Finite Group Databasebeta

← #10876271

Subgroups — #10876271

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
1#76610C211yesmaximal, characteristic
2#74358C211yesmaximal, characteristic
3C3×#2095C211yesmaximal, characteristic, complemented
4#70633C212yesmaximal
5#70624C212yesmaximal, complemented
6#70266C212yesmaximal
7#70241C212yesmaximal, complemented
8#69671C211yesmaximal, characteristic, complemented
9#69591C211yesmaximal, characteristic, complemented
10#68665C211yesmaximal, characteristic
11#68638C211yesmaximal, characteristic, complemented
12#27596—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C2×#892C2211yescharacteristic
15C2×#872C2211yescharacteristic
16C2×C3×#107C2212yes—
17C2×C3×#103C2212yes—
18C3×#391C2211yescharacteristic
19C3×#384C2211yescharacteristic
20#5229C2211yescharacteristic
21C2×#835C2212yes—
22C2×(C2×D3)⋊C8C2212yes—
23C2×#816C2212yes—
24#4990—22no—
25#4987C2212yes—
26#4978—22no—
27Dic3×C4⋊C4C2212yes—
28#4973C2211yescharacteristic
29C2×#810C2212yes—
30C2×#808C2212yes—
31#4936C2211yescharacteristic
32#4929C2211yescharacteristic
33#4664—22no—
34#4646—22no—
35C3×#336C2211yescharacteristic
36#4551C2212yes—
37#4550C2212yes—
38#4529C2212yes—
39#4524C2211yescharacteristic
40#2314—31no—
41#2310—31no—
42#2096—31no—
43#2095D311yescharacteristic, complemented
44#2074—31no—
45#2074—31no—
46#1928—32no—
47#1927—32no—
48#1854—32no—
49#1818—31no—
50#1787—32no—
51C22×C3×Q8C2311yescharacteristic
52C2×D3×Q8—22no—
53C2×D3×Q8—22no—
54C2×D3×Q8—24no—
55C2×D3×Q8—42no—
56C22×C4×D3—21no—
57C22×C4×D3C2311yescharacteristic
58C22×Dic6—21no—
59C22×Dic6C2311yescharacteristic
60C22×C3×C8C2312yes—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4D412yes—
68C2×C4⋊Dic3C2311yescharacteristic
69C2×C4⋊Dic3C2312yes—
70C2×Dic3⋊C4D412yes—
71C2×C4×Dic3—21no—
72C2×C4×Dic3—22no—
73C2×C4×Dic3C2311yescharacteristic
74C22×C3⋊C8C2312yes—
75#892—42no—
76#872—24no—
77C3×#107—42no—
78C3×#103—42no—
79C3×#95—22no—
80#835—24no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87(C2×D3)⋊C8—24no—
88#816—24no—
89#810—42no—
90#808—42no—
91C23×Q8—31no—
92C2×#117—31no—
93C2×#117—31no—
94C2×#107C2×D312yes—
95C2×#106—32no—
96C2×#103—32no—
97C2×#103—32no—
98C2×#103—32no—
99C2×#103C2×D312yes—
100C2×C22⋊C8—32no—
101#391—31no—
102#391—31no—
103#391—31no—
104#391—62no—
105#391C2×D311yescharacteristic
106#388—62no—
107#387—62no—
108#384C2×D311yescharacteristic
109#382—31no—
110#381—62no—
111#380—32no—
112C4×C4⋊C4—32no—
113#340—32no—
114#340—32no—
115#336—32no—
116#336—31no—
117#336C2×D311yescharacteristic
118C23×D3C2×D411yescharacteristic
119C2×C3×Q8—21no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8#4112yes—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2411yescharacteristic
128C22×Dic3—21no—
129C22×Dic3—22no—
130C22×Dic3—21no—
131C22×Dic3C2×D412yes—
132C22×Dic3C2×D411yescharacteristic
133D3×Q8—48no—
134D3×Q8—48no—
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—22no—
138C2×C4×D3—24no—
139C2×C4×D3—42no—
140C2×C4×D3—42no—
141C2×C4×D3#4114yes—
142C2×Dic6—22no—
143C2×Dic6—22no—
144C2×Dic6—24no—
145C2×Dic6—21no—
146C2×Dic6—42no—
147C2×Dic6—42no—
148C2×Dic6#4112yes—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—82no—
156C3×C42—22no—
157(C2×D3)⋊C4—24no—
158(C2×D3)⋊C4—42no—
159(C2×D3)⋊C4—82no—
160C4⋊Dic3—22no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163Dic3⋊C4—24no—
164C4×Dic3—22no—
165C4×Dic3—42no—
166C4×Dic3—82no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170C22×Q8—31no—
171C22×Q8—61no—
172C22×Q8—64no—
173C22×Q8—62no—
174C22×Q8—62no—
175C22×Q8—122no—
176C22×Q8C22×D311yescharacteristic
177C23×C4—31no—
178C23×C4—61no—
179C22×C8—32no—
180C22×C8C22×D312yes—
181#117—64no—
182#117—122no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4C22×D311yescharacteristic
188C2×C4⋊C4C22×D312yes—
189C2×C22⋊C4—32no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—62no—
