Finite Group Databasebeta

← #10690745

Subgroups — #10690745

Order 768.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82075C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic
3C3×#1434C211yesmaximal, characteristic
4C2×#4452C212yesmaximal
5#67612C212yesmaximal
6#9112—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#924C2211yescharacteristic
9C2×C3×#95—21no—
10C2×C3×#95—21no—
11C2×C3×#95C2212yes—
12#5212C2211yescharacteristic
13#5211—21no—
14#5210—21no—
15C2×#830C2212yes—
16#4933—21no—
17#4933—21no—
18#4929C411yescharacteristic, complemented
19#4929C411yescharacteristic, complemented
20#4928C411yescharacteristic, complemented
21#4928C411yescharacteristic, complemented
22#4925—24no—
23C2×C4⋊(C3⋊C8)—21no—
24C2×C4×C3⋊C8—21no—
25#4452—24no—
26C2×#431—31no—
27#2855—31no—
28C2×#328—32no—
29#1434D311yescharacteristic
30#1294—32no—
31C23×C3×C4—22no—
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C22×Dic6C2×C411yescharacteristic
36C2×C3×C42—22no—
37C2×C3×C42—22no—
38C2×C3×C42C2×C411yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×#217—21no—
43C2×#217—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47#924—22no—
48#924—22no—
49C2×Dic3⋊C4—21no—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C22×C3⋊C8—22no—
53C3×#95—24no—
54C3×#95—22no—
55C3×#95—22no—
56C3×#95—42no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64#830—22no—
65#830D414yes—
66C4⋊(C3⋊C8)—24no—
67C4×C3⋊C8—24no—
68C2×C22⋊Q8—31no—
69C22×C42C2×D311yescharacteristic
70#431—64no—
71C2×C4⋊C8—61no—
72C2×C22⋊C8—32no—
73C2×C4×C8—61no—
74#391—31no—
75#391—31no—
76#389—31no—
77#389—31no—
78#387—61no—
79#387—61no—
80#385—61no—
81#381—61no—
82#380—31no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95C2×D312yes—
86#328—64no—
87C24×C3C2×D411yescharacteristic
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—24no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4—42no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4SD1612yes—
107C22×C3×C4Q1612yes—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D412yes—
111C22×C3×C4#4112yes—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—21no—
116C2×Dic6—22no—
117C2×Dic6—41no—
118C2×Dic6—41no—
119C3×C42—22no—
120C3×C42—22no—
121C3×C42—21no—
122C3×C42—21no—
123C3×C42—21no—
124C3×C42—21no—
125#217—42no—
126#217—42no—
127C4⋊Dic3—22no—
128C4⋊Dic3—41no—
129C4⋊Dic3—41no—
130Dic3⋊C4—42no—
131Dic3⋊C4—42no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134C2×C3⋊C8—24no—
135C2×C3⋊C8—42no—
136C2×C3⋊C8—42no—
137C22×Q8—31no—
138C23×C4—22no—
139C23×C4#6711yescharacteristic
140C23×C4#6711yescharacteristic
141C23×C4C22×D311yescharacteristic
142C22×C8—62no—
143C22⋊Q8—62no—
144C22⋊Q8—62no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C22⋊C4—61no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42C4×D311yescharacteristic
156C2×C42C4×D311yescharacteristic
157C2×C42D1211yescharacteristic
158C2×C42D1211yescharacteristic
159C4⋊C8—64no—
160C22⋊C8—34no—
161C22⋊C8—62no—
162C4×C8—64no—
163#95—24no—
164#95—22no—
165#95—22no—
166#95—42no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—61no—
173#95—61no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3C2×C22⋊C411yescharacteristic
179C23×C3#12112yes—
180C23×C3C2×SD1611yescharacteristic
181C23×C3C2×Q1611yescharacteristic
