Finite Group Databasebeta

← #10945086

Subgroups — #10945086

Order 768.

1314 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 203 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#87260C211yesmaximal, characteristic
2#84090C211yesmaximal, characteristic
3#74074C212yesmaximal, complemented
4C3×#1782C211yesmaximal, characteristic
5#70023C214yesmaximal, complemented
6#70021C214yesmaximal, complemented
7#69534C212yesmaximal, complemented
8#33023—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10#5827—22no—
11#5827—22no—
12Dic3×#41—24no—
13#5812—21no—
14#5798—21no—
15#5798C2214yes—
16C2×#935C2212yes—
17C2×#926—22no—
18C2×#926C2212yes—
19C22×Dic3⋊C4C2211yescharacteristic
20C2×C3×#99C2214yes—
21C3×#381C2212yes—
22#5230—21no—
23#5222—21no—
24#5220C2212yes—
25#5211C2211yescharacteristic
26#5210—22no—
27#4950C2214yes—
28#4943C2214yes—
29#4661—22no—
30#4659—22no—
31#4653—24no—
32#4527C2218yes—
33#3428—31no—
34#2484—32no—
35#2406—31no—
36#1890—34no—
37#1867—34no—
38#1820—32no—
39#1782D311yescharacteristic
40C2×C3×#41—28no—
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C22×C3×Q8D411yescharacteristic
44C22×C3×D4D411yescharacteristic
45C22×C3×D4C2312yes—
46C23×C3×C4D412yes—
47C23×C3×C4C2311yescharacteristic
48C23×Dic3C2311yescharacteristic
49C23×Dic3C2311yescharacteristic
50C2×C3×C22⋊C4C2314yes—
51C2×#217—24no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217C2312yes—
55C2×#217C2314yes—
56Q8×Dic3—24no—
57Q8×Dic3—24no—
58#935—28no—
59D4×Dic3—24no—
60D4×Dic3—24no—
61D4×Dic3—28no—
62D4×Dic3—28no—
63#926—24no—
64#926—24no—
65#926—22no—
66#926—24no—
67#926—22no—
68#926—22no—
69#926—22no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—24no—
73C2×C4⋊Dic3D412yes—
74C2×C4⋊Dic3D412yes—
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—21no—
78C2×Dic3⋊C4—22no—
79C2×Dic3⋊C4—21no—
80C2×C4×Dic3—24no—
81C2×C4×Dic3D414yes—
82C3×#99—28no—
83C3×#95—22no—
84C3×#95—22no—
85C3⋊#95—21no—
86C3⋊#95—22no—
87C3⋊#95—21no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90C3⋊#95—24no—
91C22×#41C2×D311yescharacteristic
92C2×#123—32no—
93#520—62no—
94#520—62no—
95#519—61no—
96#518—34no—
97#518—61no—
98C4×#41—64no—
99C2×#117—32no—
100C2×#117—62no—
101C22×C4⋊C4—31no—
102C2×#99C2×D314yes—
103#400—62no—
104#398—62no—
105#395—61no—
106#394—61no—
107#387—64no—
108#386—32no—
109#385—34no—
110#385—62no—
111#381—31no—
112#381—34no—
113#381C2×D312yes—
114#342—38no—
115C24×C3C2×D412yes—
116C24×C3C2411yescharacteristic
117C3×#41—216no—
118C3×#41—216no—
119C2×C3×Q8—22no—
120C2×C3×Q8—24no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—22no—
125C2×C3×D4#4118yes—
126C22×C3×C4—24no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D412yes—
136C22×C3×C4C2×D412yes—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C22×Dic3—24no—
140C22×Dic3—24no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—24no—
145C22×Dic3—42no—
146C22×Dic3C2×D414yes—
147C22×Dic3C2×D414yes—
148C3×C22⋊C4—28no—
149#217—28no—
150#217—24no—
151#217—24no—
152#217—24no—
153#217—44no—
154#217—44no—
155C4⋊Dic3—24no—
156C4⋊Dic3—24no—
157C4⋊Dic3—22no—
158C4⋊Dic3—22no—
159C4⋊Dic3—22no—
160C4⋊Dic3—22no—
161C4⋊Dic3—48no—
162Dic3⋊C4—41no—
163Dic3⋊C4—42no—
164Dic3⋊C4—41no—
165Dic3⋊C4—44no—
166C4×Dic3—28no—
167C4×Dic3—24no—
168C4×Dic3—24no—
169C2×#41—28no—
170C2×#41—22no—
171C2×#41—22no—
172C22×Q8D1211yescharacteristic
173C22×D4D1211yescharacteristic
174C22×D4C22×D312yes—
175C23×C4—31no—
176C23×C4—31no—
177C23×C4D1212yes—
178C23×C4C22×D311yescharacteristic
179#123—68no—
180C4×Q8—64no—
181C4×Q8—64no—
182C4×D4—68no—
183C4×D4—68no—
184C4×D4—64no—
185C4×D4—64no—
186#117—64no—
187#117—64no—
188#117—62no—
189#117—62no—
190#117—64no—
191#117—62no—
192#117—62no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4—64no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—62no—
203C2×C22⋊C4—34no—
204C2×C22⋊C4—32no—
