Finite Group Databasebeta

← #10828572

Subgroups — #10828572

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×C3×#541C211yesmaximal, characteristic
2C2×#5045C212yesmaximal
3#82232C211yesmaximal, characteristic
4#71781C212yesmaximal, complemented
5#71739C212yesmaximal, complemented
6#71692C212yesmaximal
7#71629C212yesmaximal, complemented
8#71628C212yesmaximal, complemented
9#71584C211yesmaximal, characteristic
10#24072—31nomaximal, Sylow
11C3×#541—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#126—22no—
14C2×C3×#125—22no—
15C2×C3×#125C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×C4×Q8C2212yes—
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C2×#835C2212yes—
22C2×#835C2212yes—
23C2×#835C2212yes—
24C2×#835C2212yes—
25C2×#830C2212yes—
26#5045—28no—
27#4972—22no—
28C2×#808C2212yes—
29C2×#807C2212yes—
30#4942—22no—
31#4928C2212yes—
32#4928C2212yes—
33#4529C2212yes—
34#4529C2212yes—
35#4524C2212yes—
36#4524C2211yescharacteristic
37#4524C2211yescharacteristic
38C2×#541D311yescharacteristic
39C2×#485—32no—
40#2316—31no—
41#2059—32no—
42#2043—32no—
43#1872—32no—
44#1865—32no—
45#1863—32no—
46#1818—31no—
47C22×C3×Q8C2312yes—
48C23×C3×C4C2311yescharacteristic
49C23×Dic3—21no—
50C3×C4⋊Q8—24no—
51C3×#126—216no—
52C3×#125—216no—
53C3×#125—24no—
54C3×C22⋊Q8—24no—
55C3×C22⋊Q8D418yes—
56C3×C4×Q8—24no—
57C3×C4×Q8—24no—
58C3×#117—24no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4D412yes—
67C2×C3×C42—22no—
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×#217—21no—
71#926—24no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—24no—
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—21no—
79C22×C3⋊C8C2312yes—
80C22×C3⋊C8C2312yes—
81#835—24no—
82#835—24no—
83#835—24no—
84#835—24no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87#830—24no—
88#808—24no—
89#807—24no—
90#541—28no—
91C2×C4⋊Q8C2×D311yescharacteristic
92C2×#126—22no—
93C2×#125—22no—
94C2×#125C2×D311yescharacteristic
95C2×C22⋊Q8C2×D312yes—
96C2×C4×Q8C2×D312yes—
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⋊C8—32no—
108#389—32no—
109#389—32no—
110#389—62no—
111#380—62no—
112#340—32no—
113#340—32no—
114#336—32no—
115#336—31no—
116#336—31no—
117C24×C3C2×D411yescharacteristic
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8C2×D414yes—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D414yes—
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—24no—
136C22×Dic3—22no—
137C22×Dic3—82no—
138C3×C4⋊C4—28no—
139C3×C4⋊C4—28no—
140C3×C4⋊C4—28no—
141C3×C4⋊C4—28no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4C2×D414yes—
145C3×C22⋊C4—28no—
146C3×C22⋊C4—24no—
147C3×C42—28no—
148C3×C42—22no—
149C3×C42—22no—
150#217—42no—
151C4⋊Dic3—24no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3#4114yes—
156Dic3⋊C4—82no—
157Dic3⋊C4—82no—
158C4×Dic3—42no—
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×Q8C22×D312yes—
166C23×C4—61no—
167C23×C4C22×D311yescharacteristic
168C22×C8—32no—
169C22×C8—32no—
170C4⋊Q8—24no—
171#126—216no—
172#125—216no—
173#125—24no—
174C22⋊Q8—24no—
175C22⋊Q8#6718yes—
176C4×Q8—24no—
177C4×Q8—24no—
178#117—24no—
179#117—64no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—64no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4#6712yes—
191C2×C4⋊C4C22×D312yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4#6712yes—
195C2×C42—22no—
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—64no—
206#95—62no—
207#95—62no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—22no—
211C23×C3—22no—
212C23×C3#12012yes—
213C23×C3C22×D411yescharacteristic
214C3×Q8—24no—
215C3×Q8—24no—
216C3×Q8—42no—
217C3×Q8—42no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—24no—
221C2×C3×C4—24no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4#12012yes—
235C2×C3×C4C22×D412yes—
236C2×C3×C4C2×#4112yes—
237C2×C3×C42−1+411yescharacteristic
238C2×C3×C42−1+411yescharacteristic
239C2×Dic3—24no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—84no—
246C3⋊C8—42no—
247C3⋊C8—42no—
248C24C2×#6711yescharacteristic
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8C2×#6714yes—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—62no—
261C22×C4—64no—
262C22×C4—62no—
263C22×C4—242no—
264C22×C4C2×#6712yes—
265C22×C4C2×#6712yes—
266C22×C4C2×#6711yescharacteristic
267C22×C4C2×#6714yes—
268C22×C4C23×D311yescharacteristic
269C2×C8—62no—
270C2×C8—62no—
271C2×C8—62no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8—62no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—34no—
282C4⋊C4—64no—
283C4⋊C4—62no—
284C4⋊C4—62no—
285C4⋊C4—62no—
286C4⋊C4—242no—
287C4⋊C4—242no—
288C4⋊C4C2×#6714yes—
289C22⋊C4—28no—
290C22⋊C4—24no—
291C22⋊C4—122no—
292C42—28no—
293C42—22no—
294C42—22no—
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—44no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—44no—
311C3×C4#54712yes—
312C3×C4#54712yes—
313Dic3—44no—
314Dic3—82no—
315C23—21no—
316C23—21no—
317C23—22no—
318C23—22no—
319C23#95312yes—
320C23C22×#6711yescharacteristic
321Q8—24no—
322Q8—24no—
323Q8—42no—
324Q8—42no—
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—24no—
336C2×C4—24no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—64no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4—244no—
348C2×C4#95312yes—
349C2×C4D3×#4112yes—
350C2×C4#101011yescharacteristic
351C2×C4#101011yescharacteristic
352C2×C4C22×#6712yes—
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#582511yescharacteristic
367C22#582811yescharacteristic
368C22#583712yes—
369C22#583812yes—
370C22C2×#95311yescharacteristic
371C4—22no—
372C4—44no—
373C4—42no—
374C4—42no—
375C4—44no—
376C4—124no—
377C4—242no—
378C4#582412yes—
379C4#582412yes—
380C3#2407211yescharacteristic, complemented, Sylow
381C2—42no—
382C2#8344212yes—
383C2#8344711yescharacteristic
384C2#8345111yescharacteristic
385C2#8409011yescharacteristic
386C2#8411412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#541C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#541D3
hypocenterC3#24072
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
4. #71781C2semidirect
5. #71739C2semidirect
7. #71629C2semidirect
8. #71628C2semidirect
380. C3#24072semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×#50452
#822322
#717812
#717392
#716922
#716292
#716282
#715842
#240723