Finite Group Databasebeta

← #10876272

Subgroups — #10876272

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#82231C211yesmaximal, characteristic
2#82230C211yesmaximal, characteristic
3C3×#1818C211yesmaximal, characteristic
4#69963C212yesmaximal
5#69929C212yesmaximal
6#68692C211yesmaximal, characteristic, complemented
7#68691C211yesmaximal, characteristic, complemented
8#68638C211yesmaximal, characteristic, complemented
9#68638C211yesmaximal, characteristic, complemented
10#68628C212yesmaximal, complemented
11#68627C212yesmaximal, complemented
12#27596—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C2×C3×C22⋊C8C2212yes—
17#5212C2212yes—
18C2×#818C2212yes—
19C2×#817C2212yes—
20C2×#816C2212yes—
21C2×#816C2212yes—
22C2×#816C2212yes—
23C2×#816C2212yes—
24#4980—22no—
25#4973—22no—
26#4951—22no—
27#4943—22no—
28#4936C2211yescharacteristic
29#4935C2211yescharacteristic
30C4×C4⋊Dic3C2212yes—
31#4929C2211yescharacteristic
32#4929C2211yescharacteristic
33#4929C2211yescharacteristic
34#4929C2211yescharacteristic
35C3×#336C2212yes—
36C3×#336C2211yescharacteristic
37C3×#336C2211yescharacteristic
38#4551C2212yes—
39#4551C2212yes—
40#2314—31no—
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#1818D311yescharacteristic
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
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×#117—24no—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C22⋊C4D412yes—
65C2×C3×C42C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×#217—21no—
68C2×#217—22no—
69#926—42no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3C2311yescharacteristic
72C2×C4⋊Dic3C2311yescharacteristic
73C2×C4⋊Dic3C2312yes—
74C2×C4⋊Dic3C2312yes—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—22no—
77C3×C22⋊C8—24no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85#818—42no—
86#817—42no—
87#816—24no—
88#816—24no—
89#816—42no—
90#816—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⋊C8C2×D312yes—
101#391—31no—
102#391—31no—
103#391—31no—
104#391—31no—
105#391—62no—
106#388—62no—
107#387—62no—
108#384—31no—
109#382—31no—
110#381—62no—
111#380—32no—
112C4×C4⋊C4—32no—
113#340—32no—
114#340—32no—
115#336C2×D311yescharacteristic
116#336C2×D311yescharacteristic
117#336C2×D312yes—
118C24×C3C2×D411yescharacteristic
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2×D412yes—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4114yes—
126C22×C3×C4C2411yescharacteristic
127C22×Dic3—24no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3—21no—
132C22×Dic3—21no—
133C22×Dic3—22no—
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—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C3×C4⋊C4—24no—
156C3×C22⋊C4—24no—
157C3×C42—22no—
158C3×C42—22no—
159#217—42no—
160#217—82no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—82no—
168C4×Dic3—42no—
169C4×Dic3—82no—
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×C8C22×D312yes—
180C22×C8C22×D312yes—
181#117—24no—
182#117—122no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—32no—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4D1212yes—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4D1212yes—
192C2×C42—61no—
193C2×C42—62no—
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—24no—
203#95—62no—
204#95—62no—
205#95—62no—
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#12312yes—
227C2×C3×C4C2×#4112yes—
228C2×C3×C4C2×#4111yescharacteristic
229C2×C3×C4C2×#4111yescharacteristic
230C2×C3×C4H(5,𝔽2)12yes—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—48no—
234C2×Dic3—41no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248Dic6—24no—
249Dic6—24no—
250Dic6—48no—
251Dic6—42no—
252Dic6—42no—
253C3×C8—42no—
254C3×C8—42no—
255C24C2×D1211yescharacteristic
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—64no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—62no—
279C22×C4—122no—
280C22×C4—122no—
281C22×C4C2×D1212yes—
282C22×C4C2×D1212yes—
283C22×C4C2×D1211yescharacteristic
284C22×C4#23714yes—
285C22×C4C23×D311yescharacteristic
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C4⋊C4—24no—
293C4⋊C4—62no—
294C4⋊C4—62no—
295C4⋊C4—62no—
296C4⋊C4—62no—
297C4⋊C4—62no—
298C4⋊C4—62no—
299C4⋊C4—242no—
300C22⋊C4—24no—
301C22⋊C4—122no—
302C22⋊C4—242no—
303C42—22no—
304C42—22no—
305C42—122no—
306C42—242no—
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#55112yes—
321C3×C4#55112yes—
322Dic3—42no—
323Dic3—42no—
324Dic3—82no—
325Dic3—81no—
326Dic3—82no—
327Dic3—81no—
328C23—22no—
329C23—22no—
330C23—21no—
331C23—21no—
332C23#88612yes—
333C23C22×D1211yescharacteristic
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—62no—
350C2×C4—62no—
351C2×C4—128no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—122no—
366C2×C4#88612yes—
367C2×C4C2×#23712yes—
368C2×C4D3×#4111yescharacteristic
369C2×C4D3×#4111yescharacteristic
370C2×C4#100912yes—
371C8—42no—
372C8—42no—
373C2×C3—42no—
374C2×C3#248211yescharacteristic
375C2×C3#309712yes—
376C2×C3#312112yes—
377C2×C3#322111yescharacteristic
378C2×C3#322311yescharacteristic
379C22—24no—
380C22—22no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22C2×#88611yescharacteristic
385C22#552411yescharacteristic
386C22#552511yescharacteristic
387C22#575112yes—
388C22#575312yes—
389C4—22no—
390C4—42no—
391C4—42no—
392C4—122no—
393C4—122no—
394C4—242no—
395C4—242no—
396C4—241no—
397C4—241no—
398C4#565812yes—
399C4#565812yes—
400C3#2759611yescharacteristic, complemented, Sylow
401C2—42no—
402C2#7412911yescharacteristic
403C2#7865212yes—
404C2#7939412yes—
405C2#7939711yescharacteristic
406C2#7939811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1818C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1818D3
hypocenterC3#27596
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
6. #68692C2semidirect
7. #68691C2semidirect
8. #68638C2semidirect
9. #68638C2semidirect
10. #68628C2semidirect
11. #68627C2semidirect
400. C3#27596semidirect

Maximal subgroups

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

subgroupindex
#822312
#822302
C3×#18182
#699632
#699292
#686922
#686912
#686382
#686282
#686272
#275963