185#95—68no—
186C23×C3—41no—
187C23×C3#12011yescharacteristic
188C3×Q8—42no—
189C3×D4—42no—
190C3×D4—42no—
191C3×D4—42no—
192C3×D4—82no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—24no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—41no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12011yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12111yescharacteristic
205C2×C3×C4#12111yescharacteristic
206C2×C3×C4#12711yescharacteristic
207C2×C3×C4#13512yes—
208C2×C3×C4AGL(1,ℤ/8)12yes—
209C2×Dic3—44no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—44no—
213Dic6—44no—
214Dic6—82no—
215Dic6—81no—
216Dic6—81no—
217Dic6—82no—
218C3×C8—81no—
219C3⋊C8—42no—
220C3⋊C8—42no—
221#41—42no—
222#41—42no—
223C2×Q8—21no—
224C2×Q8—62no—
225C2×Q8—62no—
226C2×Q8—124no—
227C2×Q8—121no—
228C2×Q8—122no—
229C2×Q8—121no—
230C2×Q8—122no—
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—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C42—24no—
260C42—22no—
261C42—22no—
262C42—42no—
263C42C2×D1211yescharacteristic
264C42#23511yescharacteristic
265C42D3×D411yescharacteristic
266C42D3×D411yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3#39211yescharacteristic
273C22×C3#44912yes—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—44no—
277C3×C4—81no—
278C3×C4#45512yes—
279C3×C4#45612yes—
280C3×C4#46312yes—
281C3×C4#49511yescharacteristic
282C3×C4#49611yescharacteristic
283Dic3—82no—
284Dic3—82no—
285C23—41no—
286C23#93711yescharacteristic
287Q8—42no—
288Q8—124no—
289Q8—242no—
290Q8—242no—
291Q8—241no—
292Q8—241no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—82no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—124no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—124no—
309C2×C4#88211yescharacteristic
310C2×C4#88211yescharacteristic
311C2×C4#88411yescharacteristic
312C2×C4#88411yescharacteristic
313C2×C4#93011yescharacteristic
314C2×C4#93212yes—
315C2×C4#94011yescharacteristic
316C2×C4#95012yes—
317C8—81no—
318C8—122no—
319C8—122no—
320C2×C3—22no—
321C2×C3—161no—
322C2×C3#200411yescharacteristic
323C2×C3#200511yescharacteristic
324C2×C3#201211yescharacteristic
325C22—21no—
326C22—21no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22#496011yescharacteristic
331C22#503712yes—
332C4—21no—
333C4—21no—
334C4—44no—
335C4—81no—
336C4—242no—
337C4—242no—
338C4#482512yes—
339C4#482612yes—
340C4#501612yes—
341C4#505911yescharacteristic
342C4#507411yescharacteristic
343C3#1028811yescharacteristic, complemented, Sylow
344C2—22no—
345C2—161no—
346C2#6965911yescharacteristic
347C2#7038611yescharacteristic
348C2#7038911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1962D3
hypocenterC3#10288
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. #73308C2semidirect
5. #70507C2semidirect
6. #70506C2semidirect
7. #67611C2semidirect
343. C3#10288semidirect

Maximal subgroups

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

subgroupindex
#833172
#832702
#733082
C3×#19622
#705072
#705062
#676112
#102883