Finite Group Databasebeta

← #10690177

Subgroups — #10690177

Order 768.

908 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#393C211yesmaximal, characteristic
2#82094C211yesmaximal, characteristic
3#72132C212yesmaximal, complemented
4C2×#4525C211yesmaximal, characteristic
5#67612C212yesmaximal, complemented
6#9013—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C22×C3×C22⋊C4—21no—
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×#217—21no—
11C3×#393—22no—
12C3×#393C414yescomplemented
13C2×C3×#95—21no—
14C2×C3×#95C2211yescharacteristic
15C2×#835—24no—
16#5212—21no—
17#5212C2212yes—
18C2×#830C2212yes—
19#4950—24no—
20#4525—22no—
21C2×#393D311yescharacteristic
22#2315—31no—
23#1787—32no—
24C2×#323—31no—
25#1294—32no—
26C25×C3D411yescharacteristic
27C22×C3×Q8—21no—
28C22×C3×Q8—22no—
29C22×C3×Q8—22no—
30C22×C3×Q8—44no—
31C22×C3×Q8C2×C412yescomplemented
32C23×C3×C4—21no—
33C23×C3×C4—21no—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C23×Dic3—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4—44no—
41C2×C3×C22⋊C4C2×C412yes—
42C2×C3×C22⋊C4D412yes—
43C2×#217—22no—
44C2×#217—22no—
45C2×#217—42no—
46C2×#217—42no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3D414yes—
49C22×C3⋊C8—22no—
50C3×#95—24no—
51C3×#95—21no—
52C3×#95C2×C412yescomplemented
53#835—48no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56#830—22no—
57#830—22no—
58C23×Q8C2×D311yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—61no—
61C22×C22⋊C4C2×D311yescharacteristic
62C2×#103—64no—
63C2×C22⋊C8—32no—
64#393—22no—
65#393Dic314yescomplemented
66#385—64no—
67#380—32no—
68#380—61no—
69C2×#95—21no—
70C2×#95C2×D311yescharacteristic
71#323—62no—
72C24×C3—21no—
73C24×C3—21no—
74C24×C3—21no—
75C24×C3—21no—
76C24×C3—42no—
77C24×C3—42no—
78C24×C3C22⋊C412yes—
79C24×C3SD1612yes—
80C24×C3Q1612yes—
81C24×C3C2×D411yescharacteristic
82C2×C3×Q8—28no—
83C2×C3×Q8—24no—
84C2×C3×Q8—44no—
85C2×C3×Q8—44no—
86C2×C3×Q8—44no—
87C2×C3×Q8—48no—
88C2×C3×Q8—42no—
89C22×C3×C4—24no—
90C22×C3×C4—22no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4C22⋊C412yes—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4112yes—
108C22×Dic3—24no—
109C22×Dic3—44no—
110C22×Dic3—82no—
111C22×Dic3—82no—
112C3×C22⋊C4—24no—
113C3×C22⋊C4—24no—
114C3×C22⋊C4—48no—
115#217—44no—
116#217—42no—
117#217—42no—
118C4⋊Dic3—44no—
119C4⋊Dic3—82no—
120C2×C3⋊C8—44no—
121C2×C3⋊C8—42no—
122C25#6711yescharacteristic
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—44no—
127C22×Q8C2×Dic312yescomplemented
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—62no—
131C23×C4#6711yescharacteristic
132C23×C4C22×D311yescharacteristic
133C22×C8—62no—
134C2×C4⋊C4—34no—
135C2×C4⋊C4—62no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4—122no—
143C2×C22⋊C4—122no—
144C2×C22⋊C4C2×Dic312yes—
145C2×C22⋊C4#6712yes—
146#103—128no—
147C22⋊C8—62no—
148C22⋊C8—62no—
149#95—24no—
150#95—21no—
151#95—62no—
152#95—62no—
153#95C2×Dic312yescomplemented
154C23×C3—22no—
155C23×C3—22no—
156C23×C3—22no—
157C23×C3—22no—
158C23×C3—22no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—42no—
163C23×C3—42no—
164C23×C3—44no—
165C23×C3—44no—
166C23×C3—42no—
167C23×C3—42no—
168C23×C3—82no—
169C23×C3—82no—
170C23×C3—81no—
171C23×C3—81no—
172C23×C3#10314yescomplemented
173C23×C3C2×C22⋊C411yescharacteristic
174C23×C3#12011yescharacteristic
175C23×C3#12311yescharacteristic
176C23×C3C2×SD1611yescharacteristic
177C23×C3C2×Q1611yescharacteristic
178C23×C3AGL(1,ℤ/8)11yescharacteristic
179C23×C3#13711yescharacteristic
180C3×Q8—28no—
181C3×Q8—48no—
182C3×Q8—44no—
183C2×C3×C4—24no—
