Finite Group Databasebeta

← #10823302

Subgroups — #10823302

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×C3×#544C211yesmaximal, characteristic
2#82207C212yesmaximal
3#82157C211yesmaximal, characteristic
4#71901C214yesmaximal, complemented
5#71900C214yesmaximal, complemented
6#71574C212yesmaximal
7C2×#4530C211yesmaximal, characteristic
8#23561—31nomaximal, Sylow
9C3×#544—24no—
10C3×#544—22no—
11C3×#544C2214yes—
12C2×C3×C4⋊Q8—22no—
13C2×C3×#125—22no—
14C2×C3×C22⋊Q8—22no—
15C2×C3×#117C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C2×#926C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×#208C2212yes—
20#5210—22no—
21#4998—24no—
22#4995C2214yes—
23#4995C2214yes—
24#4977C2212yes—
25#4975C2212yes—
26#4972—22no—
27#4964—22no—
28#4964—22no—
29#4532C2218yes—
30#4530C2214yes—
31C2×#544D311yescharacteristic
32#2911—32no—
33#2317—31no—
34#2061—34no—
35#2060—34no—
36C2×#343—31no—
37#1733—32no—
38C22×C3×Q8—22no—
39C23×C3×C4D412yes—
40C23×C3×C4C2311yescharacteristic
41C3×C4⋊Q8—28no—
42C3×C4⋊Q8—28no—
43C3×#125—28no—
44C3×#125—28no—
45C3×C22⋊Q8—28no—
46C3×C22⋊Q8—28no—
47C3×#117—22no—
48C3×#117—22no—
49C3×#117D414yes—
50C3×#117C2314yes—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4D412yes—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C4⋊C4D412yes—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42—22no—
61C2×#217—21no—
62C2×#217—21no—
63#926—22no—
64#926—22no—
65#926—22no—
66#926Q814yescomplemented
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3C2311yescharacteristic
69C2×C4⋊Dic3C2311yescharacteristic
70C2×Dic3⋊C4—24no—
71C2×C4×Dic3—21no—
72C2×C4×Dic3C2312yes—
73C2×#208—22no—
74C2×#208—24no—
75C2×#208Q818yescomplemented
76C2×#208C2314yes—
77C22×C3⋊C8—22no—
78C3⋊#95—24no—
79#808—24no—
80#808—24no—
81#807—24no—
82#807—24no—
83C22×M16—32no—
84#544—24no—
85#544—22no—
86#544C2×D314yes—
87C2×C4⋊Q8—22no—
88C2×#125—22no—
89C2×C22⋊Q8—22no—
90C2×#117—31no—
91C2×#117—31no—
92C2×#117C2×D312yes—
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#343—34no—
104#339—38no—
105C24×C3C2×D411yescharacteristic
106C2×C3×Q8—28no—
107C2×C3×Q8—84no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—82no—
113C22×C3×C4C2×D414yes—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×D412yes—
117C22×C3×C4C2×Q814yes—
118C22×C3×C4C2×Q812yes—
119C22×C3×C4#4112yes—
120C22×C3×C4C2411yescharacteristic
121C22×Dic3—22no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C3×C4⋊C4—28no—
125C3×C4⋊C4—216no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4C2×D414yes—
131C3×C4⋊C4C2×D414yes—
132C3×C22⋊C4—24no—
133C3×C22⋊C4—24no—
134C3×C42—24no—
135C3×C42—24no—
136#217—24no—
137#217—41no—
138#217—41no—
139C4⋊Dic3—24no—
140C4⋊Dic3—22no—
141C4⋊Dic3—22no—
142Dic3⋊C4—84no—
143C4×Dic3—22no—
144C4×Dic3—42no—
145C4×Dic3C2×Q814yescomplemented
146#208—28no—
147#208—24no—
148#208C2×Q818yescomplemented
149C2×C3⋊C8—28no—
150C2×C3⋊C8—24no—
151C22×Q8—22no—
152C23×C4#6712yes—
153C23×C4C22×D311yescharacteristic
154C2×M16—38no—
155C2×M16—34no—
156C2×M16—64no—
157C2×M16—62no—
158C22×C8—62no—
159C4⋊Q8—28no—
160C4⋊Q8—28no—
161#125—28no—
162#125—28no—
163C22⋊Q8—28no—
164C22⋊Q8—28no—
165#117—22no—
166#117—22no—
167#117—34no—
168#117—62no—
169#117—62no—
170#117—62no—
171#117#6714yes—
