Finite Group Databasebeta

← #10695821

Subgroups — #10695821

Order 768.

822 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 79 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#84082C212yesmaximal, complemented
2#74144C211yesmaximal, characteristic, complemented
3C3×#1782C211yesmaximal, characteristic, complemented
4#69531C212yesmaximal, complemented
5#67629C211yesmaximal, characteristic, complemented
6#9980—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×#949—22no—
9C2×#930C2212yes—
10C2×C3×#99—22no—
11C3×#381C2211yescharacteristic
12C3×#381C2211yescharacteristic
13#5238—24no—
14#5232—22no—
15#5227—22no—
16#5217—22no—
17#5207C2212yes—
18#4960—22no—
19#4674—22no—
20#4671—22no—
21#4451—24no—
22#3006—32no—
23#2815—31no—
24#1782D311yescharacteristic, complemented
25#1510—32no—
26#1306—31no—
27C2×C3×#41—22no—
28C2×C3×#41—22no—
29C2×C3×#41—44no—
30C22×C3×Q8D411yescharacteristic, complemented
31C22×C3×D4—21no—
32C22×C3×D4D411yescharacteristic, complemented
33C23×C3×C4—21no—
34C23×C3×C4C2311yescharacteristic
35C22×#67—22no—
36C22×D12—22no—
37C22×D12D412yes—
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4D412yes—
40#949—48no—
41C2×#215—24no—
42C2×#213—24no—
43#930—22no—
44#930—22no—
45#930—22no—
46#930—22no—
47C2×(C2×D3)⋊C4—22no—
48C2×(C2×D3)⋊C4—24no—
49C2×C4⋊Dic3D412yes—
50C2×#208—22no—
51C3×#99—44no—
52C3×#95—22no—
53C3×#95—22no—
54#835—24no—
55#832—24no—
56C3⋊#95—22no—
57#830—22no—
58#830D414yescomplemented
59C22×#41C2×D311yescharacteristic
60C2×AGL(1,ℤ/8)—62no—
61C2×#121—32no—
62#449—64no—
63#448—62no—
64#447—62no—
65#417—62no—
66C2×#99—22no—
67#407—32no—
68#397—62no—
69#392—62no—
70#392—62no—
71#381C2×D311yescharacteristic
72#381C2×D311yescharacteristic
73#327—64no—
74C24×C3—21no—
75C24×C3C2×D411yescharacteristic
76C23×D3—22no—
77C3×#41—28no—
78C3×#41—44no—
79C3×#41—44no—
80C3×#41—44no—
81C2×C3×Q8—22no—
82C2×C3×Q8—42no—
83C2×C3×D4—24no—
84C2×C3×D4—22no—
85C2×C3×D4—42no—
86C2×C3×D4—42no—
87C2×C3×D4—41no—
88C2×C3×D4—41no—
89C22×C3×C4—21no—
90C22×C3×C4—22no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4—42no—
95C22×C3×C4—41no—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D412yes—
98C22×C3×C4C2×D412yes—
99C22×C3×C4#4111yescharacteristic
100C2×#67—42no—
101C2×#67—42no—
102C2×#67—84no—
103C22×Dic3—22no—
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—44no—
107C2×D12—84no—
108C2×D12—82no—
109C3×C22⋊C4—42no—
110C3×C22⋊C4—42no—
111#215—48no—
112#213—48no—
113(C2×D3)⋊C4—42no—
114(C2×D3)⋊C4—42no—
115(C2×D3)⋊C4—84no—
116C4⋊Dic3—22no—
117C4⋊Dic3—22no—
118#208—44no—
119C2×C3⋊C8—24no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—44no—
123C22×Q8D1211yescharacteristic, complemented
124C22×D4—21no—
125C22×D4—32no—
126C22×D4—62no—
127C22×D4—62no—
128C22×D4D1211yescharacteristic, complemented
129C23×C4—21no—
130C23×C4C22×D311yescharacteristic
131AGL(1,ℤ/8)—128no—
132C2×SD16—64no—
133C2×D8—64no—
134C2×M16—62no—
135#121—62no—
136#121—62no—
137#121—62no—
138#121—62no—
139C2×C4⋊C4—32no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4—64no—
143C2×C22⋊C4#6712yes—
144#103—64no—
145#102—64no—
146#99—44no—
147C22⋊C8—34no—
148C22⋊C8—62no—
149#95—22no—
150#95—22no—
151#95—62no—
152C23×C3—22no—
153C23×C3—22no—
154C23×C3—42no—
155C23×C3—41no—
156C23×C3—41no—
157C23×C3—82no—
158C23×C3#12011yescharacteristic
159C23×C3#12111yescharacteristic