184C2×C3×C4—24no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—44no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—44no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—81no—
199C2×C3×C4—82no—
200C2×C3×C4—82no—
201C2×C3×C4C4×D412yes—
202C2×C3×C4#12112yes—
203C2×Dic3—44no—
204C2×Dic3—44no—
205C2×Dic3—44no—
206C2×Dic3—88no—
207C3⋊C8—82no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—42no—
213C24—42no—
214C24#21512yes—
215C24#21612yes—
216C24#21712yes—
217C24C2×#6711yescharacteristic
218C2×Q8—28no—
219C2×Q8—24no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—64no—
240C22×C4—124no—
241C22×C4—242no—
242C22×C4—242no—
243C22×C4#21712yes—
244C22×C4D3×D412yes—
245C22×C4#23712yes—
246C22×C4C22×Dic311yescharacteristic
247C22×C4C2×#6711yescharacteristic
248C2×C8—124no—
249C2×C8—122no—
250C4⋊C4—124no—
251C4⋊C4—242no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—48no—
255C22⋊C4—124no—
256C22⋊C4—122no—
257C22⋊C4—122no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—24no—
267C22×C3—44no—
268C22×C3—44no—
269C22×C3—44no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3—82no—
273C22×C3—82no—
274C22×C3—82no—
275C22×C3#38611yescharacteristic
276C22×C3C2×#10311yescharacteristic
277C22×C3#41711yescharacteristic
278C22×C3#45011yescharacteristic
279C22×C3#45111yescharacteristic
280C22×C3#45512yes—
281C22×C3B(4,𝔽2)12yes—
282C22×C3#48311yescharacteristic
283C22×C3#48411yescharacteristic
284C3×C4—22no—
285C3×C4—22no—
286C3×C4—42no—
287C3×C4—44no—
288C3×C4—82no—
289Dic3—84no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—42no—
299C23—44no—
300C23—44no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—81no—
305C23—82no—
306C23—82no—
307C23—81no—
308C23#83514yescomplemented
309C23#93511yescharacteristic
310C23#93711yescharacteristic
311C23C2×#21511yescharacteristic
312C23C2×#21611yescharacteristic
313C23#94911yescharacteristic
314C23#95111yescharacteristic
315C23C2×#21711yescharacteristic
316Q8—28no—
317Q8—48no—
318Q8—44no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—124no—
338C2×C4—124no—
339C2×C4—124no—
340C2×C4—248no—
341C2×C4D4×Dic312yes—
342C2×C4#93912yes—
343C8—242no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—44no—
349C2×C3#160112yes—
350C2×C3#160212yes—
351C2×C3#187211yescharacteristic
352C2×C3#188511yescharacteristic
353C2×C3#188911yescharacteristic
354C22—21no—
355C22—21no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—21no—
361C22—21no—
362C22—24no—
363C22—44no—
364C22—42no—
365C22—44no—
366C22—44no—
367C22—42no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22#503212yes—
372C22#504311yescharacteristic
373C22#504411yescharacteristic
374C22#504611yescharacteristic
375C22#505011yescharacteristic
376C22#506112yes—
377C22#522011yescharacteristic
378C22C2×#83511yescharacteristic
379C22#523211yescharacteristic
380C4—22no—
381C4—22no—
382C4—42no—
383C4—44no—
384C4—82no—
385C4—244no—
386C3#901311yescharacteristic, complemented, Sylow
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—44no—
392C2#6969912yes—
393C2#6970112yes—
394C2#7166911yescharacteristic
395C2#7169211yescharacteristic
396C2#7172311yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#393C2
socleC23×C3#120
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenterC2×#393D3
hypocenterC3#9013
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
3. #72132C2semidirect
5. #67612C2semidirect
12. C3×#393C4semidirect
31. C22×C3×Q8C2×C4semidirect
52. C3×#95C2×C4semidirect
65. #393Dic3semidirect
127. C22×Q8C2×Dic3semidirect
153. #95C2×Dic3semidirect
172. C23×C3#103semidirect
308. C23#835semidirect
386. C3#9013semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3932
#820942
#721322
C2×#45252
#676122
#90133