Finite Group Databasebeta

← #10828577

Subgroups — #10828577

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5518C211yesmaximal, characteristic
2C3×#2316C211yesmaximal, characteristic
3C2×#4996C212yesmaximal
4#71584C211yesmaximal, characteristic
5#71559C212yesmaximal
6#69633C212yesmaximal, complemented
7#69632C212yesmaximal, complemented
8#69631C212yesmaximal, complemented
9#69606C212yesmaximal, complemented
10#24072—31nomaximal, Sylow
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C2×#924C2212yes—
15#5518—28no—
16C2×#889—22no—
17C2×#879—22no—
18C2×#870C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C4×Dic6C2212yes—
21C3×#389—22no—
22C3×#389C2212yes—
23C3×#389C2212yes—
24C3×#380—22no—
25C2×#830C2212yes—
26#4996—28no—
27C2×#810C2212yes—
28C2×#810C2212yes—
29C2×#810C2212yes—
30C2×#810C2212yes—
31C2×#808C2212yes—
32C2×#807C2212yes—
33#4529C2212yes—
34#4529C2212yes—
35#4524C2211yescharacteristic
36#4524C2212yes—
37#4524C2211yescharacteristic
38C2×#541—31no—
39C2×#485—32no—
40#2316D311yescharacteristic
41#2059—32no—
42#2043—32no—
43#1872—32no—
44#1865—32no—
45#1863—32no—
46#1818—31no—
47C23×C3×C4—21no—
48C23×C3×C4C2311yescharacteristic
49C22×Dic6C2312yes—
50C3×#117—24no—
51C3×#117—24no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4D412yes—
61C2×C3×C42—21no—
62C2×C3×C42C2311yescharacteristic
63C2×C3×C42C2311yescharacteristic
64C2×#217—22no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3C2312yes—
67#924—24no—
68#924D418yes—
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×C4×Dic3—22no—
73C22×C3⋊C8C2312yes—
74C22×C3⋊C8C2312yes—
75#889—216no—
76#879—216no—
77#870—24no—
78#869—24no—
79C4×Dic6—24no—
80C4×Dic6—24no—
81C3×#95—22no—
82C3×#95—22no—
83#830—24no—
84#810—24no—
85#810—24no—
86#810—24no—
87#810—24no—
88#808—24no—
89#807—24no—
90#541—68no—
91C2×C4⋊Q8—31no—
92C2×#126—62no—
93C2×#125—31no—
94C2×#125—62no—
95C2×C22⋊Q8—32no—
96C2×C4×Q8—32no—
97C2×#117C2×D311yescharacteristic
98C2×#117C2×D311yescharacteristic
99C22×C4⋊C4C2×D311yescharacteristic
100#485—68no—
101C2×#107—32no—
102C2×#106—32no—
103C2×#103—32no—
104C2×#103—32no—
105C2×#103—32no—
106C2×#103—32no—
107C2×C22⋊C8—32no—
108#389—22no—
109#389C2×D312yes—
110#389C2×D312yes—
111#380—22no—
112#340—32no—
113#340—32no—
114#336—32no—
115#336—31no—
116#336—31no—
117C24×C3C2×D411yescharacteristic
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—24no—
123C22×C3×C4—22no—
124C22×C3×C4—82no—
125C22×C3×C4C2×D414yes—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2411yescharacteristic
130C22×Dic3—22no—
131C22×Dic3—22no—
132C2×Dic6—22no—
133C2×Dic6—22no—
134C2×Dic6—22no—
135C2×Dic6—22no—
136C2×Dic6—22no—
137C2×Dic6C2×D414yes—
138C3×C4⋊C4—24no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—82no—
144C3×C4⋊C4—82no—
145C3×C4⋊C4#4114yes—
146C3×C22⋊C4—24no—
147C3×C22⋊C4—42no—
148C3×C42—22no—
149C3×C42—22no—
150C3×C42—42no—
151#217—28no—
152C4⋊Dic3—28no—
153C4⋊Dic3—22no—
154C4⋊Dic3C2×D414yes—
155Dic3⋊C4—28no—
156Dic3⋊C4—28no—
157Dic3⋊C4—28no—
158C4×Dic3—28no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C22×Q8—32no—
166C23×C4—21no—
167C23×C4C22×D311yescharacteristic
168C22×C8—32no—
169C22×C8—32no—
170C4⋊Q8—64no—
171#126—616no—
172#125—64no—
173#125—616no—
174C22⋊Q8—38no—
175C22⋊Q8—64no—
176C4×Q8—64no—
177C4×Q8—64no—
178#117—24no—
179#117—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4#6712yes—
190C2×C4⋊C4C22×D312yes—
191C2×C4⋊C4C22×D312yes—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4#6712yes—
195C2×C42—21no—
196C2×C42—62no—
197C2×C42C22×D311yescharacteristic
198C2×C42C22×D311yescharacteristic
