Finite Group Databasebeta

← #10876281

Subgroups — #10876281

Order 768.

842 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2311C211yesmaximal, characteristic
2#82231C211yesmaximal, characteristic, complemented
3#72135C212yesmaximal
4#71584C211yesmaximal, characteristic, complemented
5#71556C212yesmaximal, complemented
6#69685C211yesmaximal, characteristic, complemented
7#69671C211yesmaximal, characteristic, complemented
8#69666C212yesmaximal, complemented
9#69656C211yesmaximal, characteristic, complemented
10#69641C211yesmaximal, characteristic, complemented
11#69637C212yesmaximal, complemented
12#27597—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C3×#391C2211yescharacteristic
17C3×#391C2211yescharacteristic
18C3×#390C2211yescharacteristic
19C3×#390C2211yescharacteristic
20C3×#386—22no—
21C3×#380—22no—
22C2×#835C2212yes—
23C2×#835C2212yes—
24C2×#832C2212yes—
25C2×#832C2212yes—
26#5212C2212yes—
27C2×#830C2212yes—
28#4980—22no—
29C2×#808C2212yes—
30C2×#808C2212yes—
31#4951—22no—
32#4936C2211yescharacteristic
33#4935C2211yescharacteristic
34C4×C4⋊Dic3C2212yes—
35#4529C2212yes—
36#4529C2212yes—
37#4524C2212yes—
38#4524C2211yescharacteristic
39#4524C2211yescharacteristic
40#2314—31no—
41#2311D311yescharacteristic
42#2096—31no—
43#2094—31no—
44#2074—31no—
45#2073—31no—
46#1927—32no—
47#1926—32no—
48#1853—32no—
49#1818—31no—
50#1788—32no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—24no—
54C2×C3×#41—42no—
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C23×C3×C4C2311yescharacteristic
61C3×#117—24no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—42no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4D412yes—
70C2×C3×C42C2311yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2312yes—
76C2×C4×Dic3—22no—
77C22×C3⋊C8C2312yes—
78C22×C3⋊C8C2312yes—
79C3×#95—22no—
80C3×#95—22no—
81#835—24no—
82#835—42no—
83#832—24no—
84#832—42no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88#830—24no—
89#808—42no—
90#808—42no—
91C22×#41C2×D311yescharacteristic
92C2×#117C2×D311yescharacteristic
93C22×C4⋊C4C2×D311yescharacteristic
94C2×#106—32no—
95C2×#106—32no—
96C2×#103—32no—
97C2×#103—32no—
98C2×#102—32no—
99C2×#102—32no—
100C2×C22⋊C8—32no—
101#391C2×D311yescharacteristic
102#391C2×D311yescharacteristic
103#390C2×D311yescharacteristic
104#390C2×D311yescharacteristic
105#388—62no—
106#387—62no—
107#386—22no—
108#384—31no—
109#382—31no—
110#380—22no—
111#380—32no—
112C4×C4⋊C4—32no—
113#340—32no—
114#340—32no—
115#336—31no—
116#336—31no—
117#336—32no—
118C24×C3—21no—
119C24×C3C2×D411yescharacteristic
120C3×#41—48no—
121C3×#41—48no—
122C2×C3×Q8—21no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8#4112yes—
126C2×C3×D4—24no—
127C2×C3×D4—22no—
128C2×C3×D4—21no—
129C2×C3×D4—22no—
130C2×C3×D4—42no—
131C2×C3×D4—42no—
132C2×C3×D4#4112yes—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4—82no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4114yes—
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—22no—
150C22×Dic3—22no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—42no—
155C3×C22⋊C4—24no—
156C3×C22⋊C4—82no—
157C3×C22⋊C4—82no—
158C3×C42—22no—
159C3×C42—22no—
160#217—82no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—82no—
166C4×Dic3—82no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—24no—
176C2×#41—42no—
177C22×Q8C22×D311yescharacteristic
178C22×D4—21no—
179C22×D4C22×D311yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C22×D311yescharacteristic
183C22×C8—32no—
184C22×C8—32no—
185#117—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4#6712yes—
192C2×C4⋊C4C22×D311yescharacteristic
193C2×C4⋊C4C22×D311yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4#6712yes—
198C2×C42—62no—
199C2×C42C22×D311yescharacteristic
200C2×C42C22×D311yescharacteristic
201#106—122no—
202#106—122no—
203#103—64no—
