Finite Group Databasebeta

← #10690938

Subgroups — #10690938

Order 768.

1088 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 151 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×#2847C211yesmaximal, characteristic
2C2×C4×#217C211yesmaximal, characteristic, complemented
3#72126C211yesmaximal, characteristic, complemented
4C2×#4537C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9150—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×C3×#121C412yescomplemented
9C2×C3×#121C2211yescharacteristic
10C2×C3×#120—21no—
11C22×#217—21no—
12C22×C4×Dic3—21no—
13C3×#386—21no—
14C3×#386—22no—
15C3×#380C412yescomplemented
16C3×#380C2211yescharacteristic
17C4×#217—24no—
18C2×C3⋊#95C2212yes—
19C2×#830—22no—
20C2×#830C2212yes—
21#4537—24no—
22#2847D311yescharacteristic
23C2×C4×C22⋊C4—31no—
24#1779—31no—
25C2×#327—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C22×C3×D4—22no—
29C22×C3×D4—21no—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4—22no—
32C23×C3×C4—22no—
33C23×C3×C4—44no—
34C23×C3×C4C2×C412yes—
35C23×C3×C4D411yescharacteristic
36C23×C3×C4D412yes—
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—22no—
39C3×#121—22no—
40C3×#121—22no—
41C3×#121—44no—
42C3×#120—44no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C412yes—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—21no—
49C2×#217—24no—
50C2×#217—28no—
51C2×C4×Dic3—24no—
52C2×C4×Dic3—44no—
53C22×C3⋊C8—22no—
54C3×#95—22no—
55C3×#95—21no—
56C3⋊#95—22no—
57C3⋊#95D414yescomplemented
58#830—22no—
59#830—22no—
60#830—44no—
61C24×C4C2×D311yescharacteristic
62C2×#121Dic312yescomplemented
63C2×#121C2×D311yescharacteristic
64C2×#120—21no—
65C22×C22⋊C4—61no—
66C22×C42—61no—
67C2×C22⋊C8—32no—
68C2×C22⋊C8—62no—
69#386—21no—
70#386—22no—
71#380Dic312yescomplemented
72#380C2×D311yescharacteristic
73C4×C22⋊C4—64no—
74C2×#95—32no—
75#327—64no—
76C24×C3—24no—
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—22no—
80C24×C3—22no—
81C24×C3—21no—
82C24×C3—42no—
83C24×C3C22⋊C412yes—
84C24×C3C2×D411yescharacteristic
85C2×C3×D4—22no—
86C2×C3×D4—42no—
87C2×C3×D4—44no—
88C2×C3×D4—44no—
89C2×C3×D4—44no—
90C2×C3×D4—41no—
91C2×C3×D4—41no—
92C2×C3×D4—42no—
93C2×C3×D4—42no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—24no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4—42no—
104C22×C3×C4—42no—
105C22×C3×C4—42no—
106C22×C3×C4—48no—
107C22×C3×C4—48no—
108C22×C3×C4C22⋊C412yes—
109C22×C3×C4C22×C411yescharacteristic
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4112yes—
113C22×Dic3—28no—
114C22×Dic3—24no—
115C22×Dic3—44no—
116C22×Dic3—42no—
117C22×Dic3—42no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—42no—
120C3×C22⋊C4—44no—
121C3×C22⋊C4—42no—
122C3×C22⋊C4—44no—
123C3×C22⋊C4—42no—
124#217—216no—
125C4×Dic3—48no—
126C2×C3⋊C8—44no—
127C2×C3⋊C8—42no—
128C25#6711yescharacteristic
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×Dic311yescharacteristic
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—44no—
135C23×C4—62no—
136C23×C4C2×Dic312yes—
137C23×C4#6711yescharacteristic
138C23×C4#6712yes—
139C23×C4C22×D311yescharacteristic
140C22×C8—62no—
141#121—22no—
142#121—22no—
143#121—44no—
144#120—44no—
145C2×C4⋊C4C2×Dic311yescharacteristic
146C2×C4⋊C4C2×Dic312yes—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—64no—
152C2×C22⋊C4—68no—
153C2×C42—64no—
154C2×C42—124no—
155C22⋊C8—62no—
156C22⋊C8—62no—
157C22⋊C8—124no—
158#95—22no—
159#95—21no—
160#95—34no—
161#95—62no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—24no—
165C23×C3—24no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—22no—
171C23×C3—22no—
172C23×C3—48no—
173C23×C3—48no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—41no—
179C23×C3—42no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3—42no—
183C23×C3#10414yes—
