Finite Group Databasebeta

← #10896436

Subgroups — #10896436

Order 768.

726 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76297C211yesmaximal, characteristic
2#74322C211yesmaximal, characteristic, complemented
3C3×#2094C211yesmaximal, characteristic
4#70588C212yesmaximal, complemented
5#70563C212yesmaximal, complemented
6#70169C212yesmaximal
7#70162C212yesmaximal
8#69639C211yesmaximal, characteristic
9#69591C211yesmaximal, characteristic, complemented
10#68690C211yesmaximal, characteristic, complemented
11#68606C211yesmaximal, characteristic, complemented
12#29269—31nomaximal, Sylow
13C2×#936C2211yescharacteristic
14C2×D4×Dic3C2211yescharacteristic
15C2×#887C2211yescharacteristic
16C2×#870C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×#102C2212yes—
19C3×#390C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C2×#832C2212yes—
22C2×#816C2212yes—
23C2×Dic3⋊C8C2212yes—
24#4978—22no—
25#4976—22no—
26#4975C2211yescharacteristic
27Dic3×C4⋊C4C2212yes—
28#4972C2212yes—
29C2×#810C2212yes—
30C2×#808C2212yes—
31#4946—22no—
32#4945—22no—
33#4932C2211yescharacteristic
34#4929C2211yescharacteristic
35C3×#336C2211yescharacteristic
36#4551C2212yes—
37#4550C2212yes—
38#4529C2212yes—
39#4524C2211yescharacteristic
40#2330—31no—
41#2326—31no—
42#2096—31no—
43#2094D311yescharacteristic
44#2026—31no—
45#2025—31no—
46#1983—32no—
47#1948—32no—
48#1865—32no—
49#1864—32no—
50#1832—31no—
51C22×C3×D4C2311yescharacteristic
52C23×Dic3—21no—
53C22×Dic6C2311yescharacteristic
54C22×C3×C8C2312yes—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C4⋊C4C2311yescharacteristic
58C2×C3×C22⋊C4—22no—
59C2×C3×C42C2311yescharacteristic
60C2×#217—22no—
61C2×#217—21no—
62#936—22no—
63#936Q814yes—
64D4×Dic3—42no—
65C2×C4⋊Dic3C2311yescharacteristic
66C2×C4⋊Dic3C2312yes—
67C2×Dic3⋊C4—22no—
68C2×Dic3⋊C4—21no—
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4—22no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—22no—
73C2×C4×Dic3C2311yescharacteristic
74C2×C4×Dic3C2311yescharacteristic
75C22×C3⋊C8C2312yes—
76#887—42no—
77#870—24no—
78C3×#106—42no—
79C3×#102—42no—
80#832—24no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95D412yes—
85C3⋊#95D412yes—
86#816—24no—
87Dic3⋊C8—24no—
88#810—42no—
89#808—42no—
90C2×#125—31no—
91C2×#124—31no—
92C2×C4×Q8—31no—
93C2×C4×D4—31no—
94C2×#106—32no—
95C2×#106C2×D312yes—
96C2×C4⋊C8—32no—
97C2×#103—32no—
98C2×#103—32no—
99C2×#102—32no—
100C2×#102C2×D312yes—
101#391—31no—
102#390C2×D311yescharacteristic
103#389—32no—
104#389—62no—
105#388—62no—
106#387—62no—
107#385—62no—
108#384—31no—
109#384C2×D311yescharacteristic
110C4×C4⋊C4—32no—
111#376—31no—
112#340—32no—
113#340—32no—
114#336—32no—
115#336—31no—
116#336C2×D311yescharacteristic
117C24×C3—21no—
118C2×C3×D4—22no—
119C2×C3×D4—21no—
120C2×C3×D4—22no—
121C2×C3×D4C2×Q812yes—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4C2×D412yes—
125C22×C3×C4C2411yescharacteristic
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—42no—
130C22×Dic3—82no—
131C22×Dic3C2×D411yescharacteristic
132C22×Dic3C2×D411yescharacteristic
133C22×Dic3C2×D412yes—
134C2×Dic6—22no—
135C2×Dic6—22no—
136C2×Dic6—21no—
137C2×Dic6C2×Q812yes—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—82no—
145C3×C22⋊C4—82no—
146C3×C42—22no—
147#217—28no—
148#217—42no—
149C4⋊Dic3—22no—
150C4⋊Dic3—22no—
151C4⋊Dic3#4114yes—
152Dic3⋊C4—28no—
153Dic3⋊C4—42no—
154Dic3⋊C4—82no—
155Dic3⋊C4—82no—
156C4×Dic3—22no—
157C4×Dic3—42no—
158C4×Dic3—82no—
159C4×Dic3C2×Q812yes—
160C4×Dic3#4112yes—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C22×Q8—31no—
165C22×D4C22×D311yescharacteristic
166C23×C4—61no—
167C22×C8—32no—
168C22×C8C22×D312yes—
169#125—64no—
170#124—34no—
171#124—62no—
172C4×Q8—122no—
173C4×D4—122no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4C22×D311yescharacteristic
