Finite Group Databasebeta

← #10823307

Subgroups — #10823307

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5500C211yesmaximal, characteristic
2#83752C212yesmaximal
3#82157C211yesmaximal, characteristic
4C2×C3×#343C211yesmaximal, characteristic
5#71982C212yesmaximal
6#68979C214yesmaximal, complemented
7#68978C214yesmaximal, complemented
8#23561—31nomaximal, Sylow
9C22×C3×M16C2212yes—
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#926C2212yes—
12C2×#926C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#924—22no—
15#5500—22no—
16#5500—24no—
17#5500C2214yes—
18C2×#890—22no—
19C2×#888—22no—
20#5210—22no—
21#5121—22no—
22#5121—22no—
23#5121C2214yes—
24#5121C2214yes—
25#5106—24no—
26#4977C2212yes—
27#4975C2212yes—
28#4972—22no—
29C3×#343C2214yes—
30#4555C2218yes—
31C2×#544—31no—
32#2911—32no—
33#2317—31no—
34#2061—34no—
35#2060—34no—
36C2×#343D311yescharacteristic
37#1733—32no—
38C23×C3×C4D412yes—
39C23×C3×C4C2311yescharacteristic
40C22×Dic6—22no—
41C2×C3×M16—22no—
42C2×C3×M16—24no—
43C2×C3×M16Q818yescomplemented
44C2×C3×M16C2314yes—
45C22×C3×C8—22no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4—22no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C4⋊C4C2312yes—
50C2×#217—21no—
51C2×#217—22no—
52C2×#217—21no—
53#926—22no—
54#926—22no—
55#926—22no—
56#926—22no—
57#926—22no—
58#926D414yes—
59#926Q814yescomplemented
60#926C2314yes—
61C2×C4⋊Dic3—22no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3D412yes—
64C2×C4⋊Dic3D412yes—
65C2×C4⋊Dic3C2311yescharacteristic
66C2×C4⋊Dic3C2311yescharacteristic
67#924—28no—
68#924—28no—
69C2×Dic3⋊C4—24no—
70C2×Dic3⋊C4—24no—
71C2×C4×Dic3—22no—
72C2×C4×Dic3—21no—
73C2×C4×Dic3C2312yes—
74#890—28no—
75#890—28no—
76#888—28no—
77#888—28no—
78C3⋊#95—24no—
79#818—24no—
80#818—24no—
81#817—24no—
82#817—24no—
83C22×M16C2×D312yes—
84#544—34no—
85#544—64no—
86#544—62no—
87C2×C4⋊Q8—62no—
88C2×#125—62no—
89C2×C22⋊Q8—62no—
90C2×#117—31no—
91C2×#117—32no—
92C2×#117—31no—
93C22×C4⋊C4C2×D311yescharacteristic
94#428—34no—
95#428—34no—
96#428—62no—
97#428—62no—
98#427—64no—
99#389—62no—
100#385—62no—
101#384—32no—
102#382—32no—
103#343C2×D314yes—
104#339—38no—
105C24×C3C2×D411yescharacteristic
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—82no—
109C22×C3×C4C2×D414yes—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×Q814yes—
114C22×C3×C4C2×Q812yes—
115C22×C3×C4#4112yes—
116C22×C3×C4C2411yescharacteristic
117C22×Dic3—22no—
118C22×Dic3—22no—
119C22×Dic3—22no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C2×Dic6—28no—
123C2×Dic6—84no—
124C3×M16—28no—
125C3×M16—24no—
126C3×M16C2×Q818yescomplemented
127C2×C3×C8—28no—
128C2×C3×C8—24no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—24no—
131#217—24no—
132#217—24no—
133#217—24no—
134#217—41no—
135#217—41no—
136C4⋊Dic3—28no—
137C4⋊Dic3—22no—
138C4⋊Dic3—22no—
139C4⋊Dic3—24no—
140C4⋊Dic3—22no—
141C4⋊Dic3—22no—
142C4⋊Dic3C2×D414yes—
143C4⋊Dic3C2×D414yes—
144Dic3⋊C4—216no—
145Dic3⋊C4—84no—
146C4×Dic3—24no—
147C4×Dic3—24no—
148C4×Dic3—22no—
149C4×Dic3—42no—
150C4×Dic3C2×Q814yescomplemented
151C22×Q8—62no—
152C23×C4D1212yes—
153C23×C4C22×D311yescharacteristic
154C2×M16—22no—
155C2×M16—24no—
156C2×M16Dic618yescomplemented
157C2×M16C22×D314yes—
158C22×C8—22no—
159C4⋊Q8—68no—
160C4⋊Q8—68no—
161#125—68no—
162#125—68no—
163C22⋊Q8—68no—
164C22⋊Q8—68no—
165#117—34no—
166#117—34no—
167#117—34no—
168#117—62no—
169#117—62no—
170#117—62no—
171#117—62no—
172#117—62no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—62no—
180C2×C4⋊C4—64no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—64no—
183C2×C4⋊C4D1212yes—
184C2×C4⋊C4C22×D312yes—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C42—32no—
189C2×C42—62no—
190C2×C42—61no—
191#107—64no—
192#107—64no—
193#106—64no—
194#106—64no—
