Finite Group Databasebeta

← #10896377

Subgroups — #10896377

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2331C211yesmaximal, characteristic
2#73388C211yesmaximal, characteristic
3#71794C212yesmaximal
4#69632C212yesmaximal
5#69631C212yesmaximal
6#69561C212yesmaximal
7#69486C212yesmaximal
8#69468C212yesmaximal
9#69437C211yesmaximal, characteristic
10#29264—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C4×Dic6C2212yes—
14C3×#391—22no—
15C3×#389C2212yes—
16C3×#387—22no—
17C3×#382C2211yescharacteristic
18C3×C4×C4⋊C4C2212yes—
19C3×#376C2211yescharacteristic
20C2×#810C2212yes—
21C2×#810C2212yes—
22C2×#810C2212yes—
23C2×#810C2212yes—
24C2×#808C2212yes—
25C2×#807C2212yes—
26#4936C2211yescharacteristic
27#4929C2212yes—
28C2×C4⋊(C3⋊C8)C2212yes—
29#4653—22no—
30#4653—22no—
31#4529C2212yes—
32#4529C2212yes—
33#4524C2211yescharacteristic
34#4524C2212yes—
35#4524C2211yescharacteristic
36#2331D311yescharacteristic
37#2327—31no—
38#2097—32no—
39#2024—32no—
40#1981—32no—
41#1949—32no—
42#1865—32no—
43#1863—32no—
44#1832—31no—
45C22×C3×Q8—21no—
46C22×Dic6C2312yes—
47C3×C4⋊Q8—24no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4C2312yes—
51C2×C3×C4⋊C4C2312yes—
52C2×C3×C42—22no—
53C2×C3×C42C2311yescharacteristic
54C2×C3×C42C2311yescharacteristic
55C2×C3×C42C2311yescharacteristic
56C2×C4⋊Dic3—22no—
57C2×C4⋊Dic3C2312yes—
58C2×Dic3⋊C4—22no—
59C2×C4×Dic3—22no—
60C22×C3⋊C8C2312yes—
61C22×C3⋊C8C2312yes—
62#869—22no—
63#869Q814yescomplemented
64C4×Dic6—42no—
65C4×Dic6—42no—
66C3×#95—22no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95—22no—
70C3×#95D412yes—
71C3×#95D412yes—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75#810—24no—
76#810—24no—
77#810—42no—
78#810—42no—
79#808—42no—
80#807—42no—
81C4⋊(C3⋊C8)—24no—
82C2×C4⋊Q8—31no—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×C4×Q8—32no—
85C2×#107—32no—
86C2×#106—32no—
87C2×C4⋊C8—32no—
88C2×#103—32no—
89C2×#103—32no—
90C2×#103—32no—
91C2×#103—32no—
92#391—22no—
93#391—32no—
94#389C2×D312yes—
95#387—22no—
96#387—62no—
97#387—62no—
98#382—31no—
99#382C2×D311yescharacteristic
100C4×C4⋊C4C2×D312yes—
101#376C2×D311yescharacteristic
102#340—32no—
103#340—32no—
104#336—32no—
105#336—31no—
106#336—31no—
107C2×C3×Q8—24no—
108C2×C3×Q8—82no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2411yescharacteristic
116C22×Dic3—22no—
117C22×Dic3—22no—
118C2×Dic6—22no—
119C2×Dic6—22no—
120C2×Dic6—22no—
121C2×Dic6—22no—
122C2×Dic6—22no—
123C2×Dic6C2×Q814yes—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—82no—
129C3×C4⋊C4#4114yes—
130C3×C42—22no—
131C3×C42—22no—
132C3×C42—82no—
133C3×C42C2×Q812yes—
134C3×C42#4112yes—
135C4⋊Dic3—28no—
136C4⋊Dic3—22no—
137C4⋊Dic3—22no—
138Dic3⋊C4—44no—
139C4×Dic3—44no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C22×Q8—21no—
147C22×Q8—32no—
148C22×C8—32no—
149C22×C8—32no—
150C4⋊Q8—24no—
151C4⋊Q8—34no—
152C4⋊Q8—62no—
153C4×Q8—122no—
154C4×Q8—122no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—32no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4C22×D312yes—
161C2×C4⋊C4C22×D312yes—
