Finite Group Databasebeta

← #10876130

Subgroups — #10876130

Order 768.

912 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2#82062C211yesmaximal, characteristic
3#72132C212yesmaximal
4#71556C212yesmaximal
5#71531C211yesmaximal, characteristic
6#69685C212yesmaximal, complemented
7#69670C212yesmaximal, complemented
8#69666C214yesmaximal, complemented
9#27583—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C2×#926—21no—
14C3×#391—21no—
15C3×#391—21no—
16C3×#391C2212yes—
17C3×#391C2212yes—
18C2×C3×#95—21no—
19C2×C3×#95—21no—
20C2×#835C2214yes—
21C2×#835C2214yes—
22#5212—21no—
23#5212C2212yes—
24C2×#830C2212yes—
25C2×#808C2214yes—
26#4936—21no—
27#4935C2211yescharacteristic
28#4935C2211yescharacteristic
29C4×C4⋊Dic3—21no—
30C4×C4⋊Dic3C2212yes—
31#4529C2214yes—
32#4524C2212yes—
33#4524C2212yes—
34C2×#391D311yescharacteristic
35#2285—31no—
36#2096—32no—
37#2072—32no—
38#1927—34no—
39#1853—32no—
40#1817—31no—
41#1787—32no—
42C22×C3×Q8—24no—
43C22×C3×Q8—21no—
44C22×C3×Q8—24no—
45C22×C3×Q8—42no—
46C22×C3×Q8C2312yes—
47C23×C3×C4—21no—
48C23×C3×C4—21no—
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—42no—
54C2×C3×C4⋊C4C2312yes—
55C2×C3×C42—24no—
56C2×C3×C42D411yescharacteristic
57C2×C3×C42D411yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×#217—22no—
61#926—44no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3C2312yes—
67C2×C4⋊Dic3C2312yes—
68C2×C4×Dic3—22no—
69C22×C3⋊C8C2314yes—
70C3×#95—22no—
71C3×#95—22no—
72C3×#95—22no—
73C3×#95—22no—
74#835—28no—
75#835—44no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78#830—24no—
79#808—44no—
80C23×Q8C2×D311yescharacteristic
81C2×#117—61no—
82C22×C4⋊C4C2×D311yescharacteristic
83C22×C42C2×D311yescharacteristic
84C2×#106—34no—
85C2×#103—34no—
86C2×#103—34no—
87C2×C22⋊C8—32no—
88#391—21no—
89#391—21no—
90#391C2×D312yes—
91#391C2×D312yes—
92#384—31no—
93#384—31no—
94#382—61no—
95#380—32no—
96#380—61no—
97C4×C4⋊C4—32no—
98C4×C4⋊C4—61no—
99C2×#95—21no—
100C2×#95—21no—
101#340—34no—
102#336—32no—
103#336—32no—
104C24×C3C2×D411yescharacteristic
105C2×C3×Q8—24no—
106C2×C3×Q8—22no—
107C2×C3×Q8—24no—
108C2×C3×Q8—24no—
109C2×C3×Q8—416no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C2×C3×Q8#4114yes—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—24no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—41no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4—42no—
127C22×C3×C4—42no—
128C22×C3×C4SD1614yes—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4114yes—
135C22×C3×C4C2411yescharacteristic
136C22×Dic3—22no—
137C22×Dic3—22no—
138C3×C4⋊C4—24no—
139C3×C4⋊C4—42no—
140C3×C42—22no—
141C3×C42—22no—
142C3×C42—22no—
143C3×C42—22no—
144#217—44no—
145C4⋊Dic3—24no—
146C4⋊Dic3—24no—
147C4⋊Dic3—42no—
148C4⋊Dic3—42no—
149C4⋊Dic3—42no—
150C4⋊Dic3—42no—
151C4×Dic3—44no—
152C2×C3⋊C8—24no—
153C2×C3⋊C8—24no—
154C2×C3⋊C8—24no—
155C22×Q8—24no—
156C22×Q8—21no—
157C22×Q8—24no—
158C22×Q8—42no—
159C22×Q8C22×D312yes—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4#6711yescharacteristic
163C23×C4#6711yescharacteristic
164C23×C4C22×D311yescharacteristic
165C22×C8—34no—
166#117—124no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—32no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4C22×D312yes—
176C2×C22⋊C4—62no—
177C2×C42—24no—
178C2×C42—62no—
179C2×C42#6711yescharacteristic
180C2×C42#6711yescharacteristic
181C2×C42C22×D311yescharacteristic
182C2×C42C22×D311yescharacteristic
183#106—124no—
184#103—68no—
185#103—124no—
186C22⋊C8—64no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—62no—
192#95—62no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3#12312yes—
198C23×C3C2×SD1612yes—
199C23×C3#13712yes—