195#95—64no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3#12011yescharacteristic
199C23×C3#12011yescharacteristic
200C23×C3C22⋊Q812yes—
201C23×C3C4⋊Q812yes—
202C23×C3C22×D411yescharacteristic
203C2×C3×C4—24no—
204C2×C3×C4—44no—
205C2×C3×C4—44no—
206C2×C3×C4—82no—
207C2×C3×C4#12012yes—
208C2×C3×C4C22⋊Q814yes—
209C2×C3×C4C22⋊Q814yes—
210C2×C3×C4C22⋊Q812yes—
211C2×C3×C4C4⋊Q812yes—
212C2×C3×C4C4⋊Q814yes—
213C2×C3×C4C4⋊Q814yes—
214C2×C3×C4C22×D412yes—
215C2×C3×C4C22×Q812yes—
216C2×C3×C4C22×Q811yescharacteristic
217C2×C3×C4C2×#4111yescharacteristic
218C2×Dic3—24no—
219C2×Dic3—24no—
220C2×Dic3—24no—
221C2×Dic3—42no—
222C2×Dic3—44no—
223C2×Dic3—42no—
224C2×Dic3—44no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229Dic6—88no—
230C3×C8—28no—
231C24C2×D1211yescharacteristic
232C2×Q8—68no—
233C2×Q8—244no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—82no—
242C22×C4C2×Dic614yes—
243C22×C4C2×D1214yes—
244C22×C4C2×D1211yescharacteristic
245C22×C4D3×D412yes—
246C22×C4D3×D412yes—
247C22×C4#23712yes—
248C22×C4D3×Q812yes—
249C22×C4C23×D311yescharacteristic
250M16—28no—
251M16—24no—
252M16C2×Dic618yescomplemented
253C2×C8—28no—
254C2×C8—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—34no—
258C4⋊C4—34no—
259C4⋊C4—68no—
260C4⋊C4—616no—
261C4⋊C4—62no—
262C4⋊C4—62no—
263C4⋊C4—64no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—244no—
267C22⋊C4—64no—
268C22⋊C4—64no—
269C22⋊C4—64no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C42—34no—
273C42—64no—
274C42—64no—
275C42—62no—
276C42—122no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3#39512yes—
286C22×C3#39512yes—
287C22×C3C2×#12011yescharacteristic
288C22×C3C2×C22⋊Q811yescharacteristic
289C22×C3C2×C4⋊Q811yescharacteristic
290C3×C4—44no—
291C3×C4#39514yes—
292C3×C4C2×C22⋊Q812yes—
293C3×C4C2×C4⋊Q812yes—
294Dic3—44no—
295Dic3—44no—
296Dic3—44no—
297Dic3—81no—
298Dic3—81no—
299C23—41no—
300C23—41no—
301C23#86912yes—
302C23#88211yescharacteristic
303C23#88211yescharacteristic
304C23#89712yes—
305C23C22×D1211yescharacteristic
306Q8—248no—
307C2×C4—24no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—64no—
311C2×C4—64no—
312C2×C4—64no—
313C2×C4—82no—
314C2×C4—122no—
315C2×C4—124no—
316C2×C4—122no—
317C2×C4—124no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4#86912yes—
323C2×C4#87814yes—
324C2×C4#87814yes—
325C2×C4#88212yes—
326C2×C4#88814yes—
327C2×C4#88814yes—
328C2×C4#89712yes—
329C2×C4C22×Dic612yes—
330C2×C4C2×D3×D412yes—
331C2×C4C2×#23711yescharacteristic
332C2×C4C2×D3×Q811yescharacteristic
333C8—28no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—22no—
337C2×C3—22no—
338C2×C3C2×#39511yescharacteristic
339C2×C3#301911yescharacteristic
340C2×C3#302011yescharacteristic
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—42no—
348C22—42no—
349C22#465612yes—
350C22#465612yes—
351C22C2×#86911yescharacteristic
352C22C2×#88211yescharacteristic
353C22C2×#89711yescharacteristic
354C4—44no—
355C4—124no—
356C4—124no—
357C4—124no—
358C4—241no—
359C4—241no—
360C4#465614yes—
361C4C2×#87812yes—
362C4C2×#88812yes—
363C3#2356111yescharacteristic, complemented, Sylow
364C2—21no—
365C2—22no—
366C2—22no—
367C2—21no—
368C2C2×#465611yescharacteristic
369C2#7870011yescharacteristic
370C2#7870911yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#343C2
socleC22×C3C2×#120
prime radicalC24×C3C2×D4
hypercenterC2×#343D3
hypocenterC3#23561
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
6. #68979C2semidirect
7. #68978C2semidirect
43. C2×C3×M16Q8semidirect
59. #926Q8semidirect
126. C3×M16C2×Q8semidirect
150. C4×Dic3C2×Q8semidirect
156. C2×M16Dic6semidirect
252. M16C2×Dic6semidirect
363. C3#23561semidirect

Maximal subgroups

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

subgroupindex
C2×#55002
#837522
#821572
C2×C3×#3432
#719822
#689792
#689782
#235613