Finite Group Databasebeta

← #10693129

Subgroups — #10693129

Order 768.

878 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#82403C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic, complemented
3C3×#1465C211yesmaximal, characteristic, complemented
4#69988C212yesmaximal, complemented
5#67685C212yesmaximal, complemented
6#9504—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C2×#937C412yes—
10C2×#930—21no—
11C3×#407C2212yes—
12#5212C2211yescharacteristic
13#5207—21no—
14#5207C2212yes—
15C2×#823—24no—
16#5012—24no—
17#4991—21no—
18#4960—21no—
19#4955C412yes—
20#4939—21no—
21#4925—21no—
22C3×#324—21no—
23C3×#323—21no—
24#2975—31no—
25#2408—31no—
26#1783—32no—
27#1465D311yescharacteristic, complemented
28#1305—32no—
29C24×D3C2×C412yes—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C22×#67—22no—
34C22×D12—22no—
35C22×D12—21no—
36C22×D12—21no—
37C22×D12—22no—
38C22×D12—44no—
39C2×C3×M16—22no—
40C3×C4×D4—22no—
41C3×C4×D4—22no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C22⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C2×#217C2×C412yes—
47#937—44no—
48#930—44no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—22no—
51C2×(C2×D3)⋊C4—22no—
52C2×C4⋊Dic3—21no—
53C2×#208—22no—
54C2×#208D414yescomplemented
55C3×C22⋊C8—22no—
56C3×C22⋊C8—22no—
57C3⋊#95—21no—
58#830—22no—
59#830—22no—
60#830—22no—
61#823—48no—
62C4⋊(C3⋊C8)—22no—
63C4×C3⋊C8—22no—
64C23×D4—31no—
65C2×#121—61no—
66C2×#120—32no—
67C2×C4×D4C2×D311yescharacteristic
68#436—64no—
69#431—61no—
70C2×#100—64no—
71#407—32no—
72#407—61no—
73#407C2×D312yes—
74#394—61no—
75#392—61no—
76#390—61no—
77#380—31no—
78#379—32no—
79#324—21no—
80#323—21no—
81C24×C3C22⋊C411yescharacteristic
82C24×C3C22×C411yescharacteristic
83C23×D3—22no—
84C23×D3—24no—
85C23×D3—44no—
86C23×D3—42no—
87C23×D3—42no—
88C23×D3—82no—
89C2×C3×D4—22no—
90C2×C3×D4—22no—
91C2×C3×D4—41no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4112yes—
103C2×#67—44no—
104C2×#67—44no—
105C2×#67—82no—
106C22×Dic3—22no—
107C2×D12—28no—
108C2×D12—24no—
109C2×D12—44no—
110C2×D12—42no—
111C2×D12—44no—
112C2×D12—48no—
113C2×D12—41no—
114C2×D12—42no—
115C2×D12—41no—
116C3×M16—44no—
117C2×C3×C8—42no—
118C3×C4⋊C4—22no—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—22no—
121C3×C42—21no—
122C3×C42—21no—
123#217—42no—
124(C2×D3)⋊C4—44no—
125(C2×D3)⋊C4—44no—
126(C2×D3)⋊C4—82no—
127C4⋊Dic3—42no—
128#208—44no—
129#208—82no—
130C2×C3⋊C8—24no—
131C25—32no—
132C22×D4—62no—
133C22×D4—61no—
134C22×D4—62no—
135C22×D4—62no—
136C22×D4—61no—
137C22×D4—124no—
138C22×D4C4×D311yescharacteristic
139C23×C4D1211yescharacteristic
140C23×C4C22×D311yescharacteristic
141C2×M16—22no—
142C2×M16—34no—
143C2×M16—62no—
144#121—124no—
145#120—124no—
146C4×D4—22no—
147C4×D4—22no—
148C2×C4⋊C4—61no—
149C2×C4⋊C4#6711yescharacteristic
150C2×C22⋊C4—32no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—62no—
153C2×C22⋊C4—62no—
154C2×C22⋊C4C4×D311yescharacteristic
155C2×C22⋊C4#6711yescharacteristic
156C2×C42D1211yescharacteristic
157C4⋊C8—62no—
158#100—128no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—62no—
162C22⋊C8—62no—
163C22⋊C8—62no—
164C4×C8—62no—
165#95—61no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—41no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3—41no—
175C23×C3C2×C22⋊C411yescharacteristic
176C23×C3C4×D412yes—
177C22×D3—24no—
178C22×D3—28no—
179C22×D3—42no—
180C22×D3—42no—
181C22×D3—44no—
182C22×D3—42no—
183C22×D3—42no—
184C22×D3—42no—
185C22×D3—42no—
186C22×D3—88no—
187C22×D3—88no—
188C22×D3—84no—
189C22×D3—84no—
190C3×D4—44no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—24no—
