Finite Group Databasebeta

← #10804128

Subgroups — #10804128

Order 768.

1570 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 243 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
1C22×#935C211yesmaximal, characteristic
2#73817C212yesmaximal, complemented
3C3×#1781C211yesmaximal, characteristic
4#72130C211yesmaximal, characteristic
5#70029C218yesmaximal, complemented
6#69495C212yesmaximal, complemented
7#21747—31nomaximal, Sylow
8C23×C3×D4C2211yescharacteristic
9C24×Dic3C2211yescharacteristic
10C22×#217C412yescomplemented
11C22×#217C2211yescharacteristic
12C2×#935—24no—
13C2×#935—24no—
14C2×#935C414yescomplemented
15C2×#935C2214yes—
16C22×Dic3⋊C4C412yescomplemented
17C2×#877C2214yescomplemented
18C2×C3×#99C2214yes—
19C3×#379C2212yes—
20#5247C2212yes—
21C2×#834C2214yes—
22#4943C2214yescomplemented
23#4942—24no—
24Dic3×C22⋊C4—24no—
25#4527C2218yescomplemented
26C22×#123—31no—
27#2280—32no—
28#1899—38no—
29#1819—32no—
30#1781—31no—
31#1781D311yescharacteristic
32C25×C3C2×C411yescharacteristic
33C25×C3C2311yescharacteristic
34C22×C3×D4—24no—
35C22×C3×D4—44no—
36C22×C3×D4C2×C412yes—
37C22×C3×D4C2312yes—
38C23×C3×C4C2×C411yescharacteristic
39C23×Dic3—44no—
40C23×Dic3C2×C412yes—
41C23×Dic3C2×C412yes—
42C23×Dic3C2×C412yes—
43C23×Dic3D414yes—
44C23×Dic3D414yes—
45C23×Dic3C2312yes—
46C2×C3×C22⋊C4—24no—
47C2×C3×C22⋊C4C2314yes—
48C2×#217—24no—
49C2×#217—22no—
50C2×#217—22no—
51C2×#217—24no—
52C2×#217—24no—
53C2×#217C2×C412yes—
54C2×#217C2×C418yes—
55C2×#217C2314yes—
56C2×#217C2312yes—
57#935—24no—
58#935—28no—
59#935—24no—
60#935—28no—
61#935—24no—
62#935—24no—
63C2×Dic3⋊C4—24no—
64C2×Dic3⋊C4—22no—
65C2×Dic3⋊C4—22no—
66C2×Dic3⋊C4—24no—
67C2×Dic3⋊C4C2×C418yes—
68C2×C4×Dic3—24no—
69#877—48no—
70C3×#99—44no—
71#834—44no—
72C3⋊#95—28no—
73C23×D4C2×D311yescharacteristic
74C24×C4—31no—
75C2×#123—34no—
76C2×#123—34no—
77C2×#123—64no—
78C2×#123—64no—
79C2×#117—34no—
80C22×C4⋊C4—32no—
81C22×C22⋊C4—32no—
82C22×C22⋊C4—31no—
83C2×#99—34no—
84C2×#99C2×D314yes—
85#381—34no—
86#380—64no—
87#379—32no—
88#379C2×D312yes—
89C4×C22⋊C4—64no—
90#342—38no—
91C24×C3—24no—
92C24×C3—22no—
93C24×C3—44no—
94C24×C3—42no—
95C24×C3—44no—
96C24×C3C22×C411yescharacteristic
97C24×C3C22×C412yes—
98C24×C3C22×C411yescharacteristic
99C24×C3C2×D414yes—
100C24×C3C2411yescharacteristic
101C2×C3×D4—24no—
102C2×C3×D4—24no—
103C2×C3×D4—24no—
104C2×C3×D4—24no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—416no—
108C22×C3×C4—22no—
109C22×C3×C4—24no—
110C22×C3×C4—42no—
111C22×C3×C4C22×C412yes—
112C22×Dic3—28no—
113C22×Dic3—22no—
114C22×Dic3—22no—
115C22×Dic3—24no—
116C22×Dic3—44no—
117C22×Dic3—44no—
118C22×Dic3—48no—
119C22×Dic3—48no—
120C22×Dic3—44no—
121C22×Dic3C22⋊C412yes—