204#103—122no—
205#102—64no—
206#102—122no—
207C22⋊C8—64no—
208#95—22no—
209#95—22no—
210#95—62no—
211#95—62no—
212#95—62no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3#12312yes—
223C23×C3C22×D411yescharacteristic
224C3×Q8—24no—
225C3×Q8—42no—
226C3×D4—24no—
227C3×D4—48no—
228C3×D4—42no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—48no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—84no—
248C2×C3×C4#12312yes—
249C2×C3×C4C2×#4111yescharacteristic
250C2×C3×C4C2×#4111yescharacteristic
251C2×C3×C4C2×#4112yes—
252C2×C3×C4H(5,𝔽2)12yes—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C3⋊C8—42no—
260C3⋊C8—42no—
261C24—21no—
262C24C2×#6711yescharacteristic
263#41—48no—
264#41—48no—
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8#23512yes—
269C2×D4—24no—
270C2×D4—21no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4#23512yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—62no—
287C22×C4—62no—
288C22×C4—82no—
289C22×C4#23714yes—
290C22×C4C2×#6712yes—
291C22×C4C2×#6712yes—
292C22×C4C2×#6711yescharacteristic
293C22×C4C23×D311yescharacteristic
294C2×C8—62no—
295C2×C8—62no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—62no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—42no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—242no—
309C22⋊C4—24no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—242no—
313C42—22no—
314C42—22no—
315C42—242no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—42no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—44no—
322C22×C3—42no—
323C22×C3—82no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3C2×#12311yescharacteristic
327C22×C3#52511yescharacteristic
328C22×C3#53511yescharacteristic
329C22×C3#57712yes—
330C22×C3#57712yes—
331C3×C4—22no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—42no—
335C3×C4—81no—
336C3×C4—81no—
337C3×C4#55112yes—
338C3×C4#55112yes—
339Dic3—82no—
340Dic3—82no—
341C23—21no—
342C23—21no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—42no—
349C23—42no—
350C23#93512yes—
351C23C22×#6711yescharacteristic
352Q8—24no—
353Q8—42no—
354D4—24no—
355D4—48no—
356D4—42no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—48no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—84no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4#93512yes—
383C2×C4C2×#23511yescharacteristic
384C2×C4C2×#23511yescharacteristic
385C2×C4C2×#23712yes—
386C2×C4#100412yes—
387C8—122no—
388C8—122no—
389C2×C3—42no—
390C2×C3—82no—
391C2×C3#248211yescharacteristic
392C2×C3#309912yes—
393C2×C3#312512yes—
394C2×C3#322111yescharacteristic
395C2×C3#322211yescharacteristic
396C22—22no—
397C22—24no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—44no—
402C22—42no—
403C22—81no—
404C22—81no—
405C22—82no—
406C22#579111yescharacteristic
407C22C2×#93511yescharacteristic
408C22#580511yescharacteristic
409C22#583612yes—
410C22#583712yes—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—81no—
416C4—81no—
417C4—242no—
418C4—242no—
419C4#549512yes—
420C4#549512yes—
421C3#2759711yescharacteristic, complemented, Sylow
422C2—42no—
423C2—82no—
424C2#8211311yescharacteristic
425C2#8257612yes—
426C2#8339311yescharacteristic
427C2#8340911yescharacteristic
428C2#8341912yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2311C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenter#2311D3
hypocenterC3#27597
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. #82231C2semidirect
4. #71584C2semidirect
5. #71556C2semidirect
6. #69685C2semidirect
7. #69671C2semidirect
8. #69666C2semidirect
9. #69656C2semidirect
10. #69641C2semidirect
11. #69637C2semidirect
421. C3#27597semidirect

Maximal subgroups

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

subgroupindex
C3×#23112
#822312
#721352
#715842
#715562
#696852
#696712
#696662
#696562
#696412
#696372
#275973