Finite Group Databasebeta

← #10690978

Subgroups — #10690978

Order 768.

980 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 143 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×#5211C211yesmaximal, characteristic, complemented
3#72126C211yesmaximal, characteristic, complemented
4C2×#4537C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9158—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×C3×#121C412yescomplemented
9C2×C3×#121C2211yescharacteristic
10C2×C3×#120—21no—
11C22×#217—21no—
12C3×#386—21no—
13C3×#386—22no—
14C3×#380C412yescomplemented
15C3×#380C2211yescharacteristic
16#5211—24no—
17C2×C3⋊#95—21no—
18C2×C3⋊#95C2212yes—
19C2×#830—22no—
20C2×#830C2212yes—
21#4537—24no—
22#2847D311yescharacteristic
23C2×#381—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×C4D412yes—
36C23×C3×C4D411yescharacteristic
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—44no—
51C22×C3⋊C8—22no—
52C3×#95—22no—
53C3×#95—21no—
54C3⋊#95—24no—
55C3⋊#95—22no—
56C3⋊#95D414yescomplemented
57#830—22no—
58#830—22no—
59#830—44no—
60C24×C4C2×D311yescharacteristic
61C2×#121Dic312yescomplemented
62C2×#121C2×D311yescharacteristic
63C2×#120—21no—
64C22×C22⋊C4—61no—
65C2×C22⋊C8—32no—
66C2×C22⋊C8—62no—
67#386—21no—
68#386—22no—
69#381—64no—
70#380Dic312yescomplemented
71#380C2×D311yescharacteristic
72C2×#95—32no—
73C2×#95—61no—
74#327—64no—
75C24×C3—24no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—22no—
79C24×C3—22no—
80C24×C3—21no—
81C24×C3—42no—
82C24×C3C22⋊C412yes—
83C24×C3C2×D411yescharacteristic
84C2×C3×D4—22no—
85C2×C3×D4—42no—
86C2×C3×D4—44no—
87C2×C3×D4—44no—
88C2×C3×D4—44no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
91C2×C3×D4—42no—
92C2×C3×D4—42no—
93C22×C3×C4—24no—
94C22×C3×C4—24no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—21no—
100C22×C3×C4—22no—
101C22×C3×C4—42no—
102C22×C3×C4—42no—
103C22×C3×C4—42no—
104C22×C3×C4—42no—
105C22×C3×C4—48no—
106C22×C3×C4—48no—
107C22×C3×C4C22⋊C412yes—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D412yes—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4112yes—
112C22×Dic3—24no—
113C22×Dic3—44no—
114C22×Dic3—44no—
115C22×Dic3—42no—
116C22×Dic3—42no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—42no—
119C3×C22⋊C4—44no—
120C3×C22⋊C4—44no—
121C3×C22⋊C4—42no—
122C3×C22⋊C4—42no—
123#217—48no—
124C2×C3⋊C8—44no—
125C2×C3⋊C8—42no—
126C25#6711yescharacteristic
127C22×D4—22no—
128C22×D4—21no—
129C22×D4C2×Dic311yescharacteristic
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—44no—
133C23×C4—62no—
134C23×C4C2×Dic312yes—
135C23×C4#6712yes—
136C23×C4#6711yescharacteristic
137C23×C4C22×D311yescharacteristic
138C22×C8—62no—
139#121—22no—
140#121—22no—
141#121—44no—
142#120—44no—
143C2×C4⋊C4C2×Dic311yescharacteristic
144C2×C4⋊C4C2×Dic312yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—64no—
150C2×C22⋊C4—124no—
151C22⋊C8—62no—
152C22⋊C8—62no—
153C22⋊C8—124no—
154#95—22no—
155#95—21no—
156#95—34no—
157#95—64no—
158#95—62no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—24no—
162C23×C3—24no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—48no—
170C23×C3—48no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—41no—
176C23×C3—42no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—42no—
180C23×C3#9914yes—
181C23×C3C2×C22⋊C411yescharacteristic
182C23×C3#12011yescharacteristic
183C23×C3#12311yescharacteristic
184C23×C3#13512yes—
185C23×C3AGL(1,ℤ/8)12yes—