199#107—64no—
200#106—64no—
201#103—64no—
202#103—64no—
203#103—64no—
204#103—64no—
205C22⋊C8—64no—
206#95—22no—
207#95—22no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—22no—
211C23×C3—22no—
212C23×C3#12012yes—
213C23×C3C22×D411yescharacteristic
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—24no—
217C2×C3×C4—24no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—24no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—84no—
231C2×C3×C4#12012yes—
232C2×C3×C4C22×D412yes—
233C2×C3×C4C2×#4112yes—
234C2×C3×C42−1+411yescharacteristic
235C2×C3×C42−1+411yescharacteristic
236C2×Dic3—24no—
237C2×Dic3—24no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242Dic6—24no—
243Dic6—24no—
244Dic6—42no—
245Dic6—42no—
246C3⋊C8—42no—
247C3⋊C8—42no—
248C24C2×#6711yescharacteristic
249C2×Q8—34no—
250C2×Q8—62no—
251C2×Q8—62no—
252C2×Q8—62no—
253C2×Q8—62no—
254C2×Q8—62no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—82no—
264C22×C4D3×D414yes—
265C22×C4C2×#6712yes—
266C22×C4C2×#6712yes—
267C22×C4C2×#6711yescharacteristic
268C22×C4C23×D311yescharacteristic
269C2×C8—62no—
270C2×C8—62no—
271C2×C8—62no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8—62no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—34no—
281C4⋊C4—68no—
282C4⋊C4—68no—
283C4⋊C4—68no—
284C4⋊C4—68no—
285C4⋊C4—62no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4#23514yes—
289C22⋊C4—24no—
290C22⋊C4—42no—
291C22⋊C4—68no—
292C42—22no—
293C42—22no—
294C42—42no—
295C42—68no—
296C22×C3—22no—
297C22×C3—24no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—42no—
301C22×C3C2×#12011yescharacteristic
302C22×C3#52511yescharacteristic
303C22×C3#53611yescharacteristic
304C22×C3#57712yes—
305C22×C3#57812yes—
306C3×C4—22no—
307C3×C4—44no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—82no—
311C3×C4#54712yes—
312C3×C4#54712yes—
313Dic3—44no—
314Dic3—44no—
315C23—21no—
316C23—21no—
317C23—22no—
318C23—22no—
319C23#93712yes—
320C23C22×#6711yescharacteristic
321Q8—64no—
322Q8—64no—
323Q8—122no—
324Q8—122no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—64no—
342C2×C4—64no—
343C2×C4—84no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4#93712yes—
349C2×C4C2×#23512yes—
350C2×C4C2×D3×D412yes—
351C2×C4#100711yescharacteristic
352C2×C4#100711yescharacteristic
353C8—122no—
354C8—122no—
355C2×C3—42no—
356C2×C3#240611yescharacteristic
357C2×C3#301312yes—
358C2×C3#312311yescharacteristic
359C2×C3#312711yescharacteristic
360C2×C3#324212yes—
361C22—22no—
362C22—24no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22#579111yescharacteristic
367C22C2×#93711yescharacteristic
368C22#581611yescharacteristic
369C22#583612yes—
370C22#583812yes—
371C4—22no—
372C4—44no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—124no—
377C4—124no—
378C4#550712yes—
379C4#550712yes—
380C3#2407211yescharacteristic, complemented, Sylow
381C2—42no—
382C2#8219211yescharacteristic
383C2#8254511yescharacteristic
384C2#8255511yescharacteristic
385C2#8342712yes—
386C2#8348112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2316C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2316D3
hypocenterC3#24072
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2316

Complements

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

NH ≅ G/Ntype
6. #69633C2semidirect
7. #69632C2semidirect
8. #69631C2semidirect
9. #69606C2semidirect
380. C3#24072semidirect

Maximal subgroups

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

subgroupindex
C2×#55182
C3×#23162
C2×#49962
#715842
#715592
#696332
#696322
#696312
#696062
#240723