Finite Group Databasebeta

← #10692513

Subgroups — #10692513

Order 768.

548 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2564C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3#71690C212yesmaximal, complemented
4#71619C211yesmaximal, characteristic
5#67612C212yesmaximal, complemented
6#9385—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#400C411yescharacteristic, complemented
10C3×#399C411yescharacteristic, complemented
11C3×#395C411yescharacteristic, complemented
12C3×#393C411yescharacteristic, complemented
13C3×#387—21no—
14C3×#386—21no—
15C3×#385—21no—
16C3×#380—21no—
17C2×C3×#95C2211yescharacteristic
18C2×#834—21no—
19#5212C2212yes—
20#5207—21no—
21C2×#830C2212yes—
22#5011—24no—
23#4926—21no—
24#4538—24no—
25#4526—21no—
26#2973—31no—
27#2564D311yescharacteristic
28#1799—31no—
29#1520—32no—
30#1294—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×D4D411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C3×C4×D4—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
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—42no—
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—61no—
75C2×C22⋊C8—32no—
76#400Dic311yescharacteristic, complemented
77#399Dic311yescharacteristic, complemented
78#395Dic311yescharacteristic, complemented
79#393Dic311yescharacteristic, complemented
80#387—21no—
81#386—21no—
82#385—21no—
83#380—21no—
84#380—32no—
85C2×#95C2×D311yescharacteristic
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—21no—
94C2×C3×D4—22no—
95C2×C3×D4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—42no—
103C22×C3×C4—41no—
104C22×C3×C4—41no—
105C22×C3×C4—81no—
106C22×C3×C4—81no—
107C22×C3×C4—81no—
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C22⋊C411yescharacteristic
110C22×C3×C4C22⋊C411yescharacteristic
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4112yes—
115C22×Dic3—42no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—42no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—81no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—42no—
125C3×C22⋊C4—42no—
126C3×C22⋊C4—81no—
127C3×C42—21no—
128C3×C42—21no—
129#217—42no—
130#217—42no—
131C4⋊Dic3—82no—
132#208—82no—
133C2×C3⋊C8—24no—
134C2×C3⋊C8—82no—
135C22×Q8C2×Dic311yescharacteristic
136C22×D4#6711yescharacteristic
137C23×C4—21no—
138C23×C4#6711yescharacteristic
139C23×C4C22×D311yescharacteristic
140C2×M16—62no—
141C22×C8—62no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C4×D4—22no—
145C4×D4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—62no—
150C2×C4⋊C4C2×Dic311yescharacteristic
151C2×C4⋊C4C2×Dic311yescharacteristic
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—62no—
156C2×C22⋊C4C2×Dic311yescharacteristic
157C2×C22⋊C4#6711yescharacteristic
158C2×C42#6711yescharacteristic
159C4⋊C8—62no—
160#99—122no—
161C22⋊C8—34no—
162C22⋊C8—62no—
163C22⋊C8—62no—
164#97—62no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—62no—
171#95C2×Dic311yescharacteristic
172#95C2×Dic311yescharacteristic
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—41no—
180C23×C3C2×C22⋊C411yescharacteristic
181C23×C3#12112yes—
182C23×C3AGL(1,ℤ/8)11yescharacteristic
183C23×C3#13712yes—
184C23×C3#13711yescharacteristic
185C3×Q8—82no—
186C3×D4—42no—
187C3×D4—42no—
188C2×C3×C4—22no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
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—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—81no—
213C2×C3×C4—82no—
214C2×C3×C4—82no—
215C2×C3×C4C4×D412yes—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12311yescharacteristic
218C2×Dic3—82no—
219C2×Dic3—82no—
220C2×Dic3—82no—
221C3⋊C8—44no—
222C24#21711yescharacteristic
223C24C2×#6711yescharacteristic
224C2×Q8—22no—
225C2×Q8—81no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—21no—
229C2×D4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4—122no—
243C22×C4#21711yescharacteristic
244C22×C4#21711yescharacteristic
245C22×C4#21711yescharacteristic
246C22×C4D3×D412yes—
247C22×C4#23712yes—
248C22×C4C22×Dic311yescharacteristic
249C22×C4C2×#6711yescharacteristic
250M16—242no—
251C2×C8—64no—
252C2×C8—242no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—81no—
259C4⋊C4—242no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—81no—
265C22⋊C4—122no—
266C22⋊C4—122no—
267C42—21no—
268C42—21no—
269C22×C3—22no—
270C22×C3—21no—
271C22×C3—22no—
272C22×C3—21no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—42no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—81no—
281C22×C3#38611yescharacteristic
282C22×C3#41711yescharacteristic
283C22×C3#41911yescharacteristic
284C22×C3#46812yes—
285C22×C3#47012yes—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—22no—
289C3×C4—42no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—81no—
294C3×C4—81no—
295C3×C4—81no—
296Dic3—162no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#93912yes—
305C23#94212yes—
306C23#94911yescharacteristic
307C23#95111yescharacteristic
308C23C2×#21711yescharacteristic
309Q8—82no—
310D4—42no—
311D4—42no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—242no—
340C2×C4—242no—
341C2×C4—242no—
342C2×C4D4×Dic312yes—
343C2×C4#93511yescharacteristic
344C2×C4#93711yescharacteristic
345C8—124no—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—22no—
350C2×C3—81no—
351C2×C3#188411yescharacteristic
352C2×C3#189411yescharacteristic
353C2×C3#191311yescharacteristic
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#504912yes—
367C22#505212yes—
368C22#522011yescharacteristic
369C22#522511yescharacteristic
370C22#523211yescharacteristic
371C4—21no—
372C4—21no—
373C4—22no—
374C4—42no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—482no—
382C3#938511yescharacteristic, complemented, Sylow
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2#7169711yescharacteristic
389C2#7181711yescharacteristic
390C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2564C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2564D3
hypocenterC3#9385
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #71690C2semidirect
5. #67612C2semidirect
9. C3×#400C4semidirect
10. C3×#399C4semidirect
11. C3×#395C4semidirect
12. C3×#393C4semidirect
76. #400Dic3semidirect
77. #399Dic3semidirect
78. #395Dic3semidirect
79. #393Dic3semidirect
382. C3#9385semidirect

Maximal subgroups

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

subgroupindex
C3×#25642
#823022
#716902
#716192
#676122
#93853