Finite Group Databasebeta

← #10692987

Subgroups — #10692987

Order 768.

958 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2404C211yesmaximal, characteristic
2#82083C211yesmaximal, characteristic, complemented
3#72138C212yesmaximal, complemented
4#71539C211yesmaximal, characteristic, complemented
5#67628C212yesmaximal, complemented
6#9475—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×#120—21no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×#217—21no—
11C3×#393C412yescomplemented
12C3×#392C412yescomplemented
13C3×#386—21no—
14C3×#386—22no—
15C3×#381C2211yescharacteristic
16C2×#837—24no—
17C4×#217—21no—
18C4×#217C2212yes—
19#5207C2212yes—
20Dic3×C22⋊C4—24no—
21#4525—22no—
22#2404D311yescharacteristic
23#2273—31no—
24#1789—32no—
25#1462—31no—
26#1284—32no—
27C25×C3D411yescharacteristic
28C2×C3×#41—22no—
29C2×C3×#41—22no—
30C2×C3×#41—44no—
31C22×C3×Q8C2×C411yescharacteristic, complemented
32C22×C3×D4—21no—
33C22×C3×D4—22no—
34C22×C3×D4C2×C411yescharacteristic, complemented
35C23×C3×C4—21no—
36C23×C3×C4D411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—22no—
39C3×#120—44no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4C2×C412yes—
46C2×C3×C22⋊C4D412yes—
47C2×#217—22no—
48C2×#217—28no—
49C2×#217—22no—
50C2×C4×Dic3—22no—
51C2×C4×Dic3D414yescomplemented
52C2×#208—22no—
53C3×#95—22no—
54C3×#95—21no—
55C3×#95C2×C412yescomplemented
56#837—48no—
57C3⋊#95—22no—
58C3⋊#95—22no—
59#830—22no—
60#830—22no—
61C22×#41C2×D311yescharacteristic
62C2×#120—21no—
63C22×C22⋊C4—61no—
64C22×C22⋊C4C2×D311yescharacteristic
65C2×#104—64no—
66#407—32no—
67#393Dic312yescomplemented
68#392Dic312yescomplemented
69#386—21no—
70#386—22no—
71#381C2×D311yescharacteristic
72C4×C22⋊C4—32no—
73C4×C22⋊C4—64no—
74C4×C22⋊C4—61no—
75#323—62no—
76C24×C3—21no—
77C24×C3—24no—
78C24×C3—21no—
79C24×C3—21no—
80C24×C3—42no—
81C24×C3—81no—
82C24×C3C22⋊C412yes—
83C24×C3C2×D411yescharacteristic
84C3×#41—28no—
85C3×#41—44no—
86C3×#41—44no—
87C3×#41—44no—
88C2×C3×Q8—22no—
89C2×C3×Q8—42no—
90C2×C3×D4—24no—
91C2×C3×D4—22no—
92C2×C3×D4—42no—
93C2×C3×D4—42no—
94C2×C3×D4—41no—
95C2×C3×D4—41no—
96C2×C3×D4—44no—
97C2×C3×D4—44no—
98C2×C3×D4—82no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4—42no—
104C22×C3×C4—42no—
105C22×C3×C4—42no—
106C22×C3×C4—42no—
107C22×C3×C4—81no—
108C22×C3×C4C22⋊C412yes—
109C22×C3×C4C22×C411yescharacteristic
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4112yes—
113C22×Dic3—24no—
114C22×Dic3—28no—
115C22×Dic3—82no—
116C22×Dic3—82no—
117C3×C22⋊C4—24no—
118C3×C22⋊C4—24no—
119C3×C22⋊C4—44no—
120C3×C22⋊C4—42no—
121C3×C22⋊C4—82no—
122#217—28no—
123#217—28no—
124C4×Dic3—44no—
125C4×Dic3—82no—
126#208—44no—
127C2×C3⋊C8—42no—
128C25#6711yescharacteristic
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—44no—
132C22×Q8C2×Dic311yescharacteristic, complemented
133C22×D4—21no—
134C22×D4—22no—
135C22×D4C2×Dic311yescharacteristic, complemented
136C23×C4—21no—
137C23×C4—62no—
138C23×C4#6711yescharacteristic
139C23×C4C22×D311yescharacteristic
140C2×M16—62no—
141#120—44no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—68no—
149C2×C22⋊C4—62no—
150C2×C22⋊C4C2×Dic312yes—
151C2×C22⋊C4#6712yes—
152C2×C42—34no—
153C2×C42—62no—
154#104—128no—
155C22⋊C8—62no—
156C22⋊C8—62no—
157#95—22no—
158#95—21no—
159#95—62no—
160#95—62no—
161#95C2×Dic312yescomplemented
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—24no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—41no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—41no—
176C23×C3—42no—
177C23×C3—41no—
178C23×C3—42no—
179C23×C3—81no—
180C23×C3—81no—
181C23×C3—88no—
182C23×C3#10414yescomplemented
183C23×C3C2×C22⋊C411yescharacteristic