192C2×C42—31no—
193C2×C42—61no—
194C2×C42—62no—
195C2×C42C22×D311yescharacteristic
196#107—42no—
197#106—122no—
198#103—42no—
199#103—64no—
200#103—64no—
201#103—122no—
202C22⋊C8—64no—
203#95—22no—
204#95—62no—
205#95—62no—
206#95—62no—
207#95—62no—
208#95—62no—
209#95—62no—
210C23×C3C22×D411yescharacteristic
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—22no—
214C22×D3—22no—
215C22×D3#12312yes—
216C3×Q8—24no—
217C3×Q8—42no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4C2×#4111yescharacteristic
230C2×C3×C4C2×#4112yes—
231C2×C3×C4C2×#4111yescharacteristic
232C2×C3×C4H(5,𝔽2)12yes—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3#12312yes—
247C4×D3—24no—
248C4×D3—24no—
249C4×D3—48no—
250Dic6—24no—
251Dic6—48no—
252Dic6—42no—
253C3×C8—42no—
254C3⋊C8—42no—
255C24—31no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—32no—
260C2×Q8—64no—
261C2×Q8—61no—
262C2×Q8—62no—
263C2×Q8—62no—
264C2×Q8—122no—
265C2×Q8—122no—
266C2×Q8—128no—
267C2×Q8—128no—
268C2×Q8#23912yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—31no—
273C22×C4—34no—
274C22×C4—32no—
275C22×C4—64no—
276C22×C4—62no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—62no—
282C22×C4—122no—
283C22×C4—122no—
284C22×C4D3×D412yes—
285C22×C4C23×D311yescharacteristic
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—62no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—64no—
296C4⋊C4—62no—
297C4⋊C4—62no—
298C4⋊C4—62no—
299C4⋊C4—82no—
300C22⋊C4—64no—
301C22⋊C4—122no—
302C22⋊C4—242no—
303C42—22no—
304C42—62no—
305C42—122no—
306C42—242no—
307C22×C3C2×#12311yescharacteristic
308C22×C3#52511yescharacteristic
309C22×C3#53511yescharacteristic
310C22×C3#57712yes—
311C22×C3#57812yes—
312C2×D3—24no—
313C2×D3—22no—
314C2×D3—42no—
315C2×D3—41no—
316C2×D3—41no—
317C3×C4—42no—
318C3×C4—42no—
319C3×C4—81no—
320C3×C4—82no—
321C3×C4#55112yes—
322C3×C4#55112yes—
323Dic3—22no—
324Dic3—42no—
325Dic3—42no—
326Dic3—82no—
327Dic3—81no—
328C23—32no—
329C23—61no—
330C23—62no—
331C23—62no—
332C23—61no—
333C23C2×D3×D411yescharacteristic
334Q8—24no—
335Q8—42no—
336Q8—64no—
337Q8—128no—
338Q8—122no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—32no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—64no—
354C2×C4—64no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—62no—
359C2×C4—128no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—122no—
367C2×C4#100412yes—
368C2×C4C2×#23911yescharacteristic
369C2×C4D3×#4112yes—
370C2×C4D3×#4111yescharacteristic
371C8—42no—
372C8—122no—
373C2×C3#248211yescharacteristic
374C2×C3#309712yes—
375C2×C3#312112yes—
376C2×C3#322111yescharacteristic
377C2×C3#322311yescharacteristic
378D3—42no—
379C22—62no—
380C22—64no—
381C22—121no—
382C22—121no—
383C22—122no—
384C22#562711yescharacteristic
385C22D3×#12311yescharacteristic
386C22#571511yescharacteristic
387C22#576612yes—
388C22#577512yes—
389C4—42no—
390C4—42no—
391C4—62no—
392C4—81no—
393C4—82no—
394C4—122no—
395C4—122no—
396C4—241no—
397C4—242no—
398C4#563712yes—
399C4#565912yes—
400C3#2759611yescharacteristic, complemented, Sylow
401C2—122no—
402C2#7810511yescharacteristic
403C2#8109612yes—
404C2#8147012yes—
405C2#8180411yescharacteristic
406C2#8185411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2095C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2095D3
hypocenterC3#27596
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2095

Complements

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

NH ≅ G/Ntype
3. C3×#2095C2semidirect
5. #70624C2semidirect
7. #70241C2semidirect
8. #69671C2semidirect
9. #69591C2semidirect
11. #68638C2semidirect
43. #2095D3semidirect
400. C3#27596semidirect

Maximal subgroups

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

subgroupindex
#766102
#743582
C3×#20952
#706332
#706242
#702662
#702412
#696712
#695912
#686652
#686382
#275963