Finite Group Databasebeta

← #10900923

Subgroups — #10900923

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#869C211yesmaximal, characteristic, complemented
2#82080C211yesmaximal, characteristic
3C2×#4736C212yesmaximal
4C3×#1817C211yesmaximal, characteristic, complemented
5#69924C212yesmaximal, complemented
6#68678C214yesmaximal, complemented
7#68632C214yesmaximal, complemented
8#29638—31nomaximal, Sylow
9C23×Dic6—21no—
10C22×C3×C42C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C2×#929—21no—
13C22×C4⋊Dic3—21no—
14C22×C4⋊Dic3C2212yes—
15C2×#869—22no—
16C2×#869—28no—
17C2×#869C2214yes—
18C2×C3×C22⋊C8C2212yes—
19#5213—21no—
20#5212C2211yescharacteristic
21C2×#821C2214yes—
22C2×#818C2214yes—
23#4940—22no—
24#4939C2212yes—
25#4928C2212yes—
26#4736—28no—
27C3×#336C2214yes—
28#4551C2218yes—
29C22×C4⋊Q8—31no—
30C2×#480—32no—
31#2854—31no—
32#2057—34no—
33#2023—34no—
34#1823—32no—
35#1817D311yescharacteristic, complemented
36C23×C3×C4D411yescharacteristic
37C23×C3×C4D411yescharacteristic
38C23×C3×C4C2311yescharacteristic
39C22×#67—21no—
40C23×Dic3—22no—
41C22×D12C2311yescharacteristic
42C22×Dic6—28no—
43C22×Dic6—24no—
44C22×Dic6—44no—
45C22×Dic6—44no—
46C22×C3×C8C2314yes—
47C2×C3×C42—24no—
48C2×C3×C42D412yes—
49C2×C3×C42C2312yes—
50C2×#217—21no—
51#930—24no—
52#929—44no—
53C2×(C2×D3)⋊C4—22no—
54C2×(C2×D3)⋊C4—22no—
55C2×(C2×D3)⋊C4—21no—
56C2×C4⋊Dic3—28no—
57C2×C4⋊Dic3—22no—
58C2×C4⋊Dic3—24no—
59C2×C4⋊Dic3—24no—
60C2×C4⋊Dic3D418yes—
61C2×C4⋊Dic3C2311yescharacteristic
62C2×C4⋊Dic3C2312yes—
63C2×C4⋊Dic3C2314yes—
64C2×Dic3⋊C4—22no—
65#869—216no—
66#869—216no—
67C3×C22⋊C8—24no—
68C3⋊#95—22no—
69C3⋊#95—21no—
70#821—28no—
71#818—28no—
72C23×Q8—61no—
73C2×C4⋊Q8—34no—
74C2×C4⋊Q8—68no—
75C2×C4⋊Q8—62no—
76C2×#123—61no—
77C2×#121—31no—
78C22×C4⋊C4—32no—
79C22×C4⋊C4—61no—
80C22×C42C2×D311yescharacteristic
81#480—68no—
82C2×#107—34no—
83C2×#102—34no—
84C2×C22⋊C8C2×D312yes—
85#390—32no—
86#389—32no—
87#388—62no—
88#386—61no—
89#380—31no—
90#340—38no—
91#336C2×D314yes—
92C24×C3C2×D411yescharacteristic
93C23×D3—21no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4D814yes—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D412yes—
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D414yes—
104C22×C3×C4C2411yescharacteristic
105C2×#67—42no—
106C2×#67—42no—
107C2×#67—42no—
108C22×Dic3—28no—
109C22×Dic3—24no—
110C22×Dic3—21no—
111C22×Dic3—22no—
112C22×Dic3—44no—
113C22×Dic3—44no—
114C2×D12—22no—
115C2×D12—24no—
116C2×Dic6—216no—
117C2×Dic6—48no—
118C2×Dic6—432no—
119C2×Dic6—48no—
120C2×C3×C8—24no—
121C2×C3×C8—28no—
122C3×C42—24no—
123C3×C42—24no—
124#217—42no—
125(C2×D3)⋊C4—42no—
126(C2×D3)⋊C4—44no—
127(C2×D3)⋊C4—44no—
128C4⋊Dic3—216no—
129C4⋊Dic3—28no—
130C4⋊Dic3—24no—
131C4⋊Dic3—22no—
132C4⋊Dic3—24no—
133C4⋊Dic3C2×D418yes—
134Dic3⋊C4—44no—
135C22×Q8—68no—
136C22×Q8—64no—
137C22×Q8—124no—
138C22×Q8—124no—
139C22×D4—31no—
140C22×D4—61no—
141C23×C4—62no—
142C23×C4D1211yescharacteristic
143C23×C4D1211yescharacteristic
144C23×C4C22×D311yescharacteristic
145C22×C8C22×D314yes—
146C4⋊Q8—616no—
147C4⋊Q8—616no—
148#123—124no—
149#121—64no—
150C2×C4⋊C4—38no—
151C2×C4⋊C4—34no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—32no—
154C2×C4⋊C4—68no—
155C2×C4⋊C4—62no—
156C2×C4⋊C4—64no—
157C2×C4⋊C4—64no—
158C2×C4⋊C4—62no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—62no—
162C2×C22⋊C4—62no—
163C2×C42—24no—
164C2×C42D1212yes—
165C2×C42C22×D312yes—
166#107—68no—
167#102—68no—
168C22⋊C8—24no—
169#95—61no—
170#95—62no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3#12012yes—
174C23×C3C2×D812yes—