184C23×C3#12011yescharacteristic
185C23×C3#12311yescharacteristic
186C3×Q8—24no—
187C3×Q8—42no—
188C3×D4—24no—
189C3×D4—42no—
190C3×D4—44no—
191C3×D4—44no—
192C3×D4—88no—
193C2×C3×C4—24no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—48no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—81no—
208C2×C3×C4C4×D412yes—
209C2×C3×C4#12112yes—
210C2×Dic3—28no—
211C2×Dic3—44no—
212C2×Dic3—44no—
213C2×Dic3—88no—
214C3⋊C8—82no—
215C24—21no—
216C24—24no—
217C24—21no—
218C24—21no—
219C24—42no—
220C24—81no—
221C24#21712yes—
222C24C2×#6711yescharacteristic
223#41—28no—
224#41—44no—
225#41—44no—
226#41—44no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—82no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—64no—
247C22×C4—68no—
248C22×C4—81no—
249C22×C4—242no—
250C22×C4—242no—
251C22×C4#21712yes—
252C22×C4D3×D412yes—
253C22×C4#23712yes—
254C22×C4C22×Dic311yescharacteristic
255C22×C4C2×#6711yescharacteristic
256M16—124no—
257C2×C8—122no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—68no—
263C22⋊C4—68no—
264C22⋊C4—82no—
265C42—124no—
266C42—242no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—24no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—44no—
276C22×C3—44no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—44no—
280C22×C3—42no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3—84no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3#38611yescharacteristic
289C22×C3C2×#10412yes—
290C22×C3B(4,𝔽2)12yes—
291C3×C4—22no—
292C3×C4—22no—
293C3×C4—42no—
294C3×C4—42no—
295C3×C4—82no—
296Dic3—48no—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—41no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—41no—
311C23—42no—
312C23—41no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23—88no—
317C23#83714yescomplemented
318C23#93511yescharacteristic
319C23#93711yescharacteristic
320C23C2×#21711yescharacteristic
321Q8—24no—
322Q8—42no—
323D4—24no—
324D4—42no—
325D4—44no—
326D4—44no—
327D4—88no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—68no—
343C2×C4—81no—
344C2×C4—124no—
345C2×C4—124no—
346C2×C4—248no—
347C2×C4D4×Dic312yes—
348C2×C4#93912yes—
349C8—242no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—41no—
353C2×C3—44no—
354C2×C3—82no—
355C2×C3#188911yescharacteristic
356C2×C3#189611yescharacteristic
357C2×C3#191111yescharacteristic
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—24no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—44no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—82no—
375C22—82no—
376C22—84no—
377C22—81no—
378C22—81no—
379C22#503212yes—
380C22#522011yescharacteristic
381C22C2×#83712yes—
382C4—22no—
383C4—22no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4—128no—
388C3#947511yescharacteristic, complemented, Sylow
389C2—22no—
390C2—22no—
391C2—41no—
392C2—44no—
393C2—82no—
394C2#7166911yescharacteristic
395C2#7171111yescharacteristic
396C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2404C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2404D3
hypocenterC3#9475
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
2. #82083C2semidirect
3. #72138C2semidirect
4. #71539C2semidirect
5. #67628C2semidirect
11. C3×#393C4semidirect
12. C3×#392C4semidirect
31. C22×C3×Q8C2×C4semidirect
34. C22×C3×D4C2×C4semidirect
51. C2×C4×Dic3D4semidirect
55. C3×#95C2×C4semidirect
67. #393Dic3semidirect
68. #392Dic3semidirect
132. C22×Q8C2×Dic3semidirect
135. C22×D4C2×Dic3semidirect
161. #95C2×Dic3semidirect
182. C23×C3#104semidirect
317. C23#837semidirect
388. C3#9475semidirect

Maximal subgroups

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

subgroupindex
C3×#24042
#820832
#721382
#715392
#676282
#94753