Finite Group Databasebeta

← #10695741

Subgroups — #10695741

Order 768.

1020 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 121 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×#2659C211yesmaximal, characteristic, complemented
2C2×#5046C212yesmaximal
3C2×#4538C212yesmaximal, complemented
4#69426C211yesmaximal, characteristic
5#67613C211yesmaximal, characteristic, complemented
6#9966—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×C3×C22⋊Q8C2212yes—
9C22×#215—22no—
10C22×(C2×D3)⋊C4—22no—
11C3×#398—22no—
12C3×#395—22no—
13C3×#380—22no—
14C2×C3×#95C2211yescharacteristic
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#5046—24no—
18#5046—24no—
19C2×#809—22no—
20#4538—24no—
21C2×#450—32no—
22#2659D311yescharacteristic, complemented
23C2×#328—32no—
24#1438—31no—
25#1280—31no—
26C24×D3—21no—
27C22×C3×Q8D412yes—
28C23×C3×C4D412yes—
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—21no—
31C22×D12—22no—
32C22×D12—22no—
33C22×D12—21no—
34C22×D12—44no—
35C22×D12D412yes—
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C3×C22⋊Q8—22no—
39C3×C22⋊Q8—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4—22no—
43C2×C3×C4⋊C4D412yes—
44C2×C3×C22⋊C4—22no—
45C2×#215—28no—
46C2×#215—48no—
47C2×(C2×D3)⋊C4—44no—
48C2×(C2×D3)⋊C4—44no—
49C2×(C2×D3)⋊C4—44no—
50C22×C3⋊C8—22no—
51C3×#95—24no—
52C3×#95—22no—
53C3⋊#95—22no—
54#830—22no—
55#830D414yes—
56#809—28no—
57C23×D4—31no—
58C22×SD16—62no—
59C2×C22⋊Q8C2×D312yes—
60C22×C22⋊C4—62no—
61#450—64no—
62#450—64no—
63C2×#102—62no—
64C2×C22⋊C8—32no—
65#398—22no—
66#395—22no—
67#380—22no—
68C2×#95—31no—
69C2×#95C2×D311yescharacteristic
70#328—64no—
71C24×C3C2×D411yescharacteristic
72C23×D3—24no—
73C23×D3—22no—
74C23×D3—42no—
75C23×D3—42no—
76C23×D3—42no—
77C23×D3—41no—
78C23×D3—41no—
79C23×D3—42no—
80C23×D3—42no—
81C2×C3×Q8—22no—
82C2×C3×Q8—22no—
83C2×C3×Q8—44no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—42no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4SD1614yescomplemented
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D412yes—
94C22×C3×C4C2×D412yes—
95C22×C3×C4#4111yescharacteristic
96C22×Dic3—22no—
97C22×Dic3—42no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3—42no—
101C2×D12—28no—
102C2×D12—24no—
103C2×D12—44no—
104C2×D12—48no—
105C2×D12—44no—
106C2×D12—44no—
107C2×D12—42no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—42no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—82no—
113C3×C4⋊C4—82no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—42no—
116#215—416no—
117(C2×D3)⋊C4—416no—
118C2×C3⋊C8—24no—
119C2×C3⋊C8—44no—
120C25—61no—
121C22×Q8#6712yes—
122C22×D4—32no—
123C22×D4—62no—
124C22×D4—61no—
125C22×D4—62no—
126C22×D4—124no—
127C23×C4—61no—
128C23×C4#6712yes—
129C23×C4C22×D311yescharacteristic
130C2×SD16—68no—
131C2×SD16—128no—
132C22×C8—62no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4#6712yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—124no—
143C2×C22⋊C4—124no—
144C2×C22⋊C4—124no—
145#102—68no—
146C22⋊C8—34no—
147C22⋊C8—62no—
148#95—24no—
149#95—22no—
150#95—62no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—22no—
154C23×C3#12011yescharacteristic
155C23×C3#12111yescharacteristic
156C23×C3#12711yescharacteristic
157C23×C3C2×SD1612yes—
158C23×C3#13712yes—
159C22×D3—24no—
160C22×D3—44no—
161C22×D3—44no—
162C22×D3—42no—
163C22×D3—44no—
164C22×D3—44no—
165C22×D3—44no—
166C22×D3—44no—
167C22×D3—42no—
168C22×D3—44no—
169C22×D3—44no—
170C22×D3—82no—
171C22×D3—82no—
172C22×D3—82no—
173C22×D3—82no—
174C22×D3—82no—
175C22×D3—82no—
176C22×D3—81no—
177C22×D3—81no—
178C22×D3—81no—
179C22×D3—81no—
180C3×Q8—44no—
181C3×Q8—82no—
182C2×C3×C4—24no—