205C2×C22⋊C4—64no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4C22×D314yes—
209C2×C42—34no—
210C2×C42—64no—
211#99—28no—
212#95—22no—
213#95—22no—
214#95—64no—
215#95—64no—
216#95—64no—
217#95—61no—
218#95—61no—
219#95—62no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3#12314yes—
227C23×C3C22×D411yescharacteristic
228C23×C3C22×D412yes—
229C23×C3C2×#4114yes—
230C3×Q8—28no—
231C3×D4—28no—
232C3×D4—216no—
233C2×C3×C4—28no—
234C2×C3×C4—28no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—44no—
242C2×C3×C4—44no—
243C2×C3×C4—44no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12012yes—
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12314yes—
248C2×Dic3—24no—
249C2×Dic3—22no—
250C2×Dic3—24no—
251C2×Dic3—24no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—44no—
256C2×Dic3—44no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—44no—
263C2×Dic3—84no—
264C2×Dic3—84no—
265C24C2×D1212yes—
266C24C23×D311yescharacteristic
267#41—216no—
268#41—216no—
269C2×Q8—22no—
270C2×Q8—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4#23718yes—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—34no—
283C22×C4—34no—
284C22×C4—61no—
285C22×C4—62no—
286C22×C4—64no—
287C22×C4—64no—
288C22×C4—62no—
289C22×C4—61no—
290C22×C4—62no—
291C22×C4—64no—
292C22×C4—122no—
293C22×C4C2×D1211yescharacteristic
294C22×C4C2×D1211yescharacteristic
295C22×C4C2×D1212yes—
296C22×C4D3×D412yes—
297C22×C4D3×D412yes—
298C4⋊C4—64no—
299C4⋊C4—64no—
300C4⋊C4—62no—
301C4⋊C4—62no—
302C4⋊C4—62no—
303C4⋊C4—62no—
304C4⋊C4—128no—
305C4⋊C4—121no—
306C4⋊C4—122no—
307C4⋊C4—121no—
308C4⋊C4—124no—
309C22⋊C4—28no—
310C22⋊C4—68no—
311C22⋊C4—64no—
312C22⋊C4—64no—
313C22⋊C4—64no—
314C22⋊C4—124no—
315C22⋊C4—124no—
316C42—68no—
317C42—64no—
318C42—64no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—21no—
323C22×C3—24no—
324C22×C3—44no—
325C22×C3—44no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—82no—
329C22×C3C2×#12011yescharacteristic
330C22×C3C2×#12312yes—
331C22×C3#52514yes—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—22no—
335C3×C4—24no—
336C3×C4—84no—
337Dic3—42no—
338Dic3—44no—
339Dic3—44no—
340Dic3—82no—
341Dic3—81no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23#88614yes—
349C23C22×D1211yescharacteristic
350C23C2×D3×D412yes—
351C23C2×#23714yes—
352Q8—28no—
353D4—28no—
354D4—216no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—62no—
367C2×C4—64no—
368C2×C4—64no—
369C2×C4—64no—
370C2×C4—122no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—124no—
374C2×C4—124no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—121no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—124no—
381C2×C4—244no—
382C2×C4—244no—
383C2×C4#88211yescharacteristic
384C2×C4#88212yes—
385C2×C4#88211yescharacteristic
386C2×C4#88614yes—
387C2×C3—22no—
388C2×C3—24no—
389C2×C3—44no—
390C2×C3#284911yescharacteristic
391C2×C3#302511yescharacteristic
392C2×C3#302811yescharacteristic
393C22—22no—
394C22—21no—
395C22—22no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—44no—
400C22—42no—
401C22—42no—
402C22—82no—
403C22C2×#88211yescharacteristic
404C22C2×#88612yes—
405C22#558614yes—
406C4—21no—
407C4—21no—
408C4—22no—
409C4—24no—
410C4—84no—
411C4—122no—
412C4—124no—
413C4—124no—
414C4—242no—
415C4—241no—
416C3#3302311yescharacteristic, complemented, Sylow
417C2—24no—
418C2—22no—
419C2—44no—
420C2#7384711yescharacteristic
421C2#7869011yescharacteristic
422C2#7869211yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#1782C2
socleC22×C3C2×#120
prime radicalC22×C3×#41C22
hypercenter#1782D3
hypocenterC3#33023
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
3. #74074C2semidirect
5. #70023C2semidirect
6. #70021C2semidirect
7. #69534C2semidirect
416. C3#33023semidirect

Maximal subgroups

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

subgroupindex
#872602
#840902
#740742
C3×#17822
#700232
#700212
#695342
#330233