Finite Group Databasebeta

← #10691151

Subgroups — #10691151

Order 768.

532 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2638C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3#71688C212yesmaximal, complemented
4C4×#834C211yesmaximal, characteristic
5#67628C212yesmaximal, complemented
6#9183—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#398C411yescharacteristic, complemented
10C3×#397—21no—
11C3×#397C411yescharacteristic, complemented
12C3×#395C411yescharacteristic, complemented
13C3×#394—21no—
14C3×#393C411yescharacteristic, complemented
15C3×#387—21no—
16C3×#385—21no—
17C3×#381C2211yescharacteristic
18C2×#834—21no—
19C4×#217C2212yes—
20#5207C2212yes—
21C2×#830—21no—
22#5011—24no—
23#4926—21no—
24#4538—24no—
25#4526—21no—
26#2973—31no—
27#2638D311yescharacteristic
28C4×#99—31no—
29#1508—32no—
30#1284—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×D4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C3×C22⋊Q8—22no—
36C3×C22⋊Q8—22no—
37C3×C4×D4—22no—
38C3×C4×D4—22no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×#217—22no—
52C2×C4×Dic3—22no—
53C2×#208—22no—
54C22×C3⋊C8—22no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95C2×C411yescharacteristic
61C3×#95C2×C411yescharacteristic
62#834—24no—
63C3⋊#95—22no—
64#830—22no—
65#830—22no—
66#830D414yes—
67C4⋊(C3⋊C8)—22no—
68#803—22no—
69C2×C22⋊Q8C2×D311yescharacteristic
70C2×C4×D4C2×D311yescharacteristic
71#435—64no—
72#432—61no—
73C2×#99—61no—
74#407—32no—
75C2×C22⋊C8—61no—
76#398Dic311yescharacteristic, complemented
77#397—21no—
78#397Dic311yescharacteristic, complemented
79#395Dic311yescharacteristic, complemented
80#394—21no—
81#393Dic311yescharacteristic, complemented
82#387—21no—
83#385—21no—
84#381C2×D311yescharacteristic
85C4×C22⋊C4—32no—
86#342—61no—
87#328—64no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C2×D411yescharacteristic
90C2×C3×Q8—22no—
91C2×C3×Q8—81no—
92C2×C3×D4—22no—
93C2×C3×D4—22no—
94C2×C3×D4—41no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—81no—
102C22×C3×C4—81no—
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D412yes—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4112yes—
110C22×Dic3—42no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—42no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C4⋊C4—81no—
117C3×C22⋊C4—22no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—42no—
120C3×C22⋊C4—42no—
121C3×C22⋊C4—81no—
122C3×C22⋊C4—81no—
123C3×C42—21no—
124C3×C42—21no—
125#217—24no—
126#217—42no—
127C4×Dic3—82no—
128#208—82no—
129C2×C3⋊C8—24no—
130C2×C3⋊C8—82no—
131C22×Q8C2×Dic311yescharacteristic
132C22×D4#6711yescharacteristic
133C23×C4#6711yescharacteristic
134C23×C4C22×D311yescharacteristic
135C2×M16—62no—
136C22×C8—62no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C4×D4—22no—
140C4×D4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4C2×Dic311yescharacteristic
145C2×C4⋊C4C2×Dic311yescharacteristic
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—62no—
151C2×C22⋊C4C2×Dic311yescharacteristic
152C2×C22⋊C4#6711yescharacteristic
153C2×C42—62no—
154C2×C42#6711yescharacteristic
155C4⋊C8—62no—
156#99—64no—
157C22⋊C8—34no—
158C22⋊C8—62no—
159C22⋊C8—62no—
160#97—62no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166#95—62no—
167#95C2×Dic311yescharacteristic
168#95C2×Dic311yescharacteristic
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3#10412yes—
178C23×C3C2×C22⋊C411yescharacteristic
179C23×C3#12112yes—
180C3×Q8—82no—
181C3×D4—44no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—81no—
207C2×C3×C4C4×D412yes—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12311yescharacteristic
210C2×Dic3—44no—
211C2×Dic3—82no—
212C2×Dic3—82no—
213C3⋊C8—44no—
214C24#21711yescharacteristic
215C24C2×#6711yescharacteristic
216C2×Q8—22no—
217C2×Q8—81no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—122no—
230C22×C4#21711yescharacteristic
231C22×C4#21711yescharacteristic
232C22×C4#21711yescharacteristic
233C22×C4D3×D412yes—
234C22×C4#23712yes—
235C22×C4C22×Dic311yescharacteristic
236C22×C4C2×#6711yescharacteristic
237M16—242no—
238C2×C8—64no—
239C2×C8—242no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—81no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—64no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C22⋊C4—122no—
254C42—21no—
255C42—21no—
256C42—242no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—42no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—81no—
267C22×C3—81no—
268C22×C3—81no—
269C22×C3#38611yescharacteristic
270C22×C3C2×#10411yescharacteristic
271C22×C3#42111yescharacteristic
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—22no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—81no—
279C3×C4—81no—
280C3×C4—81no—
281C3×C4—81no—
282Dic3—84no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23#83712yes—
292C23#93912yes—
293C23C2×#21711yescharacteristic
294Q8—82no—
295D4—44no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—124no—
322C2×C4—242no—
323C2×C4—242no—
324C2×C4D4×Dic312yes—
325C2×C4#93511yescharacteristic
326C2×C4#93711yescharacteristic
327C8—124no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—42no—
331C2×C3—42no—
332C2×C3#187111yescharacteristic
333C2×C3#190911yescharacteristic
334C2×C3#191311yescharacteristic
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22#522011yescharacteristic
348C22C2×#83711yescharacteristic
349C22#523611yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—244no—
361C3#918311yescharacteristic, complemented, Sylow
362C2—21no—
363C2—21no—
364C2—42no—
365C2—42no—
366C2#7171311yescharacteristic
367C2#7184811yescharacteristic
368C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2638C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2638D3
hypocenterC3#9183
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #71688C2semidirect
5. #67628C2semidirect
9. C3×#398C4semidirect
11. C3×#397C4semidirect
12. C3×#395C4semidirect
14. C3×#393C4semidirect
76. #398Dic3semidirect
78. #397Dic3semidirect
79. #395Dic3semidirect
81. #393Dic3semidirect
361. C3#9183semidirect

Maximal subgroups

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

subgroupindex
C3×#26382
#823022
#716882
C4×#8342
#676282
#91833