Finite Group Databasebeta

← #10695744

Subgroups — #10695744

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
1C2×#4731C212yesmaximal
2#73361C211yesmaximal, characteristic, complemented
3C2×#4467C212yesmaximal, complemented
4#69426C211yesmaximal, characteristic
5C3×#1280C211yesmaximal, characteristic, complemented
6#9966—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C22×(C2×D3)⋊C4—22no—
9C2×#924C2212yes—
10C22×#204—22no—
11C2×C3×C22⋊C8C2212yes—
12C2×C3×#95C2211yescharacteristic
13#5212—22no—
14C2×C3⋊#95C2211yescharacteristic
15C2×#821—22no—
16#4731—24no—
17#4731—24no—
18#4661—22no—
19#4656—22no—
20#4467—24no—
21C2×#450—32no—
22#2659—31no—
23C2×#328—32no—
24#1438—31no—
25#1280D311yescharacteristic, complemented
26C24×D3—21no—
27C23×C3×C4D412yes—
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C22×D12—22no—
31C22×D12—22no—
32C22×D12—21no—
33C22×D12—44no—
34C22×D12D412yes—
35C22×Dic6D412yes—
36C22×C3×C8—22no—
37C2×#217—22no—
38C2×(C2×D3)⋊C4—44no—
39C2×(C2×D3)⋊C4—44no—
40C2×(C2×D3)⋊C4—44no—
41C2×C4⋊Dic3—22no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3D412yes—
44#924—22no—
45#924—22no—
46#924—22no—
47#924—22no—
48C2×Dic3⋊C4—22no—
49C2×#204—28no—
50C2×#204—48no—
51C3×C22⋊C8—22no—
52C3×C22⋊C8D414yes—
53C3×#95—22no—
54C3⋊#95—24no—
55C3⋊#95—22no—
56#821—28no—
57C23×D4—31no—
58C22×SD16—62no—
59C2×C22⋊Q8—32no—
60C22×C22⋊C4—62no—
61#450—64no—
62#450—64no—
63C2×#102—62no—
64C2×C22⋊C8C2×D312yes—
65#398—62no—
66#395—62no—
67#380—62no—
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—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—42no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4SD1614yescomplemented
87C22×C3×C4C2×D412yes—
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4111yescharacteristic
91C22×Dic3—22no—
92C22×Dic3—22no—
93C22×Dic3—22no—
94C22×Dic3—42no—
95C22×Dic3—41no—
96C22×Dic3—41no—
97C22×Dic3—42no—
98C2×D12—28no—
99C2×D12—24no—
100C2×D12—44no—
101C2×D12—48no—
102C2×D12—44no—
103C2×D12—44no—
104C2×D12—42no—
105C2×Dic6—22no—
106C2×Dic6—22no—
107C2×Dic6—44no—
108C2×C3×C8—24no—
109C2×C3×C8—44no—
110#217—42no—
111#217—42no—
112(C2×D3)⋊C4—416no—
113C4⋊Dic3—22no—
114C4⋊Dic3—22no—
115C4⋊Dic3—82no—
116C4⋊Dic3—82no—
117Dic3⋊C4—42no—
118Dic3⋊C4—42no—
119#204—416no—
120C25—61no—
121C22×Q8—32no—
122C22×D4—32no—
123C22×D4—62no—
124C22×D4—62no—
125C22×D4—61no—
126C22×D4—124no—
127C23×C4—61no—
128C23×C4D1212yes—
129C23×C4C22×D311yescharacteristic
130C2×SD16—68no—
131C2×SD16—128no—
132C22×C8—22no—
133C22⋊Q8—62no—
134C22⋊Q8—62no—
135C22⋊Q8—62no—
136C22⋊Q8—62no—
137C2×C4⋊C4—32no—
138C2×C4⋊C4—62no—
139C2×C4⋊C4—62no—
140C2×C4⋊C4—62no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4—124no—
143C2×C22⋊C4—124no—
144C2×C22⋊C4—124no—
145#102—68no—
146C22⋊C8—22no—
147C22⋊C8D1214yes—
148#95—22no—
149#95—64no—
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—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—44no—
188C2×C3×C4—82no—
189C2×C3×C4#12012yes—
190C2×C3×C4#12112yes—
191C2×Dic3—44no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196C2×Dic3—42no—
197C2×Dic3—44no—
198C2×Dic3—82no—
199C2×Dic3—81no—
200C2×Dic3—81no—
201D12—28no—
202D12—44no—
203D12—48no—
204Dic6—44no—
205Dic6—82no—
206C3×C8—44no—
207C24—64no—
208C24—62no—
209C24—122no—
210C24—122no—
211C24—122no—
212C24—121no—
213C24—121no—
214C24—122no—
215C24—122no—
216C24C2×D1211yescharacteristic
217C2×Q8—62no—
218C2×Q8—62no—
219C2×Q8—124no—
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—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—62no—
235C22×C4—122no—
236C22×C4—121no—
237C22×C4—121no—
238C22×C4—122no—
239C22×C4#20414yescomplemented
240C22×C4C2×D1212yes—
241C22×C4D3×D412yes—
242C22×C4D3×D411yescharacteristic
243C22×C4#23911yescharacteristic
244SD16—1216no—
245C2×C8—24no—
246C2×C8—44no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—122no—
250C4⋊C4—122no—
251C4⋊C4—242no—
252C4⋊C4—242no—
253C22⋊C4—1216no—
254C22⋊C4—122no—
255C22⋊C4—122no—
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—
284Dic3—82no—
285Dic3—82no—
286Dic3—82no—
287C23—22no—
288C23—21no—
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—241no—
307C23—241no—
308C23—242no—
309C23—241no—
310C23—241no—
311C23#87411yescharacteristic
312C23#88211yescharacteristic
313C23#89511yescharacteristic
314C23C2×#20412yes—
315C23#90912yes—
316Q8—124no—
317Q8—242no—
318D4—68no—
319D4—124no—
320D4—128no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—82no—
330C2×C4—124no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—124no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—242no—
338C2×C4—241no—
339C2×C4—241no—
340C2×C4#88212yes—
341C2×C4#89512yes—
342C8—44no—
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—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—128no—
361C22—124no—
362C22—244no—
363C22—244no—
364C22—244no—
365C22—244no—
366C22—244no—
367C22—242no—
368C22—242no—
369C22#467111yescharacteristic
370C22#469311yescharacteristic
371C22#471511yescharacteristic
372C22#473812yes—
373C22#474112yes—
374C22#474711yescharacteristic
375C22#475311yescharacteristic
376C22#484112yes—
377C4—22no—
378C4—22no—
379C4—44no—
380C4—242no—
381C4—242no—
382C4—242no—
383C3#996611yescharacteristic, complemented, Sylow
384C2—21no—
385C2—22no—
386C2—21no—
387C2—244no—
388C2#6811412yes—
389C2#6813812yes—
390C2#6868711yescharacteristic
391C2#6896511yescharacteristic
392C2#6900811yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1280C2
socleC23×C3#120
prime radicalC23×D12C22
hypercenter#1280D3
hypocenterC3#9966
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
2. #73361C2semidirect
3. C2×#4467C2semidirect
5. C3×#1280C2semidirect
25. #1280D3semidirect
86. C22×C3×C4SD16semidirect
239. C22×C4#204semidirect
383. C3#9966semidirect

Maximal subgroups

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

subgroupindex
C2×#47312
#733612
C2×#44672
#694262
C3×#12802
#99663