Finite Group Databasebeta

← #10828579

Subgroups — #10828579

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
2#82232C211yesmaximal, characteristic
3C2×#4723C212yesmaximal
4C3×#1818C211yesmaximal, characteristic
5#69930C212yesmaximal
6#68684C212yesmaximal, complemented
7#68683C212yesmaximal, complemented
8#68634C212yesmaximal, complemented
9#68631C212yesmaximal, complemented
10#24072—31nomaximal, Sylow
11C2×C3×#117C2211yescharacteristic
12C2×#926C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C2×#924C2212yes—
15#5518—28no—
16C2×#889—22no—
17C2×#879—22no—
18C2×#870C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C4×Dic6C2212yes—
21C2×C3×C22⋊C8C2212yes—
22C2×#818C2212yes—
23C2×#817C2212yes—
24C2×#816C2212yes—
25C2×#816C2212yes—
26C2×#816C2212yes—
27C2×#816C2212yes—
28#4972—22no—
29#4942—22no—
30#4928C2212yes—
31#4928C2212yes—
32#4723—28no—
33C3×#336C2212yes—
34C3×#336C2211yescharacteristic
35C3×#336C2211yescharacteristic
36#4551C2212yes—
37#4551C2212yes—
38C2×#541—31no—
39C2×#485—32no—
40#2316—31no—
41#2059—32no—
42#2043—32no—
43#1872—32no—
44#1865—32no—
45#1863—32no—
46#1818D311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×Dic3—21no—
49C22×Dic6C2312yes—
50C22×C3×C8C2312yes—
51C22×C3×C8C2312yes—
52C3×#117—24no—
53C2×C3×C4⋊C4D412yes—
54C2×C3×C22⋊C4D412yes—
55C2×C3×C42C2311yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×#217—22no—
58C2×#217—21no—
59#926—24no—
60C2×C4⋊Dic3—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—24no—
63C2×C4⋊Dic3C2312yes—
64C2×C4⋊Dic3C2312yes—
65C2×C4⋊Dic3C2312yes—
66#924—24no—
67#924D418yes—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4—22no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×Dic3⋊C4—22no—
73C2×C4×Dic3—22no—
74C2×C4×Dic3—21no—
75#889—216no—
76#879—216no—
77#870—24no—
78#869—24no—
79C4×Dic6—24no—
80C4×Dic6—24no—
81C3×C22⋊C8—24no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84#818—24no—
85#817—24no—
86#816—24no—
87#816—24no—
88#816—24no—
89#816—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×#117—31no—
98C2×#117C2×D311yescharacteristic
99C22×C4⋊C4—31no—
100#485—68no—
101C2×#107—32no—
102C2×#106—32no—
103C2×#103—32no—
104C2×#103—32no—
105C2×#103—32no—
106C2×#103—32no—
107C2×C22⋊C8C2×D312yes—
108#389—32no—
109#389—32no—
110#389—62no—
111#380—62no—
112#340—32no—
113#340—32no—
114#336C2×D312yes—
115#336C2×D311yescharacteristic
116#336C2×D311yescharacteristic
117C24×C3C2×D411yescharacteristic
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D414yes—
125C22×C3×C4C2411yescharacteristic
126C22×Dic3—22no—
127C22×Dic3—22no—
128C22×Dic3—22no—
129C22×Dic3—24no—
130C22×Dic3—22no—
131C22×Dic3—82no—
132C2×Dic6—22no—
133C2×Dic6—22no—
134C2×Dic6—22no—
135C2×Dic6—22no—
136C2×Dic6—22no—
137C2×Dic6C2×D414yes—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C3×C4⋊C4—24no—
145C3×C22⋊C4—24no—
146C3×C42—22no—
147C3×C42—22no—
148#217—28no—
149#217—42no—
150C4⋊Dic3—28no—
151C4⋊Dic3—24no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4⋊Dic3C2×D414yes—
157C4⋊Dic3#4114yes—
158Dic3⋊C4—28no—
159Dic3⋊C4—28no—
160Dic3⋊C4—28no—
161Dic3⋊C4—82no—
162Dic3⋊C4—82no—
163C4×Dic3—28no—
164C4×Dic3—42no—
165C22×Q8—32no—
166C23×C4—61no—
167C23×C4C22×D311yescharacteristic
168C22×C8C22×D312yes—
169C22×C8C22×D312yes—
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—64no—
180C2×C4⋊C4—32no—
181C2×C4⋊C4—32no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—64no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4D1212yes—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4D1212yes—
195C2×C42—62no—
196C2×C42—61no—
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—24no—
206#95—62no—
207#95—62no—
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—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4#12012yes—
225C2×C3×C4C22×D412yes—
226C2×C3×C4C2×#4112yes—
227C2×C3×C42−1+411yescharacteristic
228C2×C3×C42−1+411yescharacteristic
229C2×Dic3—24no—
230C2×Dic3—24no—
231C2×Dic3—24no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—84no—
242Dic6—24no—
243Dic6—24no—
244Dic6—42no—
245Dic6—42no—
246C3×C8—42no—
247C3×C8—42no—
248C24C2×D1211yescharacteristic
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—62no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—64no—
262C22×C4—62no—
263C22×C4—242no—
264C22×C4C2×D1212yes—
265C22×C4C2×D1212yes—
266C22×C4C2×D1211yescharacteristic
267C22×C4D3×D414yes—
268C22×C4C23×D311yescharacteristic
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C4⋊C4—24no—
276C4⋊C4—34no—
277C4⋊C4—34no—
278C4⋊C4—64no—
279C4⋊C4—68no—
280C4⋊C4—68no—
281C4⋊C4—68no—
282C4⋊C4—68no—
283C4⋊C4—62no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—62no—
287C4⋊C4—242no—
288C4⋊C4—242no—
289C22⋊C4—24no—
290C22⋊C4—68no—
291C22⋊C4—122no—
292C42—22no—
293C42—22no—
294C42—68no—
295C42—122no—
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—42no—
308C3×C4—42no—
309C3×C4#54712yes—
310C3×C4#54712yes—
311Dic3—44no—
312Dic3—44no—
313Dic3—44no—
314Dic3—82no—
315C23—22no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23#88212yes—
320C23C22×D1211yescharacteristic
321Q8—64no—
322Q8—64no—
323Q8—122no—
324Q8—122no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—64no—
336C2×C4—64no—
337C2×C4—64no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—244no—
348C2×C4#88212yes—
349C2×C4C2×D3×D412yes—
350C2×C4D3×#4112yes—
351C2×C4#101011yescharacteristic
352C2×C4#101011yescharacteristic
353C8—42no—
354C8—42no—
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—
366C22C2×#88211yescharacteristic
367C22#552411yescharacteristic
368C22#552611yescharacteristic
369C22#575112yes—
370C22#575312yes—
371C4—22no—
372C4—42no—
373C4—42no—
374C4—124no—
375C4—124no—
376C4—124no—
377C4—242no—
378C4#565512yes—
379C4#565512yes—
380C3#2407211yescharacteristic, complemented, Sylow
381C2—42no—
382C2#7406311yescharacteristic
383C2#7868012yes—
384C2#7936312yes—
385C2#7938711yescharacteristic
386C2#7939611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1818C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1818D3
hypocenterC3#24072
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
6. #68684C2semidirect
7. #68683C2semidirect
8. #68634C2semidirect
9. #68631C2semidirect
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
#822322
C2×#47232
C3×#18182
#699302
#686842
#686832
#686342
#686312
#240723