172#117C22×D314yes—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—64no—
181C2×C4⋊C4#6712yes—
182C2×C4⋊C4#6712yes—
183C2×C4⋊C4#6712yes—
184C2×C4⋊C4C22×D312yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C42—22no—
189C2×C42—32no—
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
203C3×Q8—88no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—44no—
208C2×C3×C4—44no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4—82no—
212C2×C3×C4#12012yes—
213C2×C3×C4C22⋊Q814yes—
214C2×C3×C4C22⋊Q814yes—
215C2×C3×C4C22⋊Q812yes—
216C2×C3×C4C4⋊Q814yes—
217C2×C3×C4C4⋊Q814yes—
218C2×C3×C4C4⋊Q812yes—
219C2×C3×C4C22×D412yes—
220C2×C3×C4C22×Q812yes—
221C2×C3×C4C22×Q811yescharacteristic
222C2×C3×C4C2×#4111yescharacteristic
223C2×Dic3—24no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C3⋊C8—28no—
231C24C2×#6711yescharacteristic
232C2×Q8—28no—
233C2×Q8—84no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—62no—
239C22×C4—61no—
240C22×C4—61no—
241C22×C4—82no—
242C22×C4D3×Q812yes—
243C22×C4D3×Q814yes—
244C22×C4#23912yes—
245C22×C4C2×#6714yes—
246C22×C4C2×#6711yescharacteristic
247C22×C4C2×#6712yes—
248C22×C4C2×#6712yes—
249C22×C4C23×D311yescharacteristic
250M16—38no—
251M16—68no—
252M16—64no—
253C2×C8—68no—
254C2×C8—64no—
255C4⋊C4—28no—
256C4⋊C4—216no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—64no—
262C4⋊C4—62no—
263C4⋊C4—62no—
264C4⋊C4—244no—
265C4⋊C4C2×#6714yes—
266C4⋊C4C2×#6714yes—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—64no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C42—24no—
273C42—24no—
274C42—34no—
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—44no—
292C3×C4—44no—
293C3×C4#39514yes—
294C3×C4C2×C22⋊Q812yes—
295C3×C4C2×C4⋊Q812yes—
296Dic3—44no—
297Dic3—81no—
298Dic3—81no—
299C23—41no—
300C23—41no—
301C23#94412yes—
302C23#94612yes—
303C23#95311yescharacteristic
304C23#95311yescharacteristic
305C23C22×#6711yescharacteristic
306Q8—88no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—64no—
315C2×C4—82no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—121no—
321C2×C4—121no—
322C2×C4#94412yes—
323C2×C4#94414yes—
324C2×C4#94414yes—
325C2×C4#94612yes—
326C2×C4#94614yes—
327C2×C4#94614yes—
328C2×C4#95312yes—
329C2×C4C2×D3×Q811yescharacteristic
330C2×C4C2×D3×Q812yes—
331C2×C4C2×#23911yescharacteristic
332C2×C4C22×#6712yes—
333C8—68no—
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#523012yes—
350C22#523012yes—
351C22C2×#94411yescharacteristic
352C22C2×#94611yescharacteristic
353C22C2×#95311yescharacteristic
354C4—44no—
355C4—44no—
356C4—44no—
357C4—124no—
358C4—241no—
359C4—241no—
360C4#523014yes—
361C4C2×#94412yes—
362C4C2×#94612yes—
363C3#2356111yescharacteristic, complemented, Sylow
364C2—21no—
365C2—22no—
366C2—22no—
367C2—21no—
368C2C2×#523011yescharacteristic
369C2#8410411yescharacteristic
370C2#8411611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#544C2
socleC22×C3C2×#120
prime radicalC24×C3C2×D4
hypercenterC2×#544D3
hypocenterC3#23561
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
4. #71901C2semidirect
5. #71900C2semidirect
66. #926Q8semidirect
75. C2×#208Q8semidirect
145. C4×Dic3C2×Q8semidirect
148. #208C2×Q8semidirect
363. C3#23561semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5442
#822072
#821572
#719012
#719002
#715742
C2×#45302
#235613