Finite Group Databasebeta

← #10692512

Subgroups — #10692512

Order 768.

548 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2973C211yesmaximal, characteristic
2#82385C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#69946C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9385—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#924C2211yescharacteristic
9C3×#435—24no—
10C3×#432—21no—
11C3×#407—21no—
12C2×C3×C22⋊C8C2212yes—
13#5220—21no—
14C2×#834—21no—
15#5212C2212yes—
16#5210—21no—
17C2×C3⋊#95C2211yescharacteristic
18#4986C411yescharacteristic, complemented
19#4979C411yescharacteristic, complemented
20#4949C411yescharacteristic, complemented
21#4947C411yescharacteristic, complemented
22#4942—21no—
23#4933—21no—
24#4526—21no—
25#4467—24no—
26#2973D311yescharacteristic
27#2564—31no—
28#1799—31no—
29#1520—32no—
30#1294—32no—
31C22×C3×D4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C23×Dic3—21no—
35C22×Dic6C2×C411yescharacteristic
36C2×C3×M16—22no—
37C22×C3×C8—22no—
38C3×C4×D4—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4C2×C411yescharacteristic
42C2×C3×C22⋊C4D411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×#217—21no—
45C2×#217—21no—
46C2×#217—22no—
47C2×#217—21no—
48C2×C4⋊Dic3—22no—
49C2×C4⋊Dic3C2×C411yescharacteristic
50#924—22no—
51#924—22no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C3×C4⋊C8—22no—
56C3×C22⋊C8—22no—
57C3×C22⋊C8—22no—
58C3×C22⋊C8D414yes—
59C3×#97—22no—
60#834—42no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66C3⋊#95—22no—
67C3⋊#95C2×C411yescharacteristic
68C3⋊#95C2×C411yescharacteristic
69C2×C22⋊Q8—31no—
70C2×C4×D4C2×D311yescharacteristic
71#435—24no—
72#432—21no—
73C2×#99—61no—
74#407—21no—
75C2×C22⋊C8C2×D312yes—
76#400—31no—
77#399—31no—
78#395—31no—
79#393—31no—
80#387—61no—
81#386—61no—
82#385—61no—
83#380—32no—
84#380—61no—
85C2×#95—31no—
86#342—61no—
87#328—64no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C2×D411yescharacteristic
90C2×C3×D4—22no—
91C2×C3×D4—21no—
92C2×C3×D4—21no—
93C2×C3×D4—22no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—81no—
97C22×C3×C4—81no—
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4112yes—
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—42no—
110C22×Dic3—41no—
111C22×Dic3—41no—
112C22×Dic3—42no—
113C22×Dic3—81no—
114C2×Dic6—22no—
115C2×Dic6—81no—
116C3×M16—82no—
117C2×C3×C8—24no—
118C2×C3×C8—82no—
119C3×C4⋊C4—22no—
120C3×C22⋊C4—22no—
121C3×C22⋊C4—22no—
122C3×C42—21no—
123C3×C42—21no—
124#217—42no—
125#217—42no—
126#217—42no—
127#217—42no—
128#217—81no—
129C4⋊Dic3—22no—
130C4⋊Dic3—82no—
131Dic3⋊C4—42no—
132Dic3⋊C4—41no—
133Dic3⋊C4—41no—
134Dic3⋊C4—81no—
135C22×Q8—31no—
136C22×D4D1211yescharacteristic
137C23×C4—61no—
138C23×C4#6711yescharacteristic
139C23×C4C22×D311yescharacteristic
140C2×M16—22no—
141C22×C8—22no—
142C22⋊Q8—62no—
143C22⋊Q8—62no—
144C4×D4—22no—
145C4×D4—22no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—61no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—62no—
151C2×C4⋊C4C4×D311yescharacteristic
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—62no—
155C2×C22⋊C4—61no—
156C2×C22⋊C4C4×D311yescharacteristic
157C2×C22⋊C4D1211yescharacteristic
158C2×C42#6711yescharacteristic
159C4⋊C8—22no—
160#99—122no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8#6714yes—
164#97—22no—
165#95—31no—
166#95—31no—
167#95—61no—
168#95—61no—
169#95—61no—
170#95—61no—
171#95—61no—
172#95—62no—
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#13711yescharacteristic
184C23×C3#13712yes—
185C3×D4—42no—
186C3×D4—42no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—81no—
200C2×C3×C4C4×D412yes—
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12311yescharacteristic
203C2×Dic3—22no—
204C2×Dic3—41no—
205C2×Dic3—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—41no—
210C2×Dic3—41no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3—41no—
215C2×Dic3—82no—
216C2×Dic3—82no—
217C2×Dic3—82no—
218C2×Dic3—82no—
219C2×Dic3—82no—
220Dic6—82no—
221C3×C8—44no—
222C24(C2×D3)⋊C411yescharacteristic
223C24C2×D1211yescharacteristic
224C2×Q8—62no—
225C2×Q8—241no—
226C2×D4—21no—
227C2×D4—22no—
228C2×D4—21no—
229C2×D4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—122no—
239C22×C4—121no—
240C22×C4—121no—
241C22×C4—122no—
242C22×C4—241no—
243C22×C4(C2×D3)⋊C411yescharacteristic
244C22×C4(C2×D3)⋊C411yescharacteristic
245C22×C4(C2×D3)⋊C411yescharacteristic
246C22×C4C2×C4×D311yescharacteristic
247C22×C4#23512yes—
248C22×C4C2×#6712yes—
249C22×C4C2×#6711yescharacteristic
250M16—82no—
251C2×C8—24no—
252C2×C8—82no—
253C4⋊C4—22no—
254C4⋊C4—62no—
255C4⋊C4—122no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C4⋊C4—241no—
259C4⋊C4—242no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—122no—
263C22⋊C4—122no—
264C22⋊C4—122no—
265C22⋊C4—122no—
266C22⋊C4—241no—
267C42—21no—
268C42—21no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—21no—
272C22×C3—22no—
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—22no—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292Dic3—42no—
293Dic3—81no—
294Dic3—81no—
295Dic3—81no—
296Dic3—162no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#90811yescharacteristic
305C23#90912yes—
306C23#90911yescharacteristic
307C23C2×(C2×D3)⋊C411yescharacteristic
308C23#93012yes—
309Q8—242no—
310D4—42no—
311D4—42no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—62no—
325C2×C4—81no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—121no—
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—242no—
340C2×C4—242no—
341C2×C4—242no—
342C2×C4C4×#6712yes—
343C2×C4#92911yescharacteristic
344C2×C4#95311yescharacteristic
345C8—44no—
346C2×C3—21no—
347C2×C3—22no—
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#512011yescharacteristic
367C22#513311yescharacteristic
368C22#513412yes—
369C22#513712yes—
370C22#521311yescharacteristic
371C4—21no—
372C4—21no—
373C4—22no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—122no—
378C4—241no—
379C4—241no—
380C4—241no—
381C4—482no—
382C3#938511yescharacteristic, complemented, Sylow
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2#7165611yescharacteristic
389C2#7198911yescharacteristic
390C2#7202111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2973C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2973D3
hypocenterC3#9385
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
4. #69946C2semidirect
5. #67675C2semidirect
18. #4986C4semidirect
19. #4979C4semidirect
20. #4949C4semidirect
21. #4947C4semidirect
382. C3#9385semidirect

Maximal subgroups

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

subgroupindex
C3×#29732
#823852
#716192
#699462
#676752
#93853