122C22×Dic3C22⋊C414yes—
123C22×Dic3C22⋊C412yes—
124C22×Dic3C22⋊C412yes—
125C22×Dic3C22⋊C414yes—
126C22×Dic3C22⋊C412yes—
127C22×Dic3C22×C414yes—
128C22×Dic3C22×C414yes—
129C22×Dic3C2×D414yes—
130C22×Dic3C2×D412yes—
131C22×Dic3C2×D412yes—
132C3×C22⋊C4—28no—
133C3×C22⋊C4—48no—
134#217—24no—
135#217—24no—
136#217—28no—
137#217—28no—
138#217—28no—
139#217—48no—
140Dic3⋊C4—28no—
141Dic3⋊C4—28no—
142Dic3⋊C4—48no—
143C4×Dic3—48no—
144C25C4×D311yescharacteristic
145C25C22×D311yescharacteristic
146C22×D4—24no—
147C22×D4—44no—
148C22×D4C4×D312yes—
149C22×D4C22×D312yes—
150C23×C4—32no—
151C23×C4—32no—
152C23×C4—32no—
153C23×C4—34no—
154C23×C4—34no—
155C23×C4—32no—
156C23×C4—124no—
157C23×C4C4×D311yescharacteristic
158#123—64no—
159#123—68no—
160#123—64no—
161#123—68no—
162#123—64no—
163#123—64no—
164#117—128no—
165C2×C4⋊C4—38no—
166C2×C4⋊C4—62no—
167C2×C4⋊C4—64no—
168C2×C4⋊C4—62no—
169C2×C4⋊C4—64no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—38no—
172C2×C22⋊C4—32no—
173C2×C22⋊C4—32no—
174C2×C22⋊C4—34no—
175C2×C22⋊C4—62no—
176C2×C22⋊C4—64no—
177C2×C22⋊C4—62no—
178C2×C22⋊C4—64no—
179C2×C22⋊C4—64no—
180C2×C22⋊C4C22×D314yes—
181C2×C42—64no—
182#99—44no—
183#99—124no—
184#95—68no—
185C23×C3—22no—
186C23×C3—24no—
187C23×C3—22no—
188C23×C3—28no—
189C23×C3—28no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3—42no—
194C23×C3—44no—
195C23×C3—42no—
196C23×C3—416no—
197C23×C3—44no—
198C23×C3—44no—
199C23×C3—44no—
200C23×C3—44no—
201C23×C3—88no—
202C23×C3C2×C22⋊C412yes—
203C23×C3C2×C22⋊C412yes—
204C23×C3C23×C411yescharacteristic
205C23×C3C22×D412yes—
206C23×C3H(5,𝔽2)14yes—
207C3×D4—48no—
208C3×D4—48no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—44no—
213C2×C3×C4—48no—
214C2×C3×C4—44no—
215C2×Dic3—28no—
216C2×Dic3—24no—
217C2×Dic3—24no—
218C2×Dic3—28no—
219C2×Dic3—24no—
220C2×Dic3—42no—
221C2×Dic3—44no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—44no—
225C2×Dic3—42no—
226C2×Dic3—44no—
227C2×Dic3—44no—
228C2×Dic3—44no—
229C2×Dic3—48no—
230C2×Dic3—44no—
231C2×Dic3C2×C22⋊C414yes—
232C2×Dic3C2×C22⋊C414yes—
233C24—24no—
234C24—22no—
235C24—44no—
236C24—42no—
237C24—44no—
238C24C2×C4×D311yescharacteristic
239C24C2×C4×D312yes—
240C24C2×C4×D311yescharacteristic
241C24D3×D414yes—
242C24C23×D311yescharacteristic
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—416no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—34no—
253C22×C4—34no—
254C22×C4—32no—
255C22×C4—32no—
256C22×C4—32no—
257C22×C4—34no—
258C22×C4—32no—
259C22×C4—32no—
260C22×C4—34no—
261C22×C4—32no—
262C22×C4—34no—
263C22×C4—42no—
264C22×C4—68no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—64no—
268C22×C4—124no—
269C22×C4—124no—
270C22×C4—128no—