182C2×C4⋊C4C22×D312yes—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—62no—
185C2×C22⋊C4—61no—
186C2×C42—31no—
187C2×C42—31no—
188C2×C42—61no—
189C2×C42—62no—
190C2×C42C22×D311yescharacteristic
191#106—42no—
192#106—122no—
193C4⋊C8—64no—
194#103—64no—
195#103—122no—
196#102—42no—
197#102—64no—
198#95—32no—
199#95—32no—
200#95—62no—
201#95—62no—
202#95—62no—
203C23×C3—22no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3C22×D411yescharacteristic
209C3×D4—24no—
210C3×D4—42no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4C22×Q811yescharacteristic
220C2×C3×C4C2×#4112yes—
221C2×C3×C4C2×#4111yescharacteristic
222C2×C3×C4H(5,𝔽2)12yes—
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—24no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—22no—
230C2×Dic3—22no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—84no—
238C2×Dic3C22⋊Q814yes—
239Dic6—24no—
240Dic6—42no—
241C3×C8—42no—
242C3⋊C8—42no—
243C24—21no—
244C2×Q8—32no—
245C2×Q8—61no—
246C2×Q8—62no—
247C2×Q8—62no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4D3×Q812yes—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—32no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—61no—
258C22×C4—62no—
259C22×C4—61no—
260C22×C4—122no—
261C22×C4—242no—
262C22×C4D3×D412yes—
263C22×C4C23×D311yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—62no—
268C2×C8—62no—
269C2×C8—62no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—34no—
274C4⋊C4—68no—
275C4⋊C4—62no—
276C4⋊C4—62no—
277C4⋊C4—82no—
278C4⋊C4—122no—
279C4⋊C4—242no—
280C4⋊C4—242no—
281C22⋊C4—68no—
282C22⋊C4—82no—
283C22⋊C4—122no—
284C42—22no—
285C42—32no—
286C42—32no—
287C42—62no—
288C42—122no—
289C42—242no—
290C22×C3—44no—
291C22×C3—42no—
292C22×C3—82no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3C2×C22⋊Q811yescharacteristic
296C22×C3#52511yescharacteristic
297C22×C3#53511yescharacteristic
298C22×C3#57712yes—
299C22×C3#57712yes—
300C3×C4—42no—
301C3×C4—82no—
302C3×C4—81no—
303C3×C4#55412yes—
304C3×C4#55912yes—
305Dic3—24no—
306Dic3—42no—
307Dic3—44no—
308Dic3—41no—
309Dic3—41no—
310Dic3—81no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23C2×D3×D411yescharacteristic
317Q8—64no—
318Q8—122no—
319D4—24no—
320D4—42no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—34no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—64no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4—244no—
345C2×C4#100412yes—
346C2×C4C2×D3×Q811yescharacteristic
347C2×C4D3×#4112yes—
348C2×C4D3×#4111yescharacteristic
349C8—42no—
350C8—122no—
351C2×C3—82no—
352C2×C3#258011yescharacteristic
353C2×C3#308412yes—
354C2×C3#312512yes—
355C2×C3#322111yescharacteristic
356C2×C3#322211yescharacteristic
357C22—44no—
358C22—42no—
359C22—81no—
360C22—82no—
361C22—81no—
362C22D3×C22⋊Q811yescharacteristic
363C22#565611yescharacteristic
364C22#571511yescharacteristic
365C22#576212yes—
366C22#577512yes—
367C4—42no—
368C4—64no—
369C4—81no—
370C4—82no—
371C4—121no—
372C4—124no—
373C4—121no—
374C4—122no—
375C4—241no—
376C4#563612yes—
377C4#566412yes—
378C3#2926911yescharacteristic, complemented, Sylow
379C2—82no—
380C2#7804111yescharacteristic
381C2#8106312yes—
382C2#8131312yes—
383C2#8176911yescharacteristic
384C2#8185411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2094C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2094D3
hypocenterC3#29269
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2094

Complements

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

NH ≅ G/Ntype
2. #74322C2semidirect
4. #70588C2semidirect
5. #70563C2semidirect
9. #69591C2semidirect
10. #68690C2semidirect
11. #68606C2semidirect
378. C3#29269semidirect

Maximal subgroups

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

subgroupindex
#762972
#743222
C3×#20942
#705882
#705632
#701692
#701622
#696392
#695912
#686902
#686062
#292693