Finite Group Databasebeta

← #10876264

Subgroups — #10876264

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2314C211yesmaximal, characteristic
2#82230C211yesmaximal, characteristic
3#71584C211yesmaximal, characteristic
4#69591C211yesmaximal, characteristic, complemented
5#69590C211yesmaximal, characteristic, complemented
6#69560C211yesmaximal, characteristic, complemented
7#69560C211yesmaximal, characteristic, complemented
8#69525C212yesmaximal
9#69506C212yesmaximal
10#69465C212yesmaximal, complemented
11#69464C212yesmaximal, complemented
12#27596—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C3×#388—22no—
17C3×#387—22no—
18C3×#384C2211yescharacteristic
19C3×#382C2211yescharacteristic
20C3×#380C2212yes—
21C3×C4×C4⋊C4C2212yes—
22C2×#830C2212yes—
23#4973—22no—
24C2×#810C2212yes—
25C2×#810C2212yes—
26C2×#810C2212yes—
27C2×#810C2212yes—
28C2×#808C2212yes—
29C2×#807C2212yes—
30#4943—22no—
31#4929C2211yescharacteristic
32#4929C2211yescharacteristic
33#4929C2211yescharacteristic
34#4929C2211yescharacteristic
35#4529C2212yes—
36#4529C2212yes—
37#4524C2212yes—
38#4524C2211yescharacteristic
39#4524C2211yescharacteristic
40#2314D311yescharacteristic
41#2310—31no—
42#2096—31no—
43#2095—31no—
44#2074—31no—
45#2074—31no—
46#1928—32no—
47#1927—32no—
48#1854—32no—
49#1818—31no—
50#1787—32no—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3—21no—
53C22×Dic6—21no—
54C22×Dic6—22no—
55C22×Dic6—22no—
56C22×Dic6—24no—
57C22×Dic6—42no—
58C22×Dic6C2311yescharacteristic
59C22×Dic6C2311yescharacteristic
60C3×#117—24no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4D412yes—
67C2×C3×C42—22no—
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×#217—21no—
71#926—42no—
72C2×C4⋊Dic3C2311yescharacteristic
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4×Dic3—21no—
75C22×C3⋊C8C2312yes—
76C22×C3⋊C8C2312yes—
77C3×#95—22no—
78C3×#95—22no—
79C3×#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84#830—24no—
85#810—24no—
86#810—24no—
87#810—42no—
88#810—42no—
89#808—42no—
90#807—42no—
91C23×Q8—31no—
92C2×#117—31no—
93C2×#117C2×D311yescharacteristic
94C2×#107—32no—
95C2×#106—32no—
96C2×#103—32no—
97C2×#103—32no—
98C2×#103—32no—
99C2×#103—32no—
100C2×C22⋊C8—32no—
101#391—31no—
102#391—31no—
103#391—31no—
104#391—31no—
105#391—62no—
106#388—22no—
107#387—22no—
108#384C2×D311yescharacteristic
109#382C2×D311yescharacteristic
110#381—62no—
111#380C2×D312yes—
112C4×C4⋊C4C2×D312yes—
113#340—32no—
114#340—32no—
115#336—31no—
116#336—31no—
117#336—32no—
118C24×C3C2×D411yescharacteristic
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4C2×D412yes—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4114yes—
128C22×C3×C4C2411yescharacteristic
129C22×Dic3—24no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—21no—
134C22×Dic3—42no—
135C22×Dic3—42no—
136C2×Dic6—22no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6—22no—
140C2×Dic6—24no—
141C2×Dic6—21no—
142C2×Dic6—21no—
143C2×Dic6—48no—
144C2×Dic6—48no—
145C2×Dic6—42no—
146C2×Dic6—42no—
147C2×Dic6#4112yes—
148C2×Dic6#4112yes—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—82no—
155C3×C22⋊C4—24no—
156C3×C22⋊C4—82no—
157C3×C42—22no—
158C3×C42—22no—
159C3×C42—82no—
160#217—42no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4×Dic3—42no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170C22×Q8—31no—
171C22×Q8—31no—
172C22×Q8—62no—
173C22×Q8—62no—
174C22×Q8—61no—
175C22×Q8—64no—
176C22×Q8—122no—
177C23×C4—61no—
178C23×C4C22×D311yescharacteristic
179C22×C8—32no—
180C22×C8—32no—
181#117—24no—
182#117—122no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4#6712yes—
187C2×C4⋊C4C22×D312yes—
188C2×C4⋊C4C22×D312yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4#6712yes—