160C23×C3#12711yescharacteristic
161C22×D3—42no—
162C22×D3—42no—
163C22×D3—44no—
164C22×D3—42no—
165C22×D3—44no—
166C3×Q8—24no—
167C3×Q8—42no—
168C3×D4—24no—
169C3×D4—42no—
170C3×D4—44no—
171C3×D4—44no—
172C2×C3×C4—24no—
173C2×C3×C4—22no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—42no—
177C2×C3×C4—48no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—44no—
181C2×C3×C4—42no—
182C2×C3×C4—81no—
183C2×C3×C4—82no—
184C2×C3×C4#12012yes—
185C2×C3×C4#12112yes—
186#67—84no—
187#67—84no—
188C2×Dic3—44no—
189C2×Dic3—42no—
190D12—44no—
191D12—82no—
192D12—88no—
193C3⋊C8—44no—
194C24—21no—
195C24—62no—
196C24C2×#6711yescharacteristic
197#41—28no—
198#41—44no—
199#41—44no—
200#41—44no—
201C2×Q8—22no—
202C2×Q8—42no—
203C2×D4—24no—
204C2×D4—22no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—62no—
210C2×D4—62no—
211C2×D4—124no—
212C2×D4—122no—
213C2×D4—122no—
214C2×D4—244no—
215C2×D4—242no—
216C2×D4—244no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—62no—
225C22×C4C2×D1211yescharacteristic
226C22×C4#23511yescharacteristic
227C22×C4D3×D412yes—
228C22×C4D3×D412yes—
229SD16—128no—
230D8—128no—
231M16—124no—
232C2×C8—64no—
233C4⋊C4—62no—
234C4⋊C4—62no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—122no—
238C22⋊C4—122no—
239C22⋊C4—244no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—42no—
246C22×C3—82no—
247C22×C3—82no—
248C22×C3—82no—
249C22×C3—82no—
250C22×C3#39211yescharacteristic
251C22×C3B(4,𝔽2)12yes—
252C2×D3—84no—
253C2×D3—84no—
254C2×D3—82no—
255C2×D3—82no—
256C2×D3—82no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—42no—
260C3×C4—82no—
261C3×C4—161no—
262Dic3—82no—
263C23—22no—
264C23—22no—
265C23—42no—
266C23—41no—
267C23—41no—
268C23—82no—
269C23—122no—
270C23—122no—
271C23—124no—
272C23—122no—
273C23—124no—
274C23#93011yescharacteristic
275C23#93711yescharacteristic
276C23#94011yescharacteristic
277Q8—24no—
278Q8—42no—
279D4—24no—
280D4—42no—
281D4—44no—
282D4—44no—
283D4—124no—
284D4—242no—
285D4—248no—
286D4—244no—
287D4—244no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—48no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—124no—
301C2×C4—122no—
302C2×C4#88212yes—
303C2×C4#88412yes—
304C8—124no—
305C2×C3—22no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—82no—
309C2×C3#200812yes—
310C2×C3#202111yescharacteristic
311D3—162no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—244no—
323C22—244no—
324C22—242no—
325C22—242no—
326C22—242no—
327C22#496011yescharacteristic
328C22#503212yes—
329C4—22no—
330C4—22no—
331C4—42no—
332C4—82no—
333C4—161no—
334C4—242no—
335C3#998011yescharacteristic, complemented, Sylow
336C2—22no—
337C2—41no—
338C2—41no—
339C2—82no—
340C2—482no—
341C2#6950011yescharacteristic
342C2#7005012yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1782C2
socleC22×C3#392
hypercenter#1782D3
hypocenterC3#9980
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #84082C2semidirect
2. #74144C2semidirect
3. C3×#1782C2semidirect
4. #69531C2semidirect
5. #67629C2semidirect
24. #1782D3semidirect
30. C22×C3×Q8D4semidirect
32. C22×C3×D4D4semidirect
58. #830D4semidirect
123. C22×Q8D12semidirect
128. C22×D4D12semidirect
335. C3#9980semidirect

Maximal subgroups

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

subgroupindex
#840822
#741442
C3×#17822
#695312
#676292
#99803