Finite Group Databasebeta

← #10695819

Subgroups — #10695819

Order 768.

822 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2815C211yesmaximal, characteristic, complemented
2#82624C212yesmaximal, complemented
3#71674C212yesmaximal, complemented
4#69537C211yesmaximal, characteristic, complemented
5#67629C211yesmaximal, characteristic, complemented
6#9980—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C2×C3×#121C2212yes—
9C2×#932—22no—
10C3×#397—22no—
11C3×#392—22no—
12C3×#392—22no—
13C3×#381C2211yescharacteristic
14#5211C2211yescharacteristic
15#5207C2212yes—
16#5040—22no—
17#5037—24no—
18#5036—22no—
19#4966—22no—
20C2×#806—22no—
21#4537—24no—
22#3006—32no—
23#2815D311yescharacteristic, complemented
24#1782—31no—
25#1510—32no—
26#1306—31no—
27C22×C3×D4—22no—
28C22×C3×D4—22no—
29C22×C3×D4D412yes—
30C23×C3×C4C2311yescharacteristic
31C22×#67—21no—
32C2×#235—22no—
33C2×#235—22no—
34C2×#235—44no—
35C22×D12D411yescharacteristic, complemented
36C22×C4×D3—21no—
37C22×Dic6D411yescharacteristic, complemented
38C3×#121—22no—
39C3×#121—22no—
40C3×#121—22no—
41C3×#121—22no—
42C2×C3×C4⋊C4D412yes—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—24no—
45C2×C3×C22⋊C4D412yes—
46C2×#217—21no—
47C2×#214—24no—
48#932—48no—
49C2×#213—24no—
50C2×#208—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3⋊#95—22no—
54#830—22no—
55#830D414yescomplemented
56#810—24no—
57#809—24no—
58#806—44no—
59C22×#41—31no—
60C2×AGL(1,ℤ/8)—62no—
61C2×#121C2×D312yes—
62#449—64no—
63#448—62no—
64#447—62no—
65#417—62no—
66C2×#99—62no—
67#407—32no—
68#397—22no—
69#392—22no—
70#392—22no—
71#381—31no—
72#381C2×D311yescharacteristic
73#327—64no—
74C24×C3—22no—
75C24×C3C2×D411yescharacteristic
76C23×D3—21no—
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—44no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—84no—
83C2×C3×D4—84no—
84C2×C3×D4—82no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—42no—
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D412yes—
90C22×C3×C4C2×D412yes—
91C22×C3×C4#4111yescharacteristic
92C2×#67—24no—
93C2×#67—42no—
94C2×#67—41no—
95C2×#67—41no—
96C22×Dic3—21no—
97C22×Dic3—41no—
98#235—28no—
99#235—44no—
100#235—44no—
101#235—44no—
102C2×D12—22no—
103C2×D12—42no—
104C2×C4×D3—42no—
105C2×C4×D3—42no—
106C2×C4×D3—42no—
107C2×Dic6—22no—
108C2×Dic6—42no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C22⋊C4—42no—
112C3×C22⋊C4—42no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—84no—
115#217—42no—
116#214—48no—
117#213—48no—
118#208—44no—
119C2×C3⋊C8—24no—
120C2×#41—62no—
121C2×#41—62no—
122C2×#41—124no—
123C22×Q8—31no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—31no—
127C22×D4—61no—
128C22×D4#6712yes—
129C23×C4—61no—
130C23×C4C22×D311yescharacteristic
131AGL(1,ℤ/8)—128no—
132C2×SD16—64no—
133C2×D8—64no—
134C2×M16—62no—
135#121—22no—
136#121—22no—
137#121—22no—
138#121—22no—
139C2×C4⋊C4#6712yes—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4#6712yes—
144#103—64no—
145#102—64no—
146#99—124no—
147C22⋊C8—34no—
148C22⋊C8—62no—
149#95—22no—
150#95—22no—
151#95—62no—
152C23×C3—22no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—42no—
156C23×C3—44no—
157C23×C3—42no—
158C23×C3—44no—
159C23×C3#12011yescharacteristic
160C23×C3#12111yescharacteristic
161C23×C3#12711yescharacteristic
162C22×D3—22no—
163C22×D3—41no—
164C22×D3—41no—
165C22×D3—82no—
166C3×D4—44no—
167C3×D4—82no—