184C23×C3C2×C22⋊C411yescharacteristic
185C23×C3#12011yescharacteristic
186C23×C3#12311yescharacteristic
187C23×C3#13512yes—
188C23×C3AGL(1,ℤ/8)12yes—
189C3×D4—82no—
190C3×D4—88no—
191C3×D4—82no—
192C3×D4—82no—
193C2×C3×C4—24no—
194C2×C3×C4—24no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—48no—
202C2×C3×C4—48no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4#9914yes—
210C2×C3×C4C4×D412yes—
211C2×C3×C4#12112yes—
212C2×Dic3—28no—
213C2×Dic3—44no—
214C2×Dic3—48no—
215C2×Dic3—48no—
216C2×Dic3—44no—
217C3⋊C8—82no—
218C24—24no—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—22no—
223C24—21no—
224C24—42no—
225C24#21712yes—
226C24C2×#6711yescharacteristic
227C2×D4—22no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—42no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—68no—
251C22×C4—64no—
252C22×C4—124no—
253C22×C4—122no—
254C22×C4—122no—
255C22×C4#21712yes—
256C22×C4D3×D412yes—
257C22×C4#23712yes—
258C22×C4C22×Dic311yescharacteristic
259C22×C4C2×#6711yescharacteristic
260C2×C8—124no—
261C2×C8—122no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C22⋊C4—44no—
265C22⋊C4—42no—
266C22⋊C4—44no—
267C22⋊C4—42no—
268C22⋊C4—616no—
269C42—128no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—24no—
275C22×C3—24no—
276C22×C3—24no—
277C22×C3—24no—
278C22×C3—24no—
279C22×C3—48no—
280C22×C3—48no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3—82no—
289C22×C3—82no—
290C22×C3#38611yescharacteristic
291C22×C3C2×#9912yes—
292C22×C3#41611yescharacteristic
293C22×C3#41811yescharacteristic
294C22×C3C2×#10412yes—
295C22×C3#44912yes—
296C22×C3#48212yes—
297C3×C4—22no—
298C3×C4—24no—
299C3×C4—22no—
300C3×C4—81no—
301C3×C4—82no—
302Dic3—48no—
303C23—22no—
304C23—22no—
305C23—24no—
306C23—24no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—48no—
314C23—48no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—42no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23#83714yes—
325C23#93212yes—
326C23#93511yescharacteristic
327C23#93711yescharacteristic
328C23#95012yes—
329C23C2×#21711yescharacteristic
330D4—82no—
331D4—88no—
332D4—82no—
333D4—82no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—68no—
351C2×C4—124no—
352C2×C4—128no—
353C2×C4—128no—
354C2×C4—124no—
355C2×C4#83414yes—
356C2×C4D4×Dic312yes—
357C2×C4#93912yes—
358C8—242no—
359C2×C3—22no—
360C2×C3—22no—
361C2×C3—44no—
362C2×C3—161no—
363C2×C3#150112yes—
364C2×C3#150712yes—
365C2×C3#187511yescharacteristic
366C2×C3#189611yescharacteristic
367C2×C3#189811yescharacteristic
368C22—24no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—24no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—48no—
378C22—48no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22—81no—
385C22—81no—
386C22—82no—
387C22—81no—
388C22#503512yes—
389C22#503712yes—
390C22#521511yescharacteristic
391C22C2×#83412yes—
392C22#522011yescharacteristic
393C22#523311yescharacteristic
394C22C2×#83712yes—
395C4—22no—
396C4—24no—
397C4—22no—
398C4—82no—
399C4—81no—
400C4—128no—
401C3#915011yescharacteristic, complemented, Sylow
402C2—22no—
403C2—22no—
404C2—44no—
405C2—161no—
406C2#7167011yescharacteristic
407C2#7167312yes—
408C2#7167811yescharacteristic
409C2#7170712yes—
410C2#7171111yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2847C2
socleC23×C3C2×C22⋊C4
cosocleC23×C3×C4C23
hypercenter#2847D3
hypocenterC3#9150
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
2. C2×C4×#217C2semidirect
3. #72126C2semidirect
4. C2×#4537C2semidirect
5. #67624C2semidirect
8. C2×C3×#121C4semidirect
15. C3×#380C4semidirect
57. C3⋊#95D4semidirect
62. C2×#121Dic3semidirect
71. #380Dic3semidirect
401. C3#9150semidirect

Maximal subgroups

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

subgroupindex
C3×#28472
C2×C4×#2172
#721262
C2×#45372
#676242
#91503