Finite Group Databasebeta

← #10896387

Subgroups — #10896387

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73390C211yesmaximal, characteristic, complemented
2#73389C211yesmaximal, characteristic
3C3×#1832C211yesmaximal, characteristic, complemented
4#70457C212yesmaximal
5#70455C212yesmaximal, complemented
6#68682C211yesmaximal, characteristic, complemented
7#68681C211yesmaximal, characteristic, complemented
8#68679C211yesmaximal, characteristic, complemented
9#68679C211yesmaximal, characteristic, complemented
10#68671C212yesmaximal, complemented
11#68670C212yesmaximal, complemented
12#29265—31nomaximal, Sylow
13C2×#874C2211yescharacteristic
14C2×C4×D12C2211yescharacteristic
15C2×C4×D12C2211yescharacteristic
16C2×#870C2211yescharacteristic
17C2×C3×C4⋊C8C2212yes—
18C3×#376C2211yescharacteristic
19C2×#821C2212yes—
20C2×#821C2212yes—
21C2×#821C2212yes—
22C2×#821C2212yes—
23C2×#818C2212yes—
24C2×#817C2212yes—
25#4939C2211yescharacteristic
26#4939C2211yescharacteristic
27#4936C2211yescharacteristic
28#4935C2211yescharacteristic
29C4×C4⋊Dic3C2212yes—
30#4928C2212yes—
31#4665—22no—
32#4665—22no—
33#4655—22no—
34#4647—22no—
35C3×#336C2212yes—
36C3×#336C2211yescharacteristic
37C3×#336C2211yescharacteristic
38#4551C2212yes—
39#4551C2212yes—
40#2330—31no—
41#2328—31no—
42#2094—31no—
43#2093—31no—
44#2025—31no—
45#2025—31no—
46#1982—32no—
47#1949—32no—
48#1864—32no—
49#1862—32no—
50#1832D311yescharacteristic, complemented
51C22×D12—22no—
52C22×D12C2311yescharacteristic
53C22×D12C2311yescharacteristic
54C22×C4×D3—21no—
55C22×C4×D3—21no—
56C22×C3×C8C2312yes—
57C22×C3×C8C2312yes—
58C2×C3×C42C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×C3×C42C2311yescharacteristic
61C2×(C2×D3)⋊C4—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3C2312yes—
67C2×C4⋊Dic3C2311yescharacteristic
68C2×C4⋊Dic3C2311yescharacteristic
69C2×C4⋊Dic3C2312yes—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×C4×Dic3—22no—
73#874—22no—
74#874Q814yes—
75C4×D12—42no—
76C4×D12—42no—
77#870—24no—
78C3×C4⋊C8—24no—
79C3×#95D412yes—
80C3×#95D412yes—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84#821—24no—
85#821—24no—
86#821—42no—
87#821—42no—
88#818—42no—
89#817—42no—
90C2×#127—31no—
91C2×#125—31no—
92C2×C4×D4—31no—
93C2×C4×D4—31no—
94C2×#107—32no—
95C2×#106—32no—
96C2×C4⋊C8C2×D312yes—
97C2×#102—32no—
98C2×#102—32no—
99C2×#102—32no—
100C2×#102—32no—
101#390—31no—
102#390—31no—
103#389—32no—
104#389—62no—
105#387—62no—
106#386—62no—
107#386—62no—
108#384—31no—
109#382—31no—
110C4×C4⋊C4—32no—
111#376C2×D311yescharacteristic
112#340—32no—
113#340—32no—
114#336C2×D311yescharacteristic
115#336C2×D311yescharacteristic
116#336C2×D312yes—
117C23×D3—21no—
118C23×D3—21no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2411yescharacteristic
124C22×Dic3—22no—
125C22×Dic3—22no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—22no—
131C2×D12—22no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—28no—
135C2×D12—82no—
136C2×D12—82no—
137C2×D12C2×Q812yes—
138C2×D12C2×Q812yes—
139C2×C4×D3—42no—
140C2×C4×D3—42no—
141C2×C4×D3—82no—
142C2×C4×D3—82no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C3×C42—22no—
150C3×C42—22no—
151C3×C42C2×Q812yes—
152C3×C42#4112yes—
153(C2×D3)⋊C4—42no—
154(C2×D3)⋊C4—42no—
155(C2×D3)⋊C4—82no—
156(C2×D3)⋊C4—82no—
157C4⋊Dic3—22no—
158C4⋊Dic3—22no—
159C4⋊Dic3—22no—
160C4⋊Dic3—22no—
161C4⋊Dic3—22no—
162C4⋊Dic3—82no—
163C4⋊Dic3#4114yes—
164Dic3⋊C4—28no—
165Dic3⋊C4—82no—
166C4×Dic3—82no—
167C22×D4—31no—
168C22×D4—31no—
169C22×D4—62no—
170C23×C4—61no—
171C23×C4—61no—
172C22×C8C22×D312yes—
173C22×C8C22×D312yes—
174#127—34no—
175#127—62no—
176#125—64no—
177C4×D4—122no—
178C4×D4—122no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—62no—
184C2×C4⋊C4—62no—
185C2×C4⋊C4—62no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—62no—
190C2×C42—62no—
191C2×C42C22×D311yescharacteristic
192C2×C42C22×D311yescharacteristic
193C2×C42C22×D311yescharacteristic
194#107—122no—
195#106—122no—