168C3×D4—84no—
169C3×D4—84no—
170C3×D4—88no—
171C2×C3×C4—24no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—42no—
175C2×C3×C4—44no—
176C2×C3×C4—44no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4#12012yes—
180C2×C3×C4#12112yes—
181#67—44no—
182#67—44no—
183C2×Dic3—22no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—81no—
187C2×Dic3—82no—
188D12—24no—
189D12—42no—
190C4×D3—48no—
191Dic6—24no—
192Dic6—42no—
193C3⋊C8—44no—
194C24—22no—
195C24—61no—
196C24C2×#6711yescharacteristic
197#41—68no—
198#41—124no—
199#41—124no—
200#41—124no—
201C2×Q8—62no—
202C2×Q8—122no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—44no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—64no—
209C2×D4—62no—
210C2×D4—84no—
211C2×D4—84no—
212C2×D4—82no—
213C2×D4—122no—
214C2×D4—122no—
215C2×D4—121no—
216C2×D4—121no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—61no—
221C22×C4—122no—
222C22×C4—122no—
223C22×C4—122no—
224C22×C4—121no—
225C22×C4D3×D411yescharacteristic
226C22×C4D3×D412yes—
227C22×C4#23711yescharacteristic
228C22×C4C2×#6712yes—
229SD16—128no—
230D8—128no—
231M16—124no—
232C2×C8—64no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—84no—
239C22⋊C4—122no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—82no—
246C22×C3—84no—
247C22×C3—84no—
248C22×C3—82no—
249C22×C3—82no—
250C22×C3—82no—
251C22×C3#39211yescharacteristic
252C22×C3B(4,𝔽2)12yes—
253C2×D3—42no—
254C2×D3—82no—
255C2×D3—82no—
256C2×D3—82no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—82no—
260C3×C4—82no—
261Dic3—42no—
262Dic3—161no—
263C23—22no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—44no—
268C23—42no—
269C23—44no—
270C23—62no—
271C23—121no—
272C23—121no—
273C23—242no—
274C23#93811yescharacteristic
275C23#94011yescharacteristic
276C23#95311yescharacteristic
277Q8—64no—
278Q8—122no—
279D4—44no—
280D4—64no—
281D4—82no—
282D4—84no—
283D4—84no—
284D4—88no—
285D4—122no—
286D4—124no—
287D4—124no—
288C2×C4—24no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—62no—
297C2×C4—128no—
298C2×C4—121no—
299C2×C4—121no—
300C2×C4—241no—
301C2×C4—242no—
302C2×C4#93712yes—
303C2×C4#93912yes—
304C8—124no—
305C2×C3—22no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—162no—
309C2×C3#200812yes—
310C2×C3#202111yescharacteristic
311D3—82no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—82no—
318C22—84no—
319C22—84no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—122no—
324C22—242no—
325C22—242no—
326C22—242no—
327C22#522311yescharacteristic
328C22#524412yes—
329C4—22no—
330C4—22no—
331C4—82no—
332C4—82no—
333C4—122no—
334C4—481no—
335C3#998011yescharacteristic, complemented, Sylow
336C2—22no—
337C2—41no—
338C2—41no—
339C2—162no—
340C2—242no—
341C2#7185111yescharacteristic
342C2#7185812yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2815C2
socleC22×C3#392
hypercenter#2815D3
hypocenterC3#9980
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2815

Complements

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

NH ≅ G/Ntype
1. C3×#2815C2semidirect
2. #82624C2semidirect
3. #71674C2semidirect
4. #69537C2semidirect
5. #67629C2semidirect
23. #2815D3semidirect
35. C22×D12D4semidirect
37. C22×Dic6D4semidirect
55. #830D4semidirect
335. C3#9980semidirect

Maximal subgroups

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

subgroupindex
C3×#28152
#826242
#716742
#695372
#676292
#99803