Finite Group Databasebeta

← #10685708

Subgroups — #10685708

Order 768.

602 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4933C211yesmaximal, characteristic
2C3×#1749C211yesmaximal, characteristic
3C4×#830C211yesmaximal, characteristic
4C2×#4469C212yesmaximal
5#67674C212yesmaximal
6#8154—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C22×Dic3⋊C4C411yescharacteristic
9C22×Dic3⋊C4C2211yescharacteristic
10C2×C3×C22⋊C8C2212yes—
11C2×C3×#97—21no—
12C2×C3⋊#95C411yescharacteristic
13C2×C3⋊#95C411yescharacteristic
14C2×C3⋊#95C411yescharacteristic
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#4933—21no—
18#4933—21no—
19#4933—21no—
20#4933—21no—
21C2×C4×C3⋊C8—21no—
22C3×#336—21no—
23#4524—21no—
24#4469C418yescomplemented
25C2×#387—31no—
26#1749D311yescharacteristic
27C4×C22⋊C8—31no—
28C2×#324—32no—
29#1280—32no—
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C23×Dic3C2×C411yescharacteristic
34C23×Dic3C2×C411yescharacteristic
35C23×Dic3C2×C411yescharacteristic
36C23×Dic3C2×C411yescharacteristic
37C22×C3×C8—22no—
38C2×C3×C42—42no—
39C2×C3×C42D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—22no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×Dic3⋊C4—42no—
49C2×Dic3⋊C4C2×C414yes—
50C22×C3⋊C8—22no—
51C3×C22⋊C8—22no—
52C3×C22⋊C8C2×C414yes—
53C3×#97—42no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62#830—22no—
63#830C2×C414yes—
64C4×C3⋊C8—42no—
65C22×C4⋊C4—31no—
66C22×C4⋊C4—31no—
67C22×C42C2×D311yescharacteristic
68C2×C22⋊C8—32no—
69C2×C22⋊C8C2×D312yes—
70C2×#97—21no—
71C2×C4×C8—61no—
72#387—61no—
73#387—61no—
74#387—61no—
75#387—61no—
76C2×#95—31no—
77C2×#95—31no—
78C2×#95—31no—
79C2×#95—31no—
80#336—21no—
81#336—61no—
82#324—38no—
83C24×C3C22×C411yescharacteristic
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22×C412yes—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4112yes—
100C22×Dic3—21no—
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3—21no—
104C22×Dic3—22no—
105C22×Dic3—22no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—22no—
109C22×Dic3—22no—
110C22×Dic3—41no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113C22×Dic3—41no—
114C22×Dic3—41no—
115C22×Dic3—41no—
116C22×Dic3C4212yes—
117C22×Dic3C4212yes—
118C2×C3×C8—24no—
119C2×C3×C8—42no—
120C2×C3×C8—42no—
121C3×C42—22no—
122C3×C42—22no—
123C3×C42—41no—
124C3×C42—41no—
125Dic3⋊C4—24no—
126Dic3⋊C4—24no—
127Dic3⋊C4—42no—
128Dic3⋊C4—42no—
129C2×C3⋊C8—24no—
130C2×C3⋊C8—42no—
131C2×C3⋊C8—42no—
132C23×C4—31no—
133C23×C4—31no—
134C23×C4—31no—
135C23×C4—31no—
136C23×C4C4×D311yescharacteristic
137C23×C4C4×D311yescharacteristic
138C23×C4C22×D311yescharacteristic
139C22×C8—22no—
140C22×C8—62no—
141C2×C4⋊C4—34no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—122no—
148C2×C42—42no—
149C2×C42D1211yescharacteristic
150C2×C42D1211yescharacteristic
151C2×C42#6711yescharacteristic
152C2×C42#6711yescharacteristic
153C22⋊C8—22no—
154C22⋊C8—34no—
155C22⋊C8—62no—
156C22⋊C8C4×D314yes—
157#97—42no—
158C4×C8—122no—
159#95—61no—
160#95—61no—
161#95—61no—
162#95—61no—
163#95—61no—
164#95—61no—
165#95—61no—
166#95—61no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3C2×C4211yescharacteristic
172C23×C3C2×C22⋊C411yescharacteristic
173C23×C3#11711yescharacteristic
174C23×C3#13112yes—
175C23×C3#13112yes—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
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—42no—
191C2×C3×C4—42no—
192C2×C3×C4—44no—
193C2×C3×C4—44no—
194C2×C3×C4#9912yes—
195C2×C3×C4#10112yes—
196C2×C3×C4C4×D412yes—
197C2×C3×C4C4×D412yes—