271C22×C4—128no—
272C22×C4—124no—
273C22×C4C2×C4×D312yes—
274C4⋊C4—68no—
275C4⋊C4—68no—
276C4⋊C4—128no—
277C22⋊C4—28no—
278C22⋊C4—48no—
279C22⋊C4—68no—
280C22⋊C4—68no—
281C22⋊C4—64no—
282C22⋊C4—64no—
283C22⋊C4—68no—
284C22⋊C4—128no—
285C42—128no—
286C22×C3—28no—
287C22×C3—22no—
288C22×C3—24no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—24no—
293C22×C3—44no—
294C22×C3—416no—
295C22×C3—44no—
296C22×C3—42no—
297C22×C3—44no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—88no—
301C22×C3#41014yes—
302C22×C3C22×C22⋊C411yescharacteristic
303C22×C3#51812yes—
304C22×C3#53414yes—
305C3×C4—42no—
306C3×C4—84no—
307Dic3—24no—
308Dic3—24no—
309Dic3—44no—
310Dic3—84no—
311C23—22no—
312C23—24no—
313C23—22no—
314C23—28no—
315C23—28no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—42no—
320C23—44no—
321C23—42no—
322C23—416no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—88no—
328C23D3×C22⋊C412yes—
329C23D3×C22⋊C412yes—
330C23C22×C4×D311yescharacteristic
331C23C2×D3×D412yes—
332C23#100414yes—
333D4—48no—
334D4—48no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—34no—
339C2×C4—34no—
340C2×C4—44no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—68no—
344C2×C4—64no—
345C2×C4—64no—
346C2×C4—68no—
347C2×C4—64no—
348C2×C4—122no—
349C2×C4—124no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—124no—
353C2×C4—122no—
354C2×C4—124no—
355C2×C4—124no—
356C2×C4—124no—
357C2×C4—128no—
358C2×C4—124no—
359C2×C3—24no—
360C2×C3—22no—
361C2×C3—24no—
362C2×C3—42no—
363C2×C3—82no—
364C2×C3#230811yescharacteristic
365C2×C3C2×#41012yes—
366C22—28no—
367C22—24no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—24no—
372C22—22no—
373C22—44no—
374C22—416no—
375C22—44no—
376C22—44no—
377C22—42no—
378C22—82no—
379C22—82no—
380C22—88no—
381C22#474014yes—
382C22C2×D3×C22⋊C411yescharacteristic
383C22#548612yes—
384C22#558914yes—
385C4—42no—
386C4—64no—
387C4—64no—
388C4—84no—
389C4—124no—
390C4—244no—
391C3#2174711yescharacteristic, complemented, Sylow
392C2—24no—
393C2—22no—
394C2—24no—
395C2—42no—
396C2—82no—
397C2#7384911yescharacteristic
398C2C2×#474012yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1781C2
socleC22×C3C22×C22⋊C4
prime radicalC23×C3×D4C22
cosocleC24×C3C24
hypercenter#1781D3
hypocenterC3#21747
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #73817C2semidirect
5. #70029C2semidirect
6. #69495C2semidirect
10. C22×#217C4semidirect
14. C2×#935C4semidirect
16. C22×Dic3⋊C4C4semidirect
17. C2×#877C22semidirect
22. #4943C22semidirect
25. #4527C22semidirect
391. C3#21747semidirect

Maximal subgroups

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

subgroupindex
C22×#9352
#738172
C3×#17812
#721302
#700292
#694952
#217473