Finite Group Databasebeta

← #10848790

Subgroups — #10848790

Order 768.

1152 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5518C211yesmaximal, characteristic, complemented
2C3×#2619C211yesmaximal, characteristic
3C2×#5076C211yesmaximal, characteristic, complemented
4C2×#5071C211yesmaximal, characteristic, complemented
5#71685C214yesmaximal, complemented
6#71582C211yesmaximal, characteristic, complemented
7#69660C214yesmaximal, complemented
8#69554C211yesmaximal, characteristic, complemented
9#69552C211yesmaximal, characteristic, complemented
10#25607—31nomaximal, Sylow
11C2×C3×#127C2211yescharacteristic
12C2×C3×#121C2212yes—
13C2×C3×#117C2211yescharacteristic
14C22×#214C2214yes—
15C2×#924C2212yes—
16#5518—24no—
17#5518—24no—
18C2×#889—22no—
19C2×#879—22no—
20C2×#870C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2212yes—
23C3×#394—22no—
24C3×#392—22no—
25C3×#390C2212yes—
26C3×#382C2211yescharacteristic
27C2×#832C2214yes—
28C2×#830C2212yes—
29#5076—24no—
30#5071—24no—
31C2×#810C2214yes—
32C2×#803C2211yescharacteristic
33C2×C4×C3⋊C8C2211yescharacteristic
34#4529C2214yes—
35#4524C2212yes—
36C2×#541—31no—
37C2×#492—31no—
38C2×#490—31no—
39#2619D311yescharacteristic
40#2000—34no—
41#1991—31no—
42#1960—31no—
43#1880—34no—
44#1750—31no—
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C22×Dic6C2312yes—
50C3×#127—22no—
51C3×#127—22no—
52C3×#121—44no—
53C3×#117—22no—
54C3×#117—22no—
55C2×C3×C4⋊C4D412yes—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4D412yes—
60C2×C3×C42C2311yescharacteristic
61C2×C3×C42C2311yescharacteristic
62C2×#217—22no—
63C2×#214—28no—
64C2×#214—28no—
65C2×#214—44no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3C2312yes—
68#924—24no—
69#924—24no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×C4×Dic3—22no—
74C22×C3⋊C8C2312yes—
75C22×C3⋊C8C2312yes—
76#889—216no—
77#879—216no—
78#870—22no—
79#870D414yescomplemented
80#869—22no—
81#869D414yescomplemented
82C4×Dic6—24no—
83C4×Dic6—24no—
84C3×#95—22no—
85#832—28no—
86#830—24no—
87#810—44no—
88#803—22no—
89C4×C3⋊C8—22no—
90C22×SD16—34no—
91#541—64no—
92#541—64no—
93C2×C4⋊Q8—31no—
94C2×#127C2×D311yescharacteristic
95C2×#126—62no—
96C2×#125—31no—
97C2×#125—62no—
98C2×C22⋊Q8—32no—
99C2×#121C2×D312yes—
100C2×C4×Q8—32no—
101C2×#117C2×D311yescharacteristic
102#492—64no—
103#490—64no—
104C2×#103—34no—
105C2×#102—34no—
106C2×C22⋊C8—32no—
107C2×#97—31no—
108C2×C4×C8—31no—
109#394—22no—
110#392—22no—
111#390C2×D312yes—
112#382C2×D311yescharacteristic
113#340—34no—
114#336—32no—
115C24×C3—22no—
116C24×C3C2×D411yescharacteristic
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—44no—
122C2×C3×D4—44no—
123C2×C3×D4—84no—
124C2×C3×D4—84no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C2×Dic6—24no—
137C2×Dic6—24no—
138C2×Dic6—22no—
139C2×Dic6C2×D414yes—
140C3×C4⋊C4—24no—
141C3×C4⋊C4—24no—
142C3×C22⋊C4—24no—
143C3×C22⋊C4—44no—
144C3×C22⋊C4—84no—
145C3×C42—21no—
146C3×C42—21no—
147C3×C42C2×D412yes—
148C3×C42C2×D412yes—
149#217—28no—
150#214—416no—
151C4⋊Dic3—28no—
152C4⋊Dic3—22no—
153C4⋊Dic3C2×D414yes—
154Dic3⋊C4—28no—
155Dic3⋊C4—28no—
156Dic3⋊C4—28no—
157C4×Dic3—28no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8#4114yes—
164C22×Q8—32no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4C22×D312yes—
168C23×C4C22×D311yescharacteristic
169C2×SD16—68no—
170C2×SD16—68no—
171C2×SD16—124no—
172C22×C8—32no—
173C22×C8—32no—
174C4⋊Q8—34no—
175C4⋊Q8—62no—
176#127—22no—
177#127—22no—
178#126—616no—
179#125—34no—
180#125—616no—
181#125—62no—
182C22⋊Q8—64no—
183C22⋊Q8—64no—
184#121—44no—
185C4×Q8—64no—
186C4×Q8—64no—
187#117—22no—
188#117—22no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4#6712yes—