175C23×C3AGL(1,ℤ/8)12yes—
176C23×C3C22×D411yescharacteristic
177C22×D3—41no—
178C22×D3—42no—
179C22×D3—41no—
180C22×D3—41no—
181C22×D3—42no—
182C2×C3×C4—28no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—28no—
189C2×C3×C4—28no—
190C2×C3×C4#12012yes—
191C2×C3×C4C2×D814yes—
192C2×C3×C4#13714yes—
193C2×C3×C4C22×D412yes—
194C2×C3×C42−1+414yes—
195#67—84no—
196C2×Dic3—28no—
197C2×Dic3—44no—
198C2×Dic3—44no—
199C2×Dic3—416no—
200C2×Dic3—41no—
201C2×Dic3—42no—
202C2×Dic3—41no—
203C2×Dic3—42no—
204C2×Dic3—41no—
205C2×Dic3—42no—
206D12—44no—
207Dic6—432no—
208C3×C8—44no—
209C24—61no—
210C24C2×D1211yescharacteristic
211C2×Q8—616no—
212C2×Q8—128no—
213C2×Q8—1232no—
214C2×Q8—128no—
215C2×D4—62no—
216C2×D4—64no—
217C2×D4—122no—
218C2×D4—122no—
219C2×D4—122no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—68no—
226C22×C4—64no—
227C22×C4—62no—
228C22×C4—61no—
229C22×C4—124no—
230C22×C4—124no—
231C22×C4D2414yes—
232C22×C4C2×D1211yescharacteristic
233C22×C4C2×D1212yes—
234C22×C4C2×D1212yes—
235C22×C4D3×D414yes—
236C22×C4C23×D311yescharacteristic
237C2×C8—24no—
238C2×C8—28no—
239C4⋊C4—38no—
240C4⋊C4—616no—
241C4⋊C4—68no—
242C4⋊C4—64no—
243C4⋊C4—62no—
244C4⋊C4—64no—
245C4⋊C4—124no—
246C22⋊C4—122no—
247C22⋊C4—122no—
248C22⋊C4—124no—
249C22⋊C4—124no—
250C42—24no—
251C42—24no—
252C22×C3—22no—
253C22×C3—24no—
254C22×C3—24no—
255C22×C3—22no—
256C22×C3#44714yes—
257C22×C3C2×#12011yescharacteristic
258C22×C3#53612yes—
259C22×C3C22×D811yescharacteristic
260C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C3C2×#13712yes—
262C2×D3—81no—
263C2×D3—82no—
264C2×D3—82no—
265C2×D3—81no—
266C2×D3—81no—
267C3×C4—22no—
268C3×C4—24no—
269C3×C4—24no—
270C3×C4#53614yes—
271Dic3—48no—
272Dic3—82no—
273Dic3—81no—
274C23—22no—
275C23—22no—
276C23—121no—
277C23—121no—
278C23—121no—
279C23—122no—
280C23—122no—
281C23#88212yes—
282C23C2×D2412yes—
283C23#90812yes—
284C23C22×D1211yescharacteristic
285Q8—1232no—
286D4—124no—
287D4—244no—
288C2×C4—28no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—28no—
294C2×C4—22no—
295C2×C4—28no—
296C2×C4—68no—
297C2×C4—124no—
298C2×C4—124no—
299C2×C4—1216no—
300C2×C4—122no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—121no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4#88212yes—
307C2×C4C2×D2414yes—
308C2×C4#90914yes—
309C2×C4C2×D3×D412yes—
310C2×C4#101014yes—
311C8—44no—
312C2×C3—24no—
313C2×C3#261411yescharacteristic
314C2×C3C2×#44711yescharacteristic
315C2×C3#300411yescharacteristic
316C2×C3#322612yes—
317C2×C3#323812yes—
318D3—161no—
319C22—22no—
320C22—22no—
321C22—24no—
322C22—24no—
323C22—241no—
324C22—241no—
325C22—241no—
326C22—242no—
327C22—242no—
328C22#473214yes—
329C22C2×#88211yescharacteristic
330C22#552612yes—
331C22C22×D2411yescharacteristic
332C22C2×#90811yescharacteristic
333C22C2×#90912yes—
334C4—24no—
335C4—24no—
336C4—22no—
337C4—128no—
338C4—241no—
339C4—242no—
340C4#564814yes—
341C3#2963811yescharacteristic, complemented, Sylow
342C2—24no—
343C2—481no—
344C2#7413511yescharacteristic
345C2C2×#473211yescharacteristic
346C2#7864511yescharacteristic
347C2#7926812yes—
348C2#7932212yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1817D3
hypocenterC3#29638
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
1. C22×#869C2semidirect
4. C3×#1817C2semidirect
5. #69924C2semidirect
6. #68678C2semidirect
7. #68632C2semidirect
35. #1817D3semidirect
341. C3#29638semidirect

Maximal subgroups

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

subgroupindex
C22×#8692
#820802
C2×#47362
C3×#18172
#699242
#686782
#686322
#296383