Finite Group Databasebeta

← #10876126

Subgroups — #10876126

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
1C3×#2285C211yesmaximal, characteristic
2C2×#4929C211yesmaximal, characteristic
3#71531C211yesmaximal, characteristic
4#69591C212yesmaximal, complemented
5#69589C212yesmaximal, complemented
6#69525C212yesmaximal
7#69505C212yesmaximal
8#69464C214yesmaximal, complemented
9#27583—31nomaximal, Sylow
10C23×Dic6C2211yescharacteristic
11C2×C3×#117—21no—
12C22×C3×C42C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C3×#384C2211yescharacteristic
15C3×#384C2211yescharacteristic
16C3×#382—21no—
17C3×#380—21no—
18C3×#380C2212yes—
19C3×C4×C4⋊C4—21no—
20C3×C4×C4⋊C4C2212yes—
21C2×C3⋊#95—21no—
22C2×C3⋊#95—21no—
23C2×#830C2212yes—
24C2×#810C2214yes—
25C2×#810C2214yes—
26C2×#808C2214yes—
27#4929—21no—
28#4929—21no—
29#4929C2212yes—
30#4929C2212yes—
31#4529C2214yes—
32#4524C2212yes—
33#4524C2212yes—
34C2×#391—31no—
35#2285D311yescharacteristic
36#2096—32no—
37#2072—32no—
38#1927—34no—
39#1853—32no—
40#1817—31no—
41#1787—32no—
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×Dic3—21no—
46C23×Dic3—21no—
47C22×Dic6—21no—
48C22×Dic6—24no—
49C22×Dic6—24no—
50C22×Dic6—42no—
51C22×Dic6C2312yes—
52C3×#117—44no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42—24no—
61C2×C3×C42—22no—
62C2×C3×C42D411yescharacteristic
63C2×C3×C42D411yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—42no—
68C2×C4⋊Dic3C2312yes—
69C22×C3⋊C8C2314yes—
70C3×#95—22no—
71C3×#95—22no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—22no—
76#830—24no—
77#810—28no—
78#810—44no—
79#808—44no—
80C23×Q8—31no—
81C2×#117—21no—
82C22×C4⋊C4—31no—
83C22×C42C2×D311yescharacteristic
84C2×#106—34no—
85C2×#103—34no—
86C2×#103—34no—
87C2×C22⋊C8—32no—
88#391—32no—
89#391—32no—
90#391—61no—
91#391—61no—
92#384C2×D311yescharacteristic
93#384C2×D311yescharacteristic
94#382—21no—
95#380—21no—
96#380C2×D312yes—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2×D312yes—
99C2×#95—61no—
100C2×#95—61no—
101#340—34no—
102#336—32no—
103#336—32no—
104C24×C3C2×D411yescharacteristic
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—41no—
114C22×C3×C4SD1614yes—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4114yes—
121C22×C3×C4C2411yescharacteristic
122C22×Dic3—24no—
123C22×Dic3—22no—
124C22×Dic3—22no—
125C22×Dic3—42no—
126C22×Dic3—42no—
127C22×Dic3—42no—
128C22×Dic3—42no—
129C22×Dic3—42no—
130C2×Dic6—24no—
131C2×Dic6—24no—
132C2×Dic6—24no—
133C2×Dic6—22no—
134C2×Dic6—416no—
135C2×Dic6—42no—
136C2×Dic6—42no—
137C2×Dic6#4114yes—
138C3×C4⋊C4—24no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—42no—
141C3×C4⋊C4—42no—
142C3×C4⋊C4—42no—
143C3×C4⋊C4—42no—
144C3×C22⋊C4—44no—
145C3×C42—22no—
146C3×C42—22no—
147C3×C42—22no—
148C3×C42—22no—
149C3×C42—44no—
150C4⋊Dic3—24no—
151C4⋊Dic3—42no—
152C2×C3⋊C8—24no—
153C2×C3⋊C8—24no—
154C2×C3⋊C8—24no—
155C22×Q8—32no—
156C22×Q8—64no—
157C22×Q8—61no—
158C22×Q8—64no—
159C22×Q8—122no—
160C23×C4—61no—
161C23×C4—61no—
162C23×C4#6711yescharacteristic
163C23×C4#6711yescharacteristic
164C23×C4C22×D311yescharacteristic
165C22×C8—34no—
166#117—44no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—32no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—122no—
174C2×C4⋊C4C22×D312yes—
175C2×C4⋊C4C22×D312yes—
176C2×C22⋊C4—22no—
177C2×C42—24no—
178C2×C42—22no—
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—62no—
190#95—62no—
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
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—28no—
210C2×C3×C4—28no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—44no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4#12312yes—
222C2×C3×C4C2×SD1612yes—
223C2×C3×C4C2×SD1612yes—
224C2×C3×C4#13712yes—
225C2×C3×C4C2×#4112yes—
226C2×C3×C4C2×#4112yes—
227C2×C3×C4H(5,𝔽2)12yes—
228C2×Dic3—24no—
229C2×Dic3—48no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—84no—
236Dic6—28no—
237Dic6—48no—
238Dic6—44no—
239C3⋊C8—44no—
240C24C2×#6711yescharacteristic
241C2×Q8—34no—
242C2×Q8—64no—
243C2×Q8—62no—
244C2×Q8—64no—
245C2×Q8—64no—
246C2×Q8—1216no—
247C2×Q8—122no—
248C2×Q8—122no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—64no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—122no—
262C22×C4—122no—
263C22×C4—122no—
264C22×C4—122no—
265C22×C4—122no—
266C22×C4#21414yes—
267C22×C4#23514yes—
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—24no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—64no—
284C4⋊C4—122no—
285C22⋊C4—44no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—44no—
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—82no—
311C3×C4—82no—
312C3×C4#55114yes—
313Dic3—44no—
314Dic3—82no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23#92912yes—
320C23C2×#21412yes—
321C23#94212yes—
322C23C22×#6711yescharacteristic
323Q8—68no—
324Q8—128no—
325Q8—124no—
326C2×C4—28no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—28no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—64no—
347C2×C4—128no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—244no—
354C2×C4#92912yes—
355C2×C4C2×#21412yes—
356C2×C4C2×#21412yes—
357C2×C4#95112yes—
358C2×C4C2×#23512yes—
359C2×C4D3×#4112yes—
360C2×C4#100912yes—
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#497114yes—
377C22C2×#92911yescharacteristic
378C22C22×#21411yescharacteristic
379C22C2×#94211yescharacteristic
380C22#582211yescharacteristic
381C22C2×#95111yescharacteristic
382C22#582811yescharacteristic
383C22#583011yescharacteristic
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—41no—
389C4—82no—
390C4—82no—
391C4—124no—
392C4—242no—
393C4#552714yes—
394C3#2758311yescharacteristic, complemented, Sylow
395C2—24no—
396C2C2×#497111yescharacteristic
397C2#8222011yescharacteristic
398C2#8228211yescharacteristic
399C2#8331511yescharacteristic
400C2#8331711yescharacteristic
401C2#8335711yescharacteristic
402C2#8336911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2285C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2285D3
hypocenterC3#27583
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
4. #69591C2semidirect
5. #69589C2semidirect
8. #69464C2semidirect
394. C3#27583semidirect

Maximal subgroups

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

subgroupindex
C3×#22852
C2×#49292
#715312
#695912
#695892
#695252
#695052
#694642
#275833