182C23×C3#13512yes—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—44no—
201C2×C3×C4—44no—
202C2×C3×C4—44no—
203C2×C3×C4—44no—
204C2×C3×C4—82no—
205C2×C3×C4—82no—
206C2×C3×C4—82no—
207C2×C3×C4—82no—
208C2×C3×C4#10312yes—
209C2×C3×C4#10312yes—
210C2×C3×C4#10412yes—
211C2×C3×C4C4×D412yes—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226Dic6—82no—
227C3⋊C8—44no—
228C24C2×#6711yescharacteristic
229C2×Q8—62no—
230C2×Q8—121no—
231C2×Q8—121no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4(C2×D3)⋊C411yescharacteristic
251C22×C4(C2×D3)⋊C411yescharacteristic
252C22×C4(C2×D3)⋊C411yescharacteristic
253C22×C4(C2×D3)⋊C411yescharacteristic
254C22×C4#21412yes—
255C22×C4#21612yes—
256C22×C4C2×C4×D311yescharacteristic
257C22×C4C2×D1211yescharacteristic
258C22×C4D3×D412yes—
259C22×C4#23712yes—
260C2×C8—64no—
261C2×C8—122no—
262C2×C8—122no—
263C4⋊C4—62no—
264C4⋊C4—121no—
265C4⋊C4—121no—
266C4⋊C4—122no—
267C4⋊C4—122no—
268C4⋊C4—121no—
269C4⋊C4—121no—
270C22⋊C4—122no—
271C22⋊C4—122no—
272C42—22no—
273C42—22no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3#38611yescharacteristic
287C22×C3#41011yescharacteristic
288C22×C3#41011yescharacteristic
289C22×C3C2×#10311yescharacteristic
290C22×C3#41611yescharacteristic
291C22×C3C2×#10411yescharacteristic
292C22×C3#42111yescharacteristic
293C22×C3#46512yes—
294C22×C3#46712yes—
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—84no—
303Dic3—81no—
304Dic3—81no—
305Dic3—81no—
306Dic3—81no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23C2×(C2×D3)⋊C411yescharacteristic
312C23C2×#21411yescharacteristic
313C23#93912yes—
314C23C2×#21611yescharacteristic
315C23#95012yes—
316Q8—242no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4#80512yes—
355C2×C4#81012yes—
356C2×C4#81012yes—
357C2×C4#67⋊C412yes—
358C2×C4#88211yescharacteristic
359C2×C4#88611yescharacteristic
360C8—124no—
361C2×C3—22no—
362C2×C3—22no—
363C2×C3#150011yescharacteristic
364C2×C3#150311yescharacteristic
365C2×C3#151811yescharacteristic
366C2×C3#152111yescharacteristic
367C2×C3#187111yescharacteristic
368C2×C3#189211yescharacteristic
369C2×C3#189911yescharacteristic
370C22—22no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22C2×#80511yescharacteristic
379C22#495811yescharacteristic
380C22C2×#81011yescharacteristic
381C22#500211yescharacteristic
382C22#500511yescharacteristic
383C22#500611yescharacteristic
384C22#500611yescharacteristic
385C22#504112yes—
386C22#505112yes—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—21no—
391C4—22no—
392C4—21no—
393C4—41no—
394C4—84no—
395C4—241no—
396C4—241no—
397C4—241no—
398C4—241no—
399C3#911211yescharacteristic, complemented, Sylow
400C2—22no—
401C2—22no—
402C2#6943111yescharacteristic
403C2#6945311yescharacteristic
404C2#6947111yescharacteristic
405C2#6951411yescharacteristic
406C2#6953511yescharacteristic
407C2#6954611yescharacteristic
408C2#6956511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1434C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1434D3
hypocenterC3#9112
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
18. #4929C4semidirect
19. #4929C4semidirect
20. #4928C4semidirect
21. #4928C4semidirect
399. C3#9112semidirect

Maximal subgroups

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

subgroupindex
#820752
C2×#49252
C3×#14342
C2×#44522
#676122
#91123