198C2×Dic3—22no—
199C2×Dic3—22no—
200C2×Dic3—24no—
201C2×Dic3—42no—
202C2×Dic3—41no—
203C2×Dic3—41no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—82no—
215C2×Dic3—82no—
216C3×C8—44no—
217C3⋊C8—44no—
218C24C2×C4×D311yescharacteristic
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—32no—
224C22×C4—32no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—62no—
232C22×C4—62no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—121no—
240C22×C4—121no—
241C22×C4—121no—
242C22×C4—121no—
243C22×C4—121no—
244C22×C4—121no—
245C22×C4(C2×D3)⋊C411yescharacteristic
246C22×C4(C2×D3)⋊C411yescharacteristic
247C22×C4(C2×D3)⋊C411yescharacteristic
248C22×C4(C2×D3)⋊C411yescharacteristic
249C22×C4C2×C4×D312yes—
250C22×C4C2×D1211yescharacteristic
251C22×C4#23512yes—
252C22×C4C2×#6711yescharacteristic
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—64no—
257C2×C8—122no—
258C2×C8—122no—
259C4⋊C4—64no—
260C4⋊C4—64no—
261C4⋊C4—122no—
262C4⋊C4—122no—
263C42—22no—
264C42—22no—
265C42—41no—
266C42—41no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3C4×C22⋊C411yescharacteristic
276C22×C3#40512yes—
277C22×C3#40811yescharacteristic
278C22×C3#40811yescharacteristic
279C22×C3C2×#9911yescharacteristic
280C22×C3#41011yescharacteristic
281C22×C3C2×#10111yescharacteristic
282C22×C3#41311yescharacteristic
283C22×C3#43312yes—
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—41no—
289C3×C4—41no—
290Dic3—42no—
291Dic3—42no—
292Dic3—81no—
293Dic3—81no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C42×D311yescharacteristic
299C23#87211yescharacteristic
300C23#90712yes—
301C23#90712yes—
302C23C2×(C2×D3)⋊C411yescharacteristic
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—62no—
322C2×C4—62no—
323C2×C4—64no—
324C2×C4—122no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4—121no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—121no—
333C2×C4—121no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—242no—
338C2×C4—242no—
339C2×C4#80612yes—
340C2×C4#82412yes—
341C2×C4C4×D1212yes—
342C2×C4C4×#6712yes—
343C8—44no—
344C8—124no—
345C2×C3—22no—
346C2×C3—22no—
347C2×C3#146811yescharacteristic
348C2×C3#147011yescharacteristic
349C2×C3#147211yescharacteristic
350C2×C3#147311yescharacteristic
351C2×C3#175011yescharacteristic
352C2×C3C4×#9911yescharacteristic
353C2×C3C4×#10111yescharacteristic
354C22—21no—
355C22—21no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—21no—
361C22—21no—
362C22#470812yes—
363C22#470912yes—
364C22C4×(C2×D3)⋊C411yescharacteristic
365C22C2×#80611yescharacteristic
366C22#500611yescharacteristic
367C22#512911yescharacteristic
368C22#512911yescharacteristic
369C22#513211yescharacteristic
370C22C2×#82411yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—122no—
378C4—122no—
379C4—241no—
380C4—241no—
381C3#815411yescharacteristic, complemented, Sylow
382C2—22no—
383C2—22no—
384C2C4×#80611yescharacteristic
385C2#6983011yescharacteristic
386C2C4×#82411yescharacteristic
387C2#6996911yescharacteristic
388C2#6997211yescharacteristic
389C2#7082911yescharacteristic
390C2#7083111yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC23×C3C2×C42
FrattiniC23×C4C22×D3
FittingC3×#1749C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C22×C4
cosocleC23×C3×C4C23
hypercenter#1749D3
hypocenterC3#8154
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
24. #4469C4semidirect
381. C3#8154semidirect

Maximal subgroups

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

subgroupindex
C2×#49332
C3×#17492
C4×#8302
C2×#44692
#676742
#81543