194C2×C3×C4—22no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—81no—
203C2×C3×C4#10012yescomplemented
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12112yes—
206C2×C3×C4#12311yescharacteristic
207#67—88no—
208C2×Dic3—42no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211D12—48no—
212D12—44no—
213D12—44no—
214C3×C8—82no—
215C3⋊C8—44no—
216C24—62no—
217C24—64no—
218C24—122no—
219C24—122no—
220C24—124no—
221C24—242no—
222C24(C2×D3)⋊C411yescharacteristic
223C24C2×C4×D311yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—41no—
227C2×D4—68no—
228C2×D4—64no—
229C2×D4—122no—
230C2×D4—124no—
231C2×D4—128no—
232C2×D4—124no—
233C2×D4—124no—
234C2×D4—124no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—122no—
238C2×D4—242no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—62no—
244C22×C4(C2×D3)⋊C411yescharacteristic
245C22×C4(C2×D3)⋊C411yescharacteristic
246C22×C4(C2×D3)⋊C411yescharacteristic
247C22×C4C2×D1211yescharacteristic
248C22×C4D3×D412yes—
249C22×C4#23712yes—
250C22×C4C2×#6711yescharacteristic
251M16—44no—
252M16—124no—
253M16—242no—
254C2×C8—42no—
255C2×C8—64no—
256C4⋊C4—22no—
257C4⋊C4—122no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—124no—
261C22⋊C4—124no—
262C22⋊C4—122no—
263C22⋊C4—242no—
264C42—21no—
265C42—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—42no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—81no—
278C22×C3#38611yescharacteristic
279C22×C3C2×#10011yescharacteristic
280C22×C3#41311yescharacteristic
281C2×D3—44no—
282C2×D3—44no—
283C2×D3—44no—
284C2×D3—82no—
285C2×D3—82no—
286C2×D3—88no—
287C2×D3—82no—
288C2×D3—82no—
289C2×D3—88no—
290C3×C4—22no—
291C3×C4—21no—
292C3×C4—22no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4B(4,𝔽2)12yes—
296Dic3—82no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—64no—
307C23—68no—
308C23—122no—
309C23—122no—
310C23—124no—
311C23—122no—
312C23—122no—
313C23—122no—
314C23—122no—
315C23—244no—
316C23—244no—
317C23—248no—
318C23—248no—
319C23#67⋊C412yes—
320C23C2×(C2×D3)⋊C411yescharacteristic
321D4—44no—
322D4—128no—
323D4—124no—
324D4—124no—
325D4—248no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4#82312yescomplemented
342C2×C4#88211yescharacteristic
343C2×C4#88611yescharacteristic
344C2×C4#93912yes—
345C8—82no—
346C8—124no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—42no—
350C2×C3—81no—
351C2×C3#188911yescharacteristic
352C2×C3#190011yescharacteristic
353C2×C3#191411yescharacteristic
354D3—84no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22—124no—
368C22—124no—
369C22—124no—
370C22—242no—
371C22—242no—
372C22—248no—
373C22—248no—
374C22—242no—
375C22—242no—
376C22#495811yescharacteristic
377C22C2×#82311yescharacteristic
378C22#513211yescharacteristic
379C4—22no—
380C4—22no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—242no—
385C4#474112yes—
386C3#950411yescharacteristic, complemented, Sylow
387C2—21no—
388C2—21no—
389C2—42no—
390C2—81no—
391C2—244no—
392C2#6998511yescharacteristic
393C2#7002211yescharacteristic
394C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1465C2
socleC22×C3#386
prime radical#82403C2
cosocleC23×C3×C4C23
hypercenter#1465D3
hypocenterC3#9504
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #82304C2semidirect
3. C3×#1465C2semidirect
4. #69988C2semidirect
5. #67685C2semidirect
27. #1465D3semidirect
54. C2×#208D4semidirect
203. C2×C3×C4#100semidirect
341. C2×C4#823semidirect
386. C3#9504semidirect

Maximal subgroups

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

subgroupindex
#824032
#823042
C3×#14652
#699882
#676852
#95043