200C23×C3C22×D411yescharacteristic
201C3×Q8—28no—
202C3×Q8—48no—
203C3×Q8—44no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—28no—
213C2×C3×C4—28no—
214C2×C3×C4—21no—
215C2×C3×C4—24no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—44no—
219C2×C3×C4—48no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—84no—
226C2×C3×C4#12312yes—
227C2×C3×C4C2×SD1612yes—
228C2×C3×C4C2×SD1612yes—
229C2×C3×C4#13712yes—
230C2×C3×C4C2×#4112yes—
231C2×C3×C4C2×#4112yes—
232C2×C3×C4H(5,𝔽2)12yes—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C3⋊C8—44no—
240C24C2×#6711yescharacteristic
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—416no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8#23514yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—62no—
265C22×C4—62no—
266C22×C4#21514yes—
267C22×C4#23714yes—
268C22×C4C2×#6711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6711yescharacteristic
271C22×C4C2×#6711yescharacteristic
272C22×C4C2×#6711yescharacteristic
273C22×C4C23×D311yescharacteristic
274C2×C8—64no—
275C2×C8—64no—
276C2×C8—64no—
277C4⋊C4—24no—
278C4⋊C4—42no—
279C4⋊C4—64no—
280C4⋊C4—64no—
281C4⋊C4—122no—
282C4⋊C4—122no—
283C4⋊C4—122no—
284C4⋊C4—122no—
285C22⋊C4—124no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—124no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3#48314yes—
298C22×C3C2×#12311yescharacteristic
299C22×C3#52511yescharacteristic
300C22×C3#53511yescharacteristic
301C22×C3C22×SD1611yescharacteristic
302C22×C3C2×#13711yescharacteristic
303C22×C3C2×#13711yescharacteristic
304C22×C3#57511yescharacteristic
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—41no—
310C3×C4—44no—
311C3×C4—82no—
312C3×C4#55114yes—
313Dic3—82no—
314Dic3—82no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23#93512yes—
320C23C2×#21512yes—
321C23#95112yes—
322C23C22×#6711yescharacteristic
323Q8—28no—
324Q8—48no—
325Q8—44no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—28no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—28no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—48no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—84no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4#93512yes—
355C2×C4C2×#21512yes—
356C2×C4C2×#21512yes—
357C2×C4#94212yes—
358C2×C4C2×#23512yes—
359C2×C4C2×#23712yes—
360C2×C4#100412yes—
361C8—124no—
362C2×C3—24no—
363C2×C3#248211yescharacteristic
364C2×C3C2×#48311yescharacteristic
365C2×C3#309511yescharacteristic
366C2×C3#310311yescharacteristic
367C2×C3#311311yescharacteristic
368C2×C3#320611yescharacteristic
369C2×C3#321511yescharacteristic
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22#504314yes—
377C22#579111yescharacteristic
378C22C2×#93511yescharacteristic
379C22#580511yescharacteristic
380C22C22×#21511yescharacteristic
381C22C2×#94211yescharacteristic
382C22#582211yescharacteristic
383C22C2×#95111yescharacteristic
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—41no—
389C4—44no—
390C4—82no—
391C4—242no—
392C4—242no—
393C4#549514yes—
394C3#2758311yescharacteristic, complemented, Sylow
395C2—24no—
396C2#8211311yescharacteristic
397C2#8244211yescharacteristic
398C2#8244811yescharacteristic
399C2C2×#504311yescharacteristic
400C2#8272711yescharacteristic
401C2#8294911yescharacteristic
402C2#8321311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#391C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#391D3
hypocenterC3#27583
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
6. #69685C2semidirect
7. #69670C2semidirect
8. #69666C2semidirect
394. C3#27583semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3912
#820622
#721322
#715562
#715312
#696852
#696702
#696662
#275833