195C2×C4⋊C4C22×D312yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4#6712yes—
200C2×C42—62no—
201C2×C42C22×D311yescharacteristic
202C2×C42C22×D311yescharacteristic
203#103—124no—
204#102—68no—
205C22⋊C8—64no—
206#97—62no—
207C4×C8—62no—
208#95—22no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—24no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—44no—
217C23×C3—44no—
218C23×C3C22×D411yescharacteristic
219C3×D4—44no—
220C3×D4—44no—
221C3×D4—88no—
222C3×D4—88no—
223C2×C3×C4—24no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4#12014yes—
234C2×C3×C4C22×D411yescharacteristic
235C2×C3×C4C22×D411yescharacteristic
236C2×C3×C4C2×#4112yes—
237C2×C3×C4H(5,𝔽2)12yes—
238C2×Dic3—24no—
239C2×Dic3—24no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—44no—
243Dic6—28no—
244Dic6—44no—
245C3⋊C8—24no—
246C3⋊C8—42no—
247C24—22no—
248C24C2×#6711yescharacteristic
249C2×Q8—34no—
250C2×Q8—62no—
251C2×Q8—64no—
252C2×Q8—64no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—84no—
260C2×D4—84no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4C2×#6712yes—
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6712yes—
271C22×C4C23×D311yescharacteristic
272SD16—1216no—
273C2×C8—34no—
274C2×C8—62no—
275C2×C8—62no—
276C2×C8—62no—
277C2×C8—62no—
278C2×C8—62no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—34no—
282C4⋊C4—68no—
283C4⋊C4—68no—
284C4⋊C4—68no—
285C4⋊C4—68no—
286C4⋊C4—62no—
287C22⋊C4—24no—
288C22⋊C4—44no—
289C22⋊C4—68no—
290C22⋊C4—84no—
291C42—21no—
292C42—21no—
293C42—68no—
294C42D3×D412yes—
295C42D3×D412yes—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—42no—
299C22×C3—42no—
300C22×C3—84no—
301C22×C3—82no—
302C22×C3—82no—
303C22×C3—82no—
304C22×C3—84no—
305C22×C3C2×#12011yescharacteristic
306C22×C3#53211yescharacteristic
307C22×C3#53511yescharacteristic
308C22×C3#57714yes—
309C3×C4—24no—
310C3×C4—41no—
311C3×C4—42no—
312C3×C4—82no—
313C3×C4#54612yes—
314C3×C4#55012yes—
315Dic3—44no—
316Dic3—44no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—24no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—44no—
325C23—44no—
326C23C22×#6711yescharacteristic
327Q8—68no—
328Q8—124no—
329D4—44no—
330D4—44no—
331D4—88no—
332D4—88no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—64no—
344C2×C4—64no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—124no—
348C2×C4#93714yes—
349C2×C4C2×D3×D411yescharacteristic
350C2×C4C2×D3×D411yescharacteristic
351C2×C4#100412yes—
352C2×C4D3×#4112yes—
353C8—64no—
354C8—122no—
355C2×C3—81no—
356C2×C3—162no—
357C2×C3#243111yescharacteristic
358C2×C3#301012yes—
359C2×C3#317412yes—
360C2×C3#322212yes—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—42no—
365C22—84no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—84no—
370C22C2×#93711yescharacteristic
371C22#580511yescharacteristic
372C22#583211yescharacteristic
373C22#583614yes—
374C4—24no—
375C4—42no—
376C4—41no—
377C4—82no—
378C4—124no—
379C4—124no—
380C4#559912yes—
381C4#561212yes—
382C3#2560711yescharacteristic, complemented, Sylow
383C2—81no—
384C2—162no—
385C2#8265211yescharacteristic
386C2#8339012yes—
387C2#8339312yes—
388C2#8343112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2619C2
socleC23×C3C22×D4
hypercenter#2619D3
hypocenterC3#25607
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. C2×#5518C2semidirect
3. C2×#5076C2semidirect
4. C2×#5071C2semidirect
5. #71685C2semidirect
6. #71582C2semidirect
7. #69660C2semidirect
8. #69554C2semidirect
9. #69552C2semidirect
79. #870D4semidirect
81. #869D4semidirect
382. C3#25607semidirect

Maximal subgroups

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

subgroupindex
C2×#55182
C3×#26192
C2×#50762
C2×#50712
#716852
#715822
#696602
#695542
#695522
#256073