196C4⋊C8—24no—
197#102—64no—
198#102—64no—
199#102—122no—
200#102—122no—
201#95—62no—
202#95—62no—
203#95—62no—
204#95D1212yes—
205#95D1212yes—
206C23×C3C22×D411yescharacteristic
207C22×D3—22no—
208C22×D3—22no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3—41no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—42no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4C22⋊Q814yes—
227C2×C3×C4C22×Q811yescharacteristic
228C2×C3×C4C2×#4111yescharacteristic
229C2×C3×C4C2×#4112yes—
230C2×C3×C4H(5,𝔽2)12yes—
231C2×Dic3—24no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241D12—24no—
242D12—24no—
243D12—42no—
244D12—42no—
245D12—88no—
246C4×D3—84no—
247C4×D3—84no—
248C3×C8—42no—
249C3×C8—42no—
250C24—61no—
251C24—61no—
252C2×D4—32no—
253C2×D4—32no—
254C2×D4—68no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—242no—
262C2×D4—242no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—122no—
268C22×C4—122no—
269C22×C4—242no—
270C22×C4—242no—
271C22×C4C2×D1211yescharacteristic
272C22×C4C2×D1212yes—
273C22×C4C2×D1211yescharacteristic
274C22×C4C2×D1212yes—
275C22×C4C23×D311yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—22no—
282C4⋊C4—34no—
283C4⋊C4—68no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—242no—
290C4⋊C4—242no—
291C22⋊C4—122no—
292C22⋊C4—122no—
293C22⋊C4—242no—
294C22⋊C4—242no—
295C42—22no—
296C42—22no—
297C42—242no—
298C42#23712yes—
299C42D3×Q812yes—
300C22×C3C2×C22⋊Q811yescharacteristic
301C22×C3#52511yescharacteristic
302C22×C3#53511yescharacteristic
303C22×C3#57612yes—
304C22×C3#57712yes—
305C2×D3—44no—
306C2×D3—44no—
307C2×D3—42no—
308C2×D3—42no—
309C2×D3—82no—
310C2×D3—82no—
311C2×D3—81no—
312C2×D3—81no—
313C2×D3—81no—
314C2×D3—81no—
315C3×C4—24no—
316C3×C4—42no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4#55412yes—
320C3×C4#55912yes—
321Dic3—44no—
322Dic3—81no—
323Dic3—82no—
324Dic3—81no—
325C23—62no—
326C23—62no—
327C23—122no—
328C23—122no—
329C23—122no—
330C23—122no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23—121no—
335C23C22×D1211yescharacteristic
336D4—64no—
337D4—64no—
338D4—122no—
339D4—122no—
340D4—248no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—64no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—122no—
355C2×C4—122no—
356C2×C4—122no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4—244no—
361C2×C4—244no—
362C2×C4#89714yes—
363C2×C4C2×#23711yescharacteristic
364C2×C4C2×D3×Q811yescharacteristic
365C2×C4D3×#4112yes—
366C2×C4#100912yes—
367C8—42no—
368C8—42no—
369C2×C3#258011yescharacteristic
370C2×C3#308112yes—
371C2×C3#311912yes—
372C2×C3#322011yescharacteristic
373C2×C3#322211yescharacteristic
374D3—82no—
375D3—82no—
376C22—122no—
377C22—122no—
378C22—124no—
379C22—124no—
380C22—242no—
381C22—242no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22—241no—
386C22C2×#89711yescharacteristic
387C22#552411yescharacteristic
388C22#552511yescharacteristic
389C22#575112yes—
390C22#575212yes—
391C4—24no—
392C4—42no—
393C4—41no—
394C4—41no—
395C4—124no—
396C4—241no—
397C4—242no—
398C4—241no—
399C4#568112yes—
400C4#569012yes—
401C3#2926511yescharacteristic, complemented, Sylow
402C2—242no—
403C2—242no—
404C2#7419011yescharacteristic
405C2#7909012yes—
406C2#7938412yes—
407C2#7939011yescharacteristic
408C2#7939111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1832C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1832D3
hypocenterC3#29265
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #1832

Complements

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

NH ≅ G/Ntype
1. #73390C2semidirect
3. C3×#1832C2semidirect
5. #70455C2semidirect
6. #68682C2semidirect
7. #68681C2semidirect
8. #68679C2semidirect
9. #68679C2semidirect
10. #68671C2semidirect
11. #68670C2semidirect
50. #1832D3semidirect
401. C3#29265semidirect

Maximal subgroups

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

subgroupindex
#733902
#733892
C3×#18322
#704572
#704552
#686822
#686812
#686792
#686712
#686702
#292653