162C2×C42—22no—
163C2×C42—62no—
164C2×C42C22×D311yescharacteristic
165C2×C42C22×D311yescharacteristic
166C2×C42C22×D311yescharacteristic
167#107—122no—
168#106—122no—
169C4⋊C8—64no—
170#103—64no—
171#103—64no—
172#103—122no—
173#103—122no—
174#95—22no—
175#95—21no—
176#95—21no—
177#95—22no—
178#95—62no—
179#95—62no—
180#95—62no—
181#95#6712yes—
182#95#6712yes—
183C23×C3C22×D411yescharacteristic
184C3×Q8—84no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—24no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4C22⋊Q814yes—
201C2×C3×C4C22×Q811yescharacteristic
202C2×C3×C4C2×#4112yes—
203C2×C3×C4C2×#4111yescharacteristic
204C2×C3×C4H(5,𝔽2)12yes—
205C2×Dic3—24no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211Dic6—24no—
212Dic6—24no—
213Dic6—42no—
214Dic6—42no—
215C3⋊C8—42no—
216C3⋊C8—42no—
217C2×Q8—24no—
218C2×Q8—34no—
219C2×Q8—62no—
220C2×Q8—62no—
221C2×Q8—62no—
222C2×Q8—62no—
223C2×Q8—62no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6711yescharacteristic
231C22×C4C2×#6712yes—
232C22×C4C2×#6712yes—
233C22×C4C23×D311yescharacteristic
234C2×C8—62no—
235C2×C8—62no—
236C2×C8—62no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8—62no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—68no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—82no—
248C4⋊C4—124no—
249C4⋊C4#23514yes—
250C42—22no—
251C42—22no—
252C42—82no—
253C42—124no—
254C42D3×Q812yes—
255C42#23912yes—
256C22×C3C2×C22⋊Q811yescharacteristic
257C22×C3#52511yescharacteristic
258C22×C3#53511yescharacteristic
259C22×C3#57712yes—
260C22×C3#57812yes—
261C3×C4—24no—
262C3×C4—41no—
263C3×C4—44no—
264C3×C4—41no—
265C3×C4—42no—
266C3×C4—82no—
267C3×C4#55412yes—
268C3×C4#55912yes—
269Dic3—44no—
270Dic3—82no—
271C23C22×#6711yescharacteristic
272Q8—64no—
273Q8—64no—
274Q8—84no—
275Q8—122no—
276Q8—122no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—64no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4#94614yes—
299C2×C4C2×#23512yes—
300C2×C4C2×D3×Q811yescharacteristic
301C2×C4C2×#23911yescharacteristic
302C2×C4#100912yes—
303C8—122no—
304C8—122no—
305C2×C3#258011yescharacteristic
306C2×C3#308212yes—
307C2×C3#312311yescharacteristic
308C2×C3#312711yescharacteristic
309C2×C3#322512yes—
310C22#579111yescharacteristic
311C22C2×#94611yescharacteristic
312C22#583011yescharacteristic
313C22#583612yes—
314C22#583812yes—
315C4—24no—
316C4—42no—
317C4—41no—
318C4—44no—
319C4—41no—
320C4—82no—
321C4—124no—
322C4—242no—
323C4#557912yes—
324C4#558512yes—
325C3#2926411yescharacteristic, complemented, Sylow
326C2#8250111yescharacteristic
327C2#8254511yescharacteristic
328C2#8255511yescharacteristic
329C2#8346412yes—
330C2#8348012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2331C2
socle, prime radicalC23×C3C22×D4
hypercenter#2331D3
hypocenterC3#29264
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
63. #869Q8semidirect
325. C3#29264semidirect

Maximal subgroups

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

subgroupindex
C3×#23312
#733882
#717942
#696322
#696312
#695612
#694862
#694682
#694372
#292643