186C3×D4—82no—
187C3×D4—88no—
188C3×D4—82no—
189C3×D4—82no—
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—48no—
198C2×C3×C4—48no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4C4×D412yes—
208C2×C3×C4#12112yes—
209C2×Dic3—44no—
210C2×Dic3—44no—
211C2×Dic3—44no—
212C2×Dic3—84no—
213C2×Dic3—84no—
214C3⋊C8—82no—
215C24—24no—
216C24—21no—
217C24—21no—
218C24—22no—
219C24—22no—
220C24—21no—
221C24—42no—
222C24#21712yes—
223C24C2×#6711yescharacteristic
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—42no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—48no—
246C22×C4—48no—
247C22×C4—64no—
248C22×C4—124no—
249C22×C4—124no—
250C22×C4—122no—
251C22×C4—122no—
252C22×C4#21712yes—
253C22×C4D3×D412yes—
254C22×C4#23712yes—
255C22×C4C22×Dic311yescharacteristic
256C22×C4C2×#6711yescharacteristic
257C2×C8—124no—
258C2×C8—122no—
259C4⋊C4—22no—
260C4⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—128no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—24no—
271C22×C3—24no—
272C22×C3—24no—
273C22×C3—24no—
274C22×C3—24no—
275C22×C3—48no—
276C22×C3—48no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3—82no—
285C22×C3—82no—
286C22×C3#38611yescharacteristic
287C22×C3C2×#9912yes—
288C22×C3#41611yescharacteristic
289C22×C3#41811yescharacteristic
290C22×C3#42112yes—
291C22×C3#44912yes—
292C22×C3#48212yes—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—42no—
296C3×C4—81no—
297C3×C4—82no—
298Dic3—84no—
299C23—22no—
300C23—22no—
301C23—24no—
302C23—24no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—48no—
310C23—48no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—41no—
316C23—42no—
317C23—41no—
318C23—41no—
319C23—42no—
320C23#83414yes—
321C23#93212yes—
322C23#93511yescharacteristic
323C23#93711yescharacteristic
324C23#95012yes—
325C23C2×#21711yescharacteristic
326D4—82no—
327D4—88no—
328D4—82no—
329D4—82no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—124no—
348C2×C4—124no—
349C2×C4—124no—
350C2×C4—244no—
351C2×C4—244no—
352C2×C4D4×Dic312yes—
353C2×C4#93912yes—
354C8—242no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—44no—
358C2×C3—161no—
359C2×C3#151012yes—
360C2×C3#151612yes—
361C2×C3#187511yescharacteristic
362C2×C3#189711yescharacteristic
363C2×C3#189811yescharacteristic
364C22—24no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—24no—
370C22—24no—
371C22—24no—
372C22—24no—
373C22—48no—
374C22—48no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22—82no—
383C22—81no—
384C22#503512yes—
385C22#503712yes—
386C22#521511yescharacteristic
387C22C2×#83412yes—
388C22#522011yescharacteristic
389C22#523311yescharacteristic
390C22#523612yes—
391C4—22no—
392C4—22no—
393C4—42no—
394C4—82no—
395C4—81no—
396C4—244no—
397C3#915811yescharacteristic, complemented, Sylow
398C2—22no—
399C2—22no—
400C2—44no—
401C2—161no—
402C2#7167011yescharacteristic
403C2#7167412yes—
404C2#7167811yescharacteristic
405C2#7171012yes—
406C2#7171211yescharacteristic
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#9158
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×#5211C2semidirect
3. #72126C2semidirect
4. C2×#4537C2semidirect
5. #67624C2semidirect
8. C2×C3×#121C4semidirect
14. C3×#380C4semidirect
56. C3⋊#95D4semidirect
61. C2×#121Dic3semidirect
70. #380Dic3semidirect
397. C3#9158semidirect

Maximal subgroups

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

subgroupindex
C3×#28472
C2×#52112
#721262
C2×#45372
#676242
#91583