192C2×C42—22no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195C2×C42C22×D311yescharacteristic
196#107—122no—
197#106—122no—
198#103—64no—
199#103—64no—
200#103—122no—
201#103—122no—
202C22⋊C8—64no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—62no—
207#95—62no—
208#95—62no—
209#95—62no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3#12312yes—
215C23×C3C22×D411yescharacteristic
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—24no—
219C2×C3×C4—24no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4#12312yes—
233C2×C3×C4C2×#4112yes—
234C2×C3×C4C2×#4111yescharacteristic
235C2×C3×C4C2×#4111yescharacteristic
236C2×C3×C4H(5,𝔽2)12yes—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—48no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248Dic6—24no—
249Dic6—24no—
250Dic6—48no—
251Dic6—42no—
252Dic6—42no—
253C3⋊C8—42no—
254C3⋊C8—42no—
255C24C2×#6711yescharacteristic
256C2×Q8—32no—
257C2×Q8—32no—
258C2×Q8—64no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—62no—
262C2×Q8—62no—
263C2×Q8—62no—
264C2×Q8—62no—
265C2×Q8—128no—
266C2×Q8—128no—
267C2×Q8—122no—
268C2×Q8—122no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—64no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—122no—
280C22×C4—122no—
281C22×C4#23514yes—
282C22×C4C2×#6712yes—
283C22×C4C2×#6712yes—
284C22×C4C2×#6711yescharacteristic
285C22×C4C23×D311yescharacteristic
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—62no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—62no—
298C4⋊C4—62no—
299C4⋊C4—82no—
300C22⋊C4—24no—
301C22⋊C4—82no—
302C22⋊C4—122no—
303C42—22no—
304C42—22no—
305C42—82no—
306C42—122no—
307C22×C3—24no—
308C22×C3—22no—
309C22×C3—42no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3C2×#12311yescharacteristic
313C22×C3#52511yescharacteristic
314C22×C3#53511yescharacteristic
315C22×C3#57712yes—
316C22×C3#57812yes—
317C3×C4—22no—
318C3×C4—42no—
319C3×C4—42no—
320C3×C4—82no—
321C3×C4—82no—
322C3×C4#55112yes—
323C3×C4#55112yes—
324Dic3—42no—
325Dic3—42no—
326Dic3—81no—
327Dic3—81no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23#92912yes—
333C23C22×#6711yescharacteristic
334Q8—64no—
335Q8—64no—
336Q8—128no—
337Q8—122no—
338Q8—122no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—128no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4#92912yes—
367C2×C4C2×#23512yes—
368C2×C4D3×#4111yescharacteristic
369C2×C4D3×#4111yescharacteristic
370C2×C4#100912yes—
371C8—122no—
372C8—122no—
373C2×C3—42no—
374C2×C3#248211yescharacteristic
375C2×C3#309712yes—
376C2×C3#312112yes—
377C2×C3#322111yescharacteristic
378C2×C3#322311yescharacteristic
379C22—22no—
380C22—24no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22C2×#92911yescharacteristic
385C22#582811yescharacteristic
386C22#583011yescharacteristic
387C22#583612yes—
388C22#583812yes—
389C4—22no—
390C4—42no—
391C4—42no—
392C4—82no—
393C4—82no—
394C4—122no—
395C4—122no—
396C4—241no—
397C4—241no—
398C4#552712yes—
399C4#552712yes—
400C3#2759611yescharacteristic, complemented, Sylow
401C2—42no—
402C2#8228211yescharacteristic
403C2#8257312yes—
404C2#8344011yescharacteristic
405C2#8345211yescharacteristic
406C2#8346712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2314C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2314D3
hypocenterC3#27596
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
4. #69591C2semidirect
5. #69590C2semidirect
6. #69560C2semidirect
7. #69560C2semidirect
10. #69465C2semidirect
11. #69464C2semidirect
400. C3#27596semidirect

Maximal subgroups

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

subgroupindex
C3×#23142
#822302
#715842
#695912
#695902
#695602
#695252
#695062
#694652
#694642
#275963