183C2×C3×C4—24no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—44no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—44no—
195C2×C3×C4—82no—
196C2×C3×C4#12012yes—
197C2×C3×C4#12112yes—
198C2×Dic3—44no—
199C2×Dic3—42no—
200C2×Dic3—82no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203D12—28no—
204D12—44no—
205D12—48no—
206C3⋊C8—44no—
207C24—64no—
208C24—62no—
209C24—122no—
210C24—122no—
211C24—122no—
212C24—121no—
213C24—122no—
214C24—122no—
215C24—121no—
216C24C2×#6711yescharacteristic
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×D4—68no—
221C2×D4—64no—
222C2×D4—124no—
223C2×D4—124no—
224C2×D4—128no—
225C2×D4—124no—
226C2×D4—122no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—62no—
235C22×C4—122no—
236C22×C4—121no—
237C22×C4—121no—
238C22×C4—122no—
239C22×C4#21514yescomplemented
240C22×C4D3×D411yescharacteristic
241C22×C4D3×D412yes—
242C22×C4#23711yescharacteristic
243C22×C4C2×#6712yes—
244SD16—1216no—
245C2×C8—64no—
246C2×C8—124no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—1216no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#39211yescharacteristic
265C22×C3#44812yes—
266C22×C3#45311yescharacteristic
267C22×C3#45411yescharacteristic
268C22×C3B(4,𝔽2)12yes—
269C22×C3#46112yes—
270C22×C3#49211yescharacteristic
271C22×C3#49711yescharacteristic
272C2×D3—48no—
273C2×D3—44no—
274C2×D3—84no—
275C2×D3—84no—
276C2×D3—84no—
277C2×D3—84no—
278C2×D3—84no—
279C2×D3—82no—
280C2×D3—82no—
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—44no—
284C3×C4—82no—
285C3×C4—82no—
286Dic3—82no—
287C23—21no—
288C23—22no—
289C23—21no—
290C23—64no—
291C23—124no—
292C23—124no—
293C23—122no—
294C23—124no—
295C23—124no—
296C23—124no—
297C23—124no—
298C23—122no—
299C23—124no—
300C23—124no—
301C23—242no—
302C23—242no—
303C23—242no—
304C23—242no—
305C23—242no—
306C23—242no—
307C23—241no—
308C23—241no—
309C23—241no—
310C23—241no—
311C23#93811yescharacteristic
312C23#94011yescharacteristic
313C23C2×#21512yes—
314C23#94212yes—
315C23#95311yescharacteristic
316Q8—44no—
317Q8—82no—
318D4—68no—
319D4—124no—
320D4—128no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—124no—
336C2×C4—122no—
337C2×C4—241no—
338C2×C4—242no—
339C2×C4—241no—
340C2×C4#93712yes—
341C2×C4#93912yes—
342C8—124no—
343C2×C3—21no—
344C2×C3—22no—
345C2×C3—21no—
346C2×C3#159612yes—
347C2×C3#159912yes—
348C2×C3#200611yescharacteristic
349C2×C3#201311yescharacteristic
350C2×C3#202111yescharacteristic
351D3—84no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—128no—
361C22—124no—
362C22—244no—
363C22—244no—
364C22—244no—
365C22—242no—
366C22—242no—
367C22—244no—
368C22—244no—
369C22#508312yes—
370C22#508511yescharacteristic
371C22#508611yescharacteristic
372C22#522311yescharacteristic
373C22#522712yes—
374C22#524111yescharacteristic
375C22#524412yes—
376C22#524511yescharacteristic
377C4—22no—
378C4—22no—
379C4—44no—
380C4—82no—
381C4—82no—
382C4—242no—
383C3#996611yescharacteristic, complemented, Sylow
384C2—21no—
385C2—22no—
386C2—21no—
387C2—244no—
388C2#7179111yescharacteristic
389C2#7185111yescharacteristic
390C2#7187911yescharacteristic
391C2#7189212yes—
392C2#7190212yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2659C2
socleC23×C3#120
prime radicalC23×D12C22
hypercenter#2659D3
hypocenterC3#9966
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
1. C3×#2659C2semidirect
3. C2×#4538C2semidirect
5. #67613C2semidirect
22. #2659D3semidirect
91. C22×C3×C4SD16semidirect
239. C22×C4#215semidirect
383. C3#9966semidirect

Maximal subgroups

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

subgroupindex
C3×#26592
C2×#50462
C2×#45382
#694262
#676132
#99663