Finite Group Databasebeta

← #10693087

Subgroups — #10693087

Order 768.

1212 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1212 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×#2403C211yesmaximal, characteristic
2#84043C211yesmaximal, characteristic, complemented
3#72130C212yesmaximal, complemented
4#72130C211yesmaximal, characteristic, complemented
5C3⋊#1304C212yesmaximal, complemented
6#9495—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C23×C3×D4C2211yescharacteristic
9C2×C3×#120—21no—
10C2×C3×#120C412yescomplemented
11C22×#217—21no—
12C3×#392—22no—
13C3×#386—21no—
14C3×#386C412yescomplemented
15C3×#379C2211yescharacteristic
16#5247C2212yes—
17#5247C2212yes—
18#5220—24no—
19C2×#834—22no—
20C2×#834—24no—
21#5211—21no—
22#2403D311yescharacteristic
23#2308—31no—
24#1781—32no—
25#1781—31no—
26#1304—32no—
27C25×C3C2×C411yescharacteristic, complemented
28C25×C3D411yescharacteristic
29C25×C3C2311yescharacteristic
30C22×C3×D4—22no—
31C22×C3×D4—22no—
32C22×C3×D4—21no—
33C22×C3×D4—22no—
34C22×C3×D4—21no—
35C22×C3×D4—22no—
36C22×C3×D4—22no—
37C22×C3×D4—44no—
38C22×C3×D4—44no—
39C22×C3×D4C2×C412yes—
40C22×C3×D4D412yes—
41C23×C3×C4C2×C411yescharacteristic, complemented
42C23×C3×C4D411yescharacteristic
43C23×Dic3—22no—
44C3×#120—44no—
45C3×#120—44no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4C2×C412yescomplemented
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—22no—
57C2×#217—44no—
58C2×#217D414yescomplemented
59C3×#95—22no—
60#834—48no—
61#834—44no—
62C3⋊#95—24no—
63C23×D4C2×D311yescharacteristic
64C23×D4C2×D311yescharacteristic
65C2×#120—21no—
66C2×#120Dic312yescomplemented
67C22×C22⋊C4—61no—
68C2×#99—64no—
69C2×#99—62no—
70#392—22no—
71#386—21no—
72#386—64no—
73#386Dic312yescomplemented
74#381—61no—
75#379—32no—
76#379—32no—
77#379C2×D311yescharacteristic
78C24×C3—21no—
79C24×C3—24no—
80C24×C3—21no—
81C24×C3—21no—
82C24×C3—22no—
83C24×C3—42no—
84C24×C3—42no—
85C24×C3—42no—
86C24×C3—42no—
87C24×C3—42no—
88C24×C3—81no—
89C24×C3—81no—
90C24×C3—81no—
91C24×C3C22⋊C412yes—
92C24×C3C22×C411yescharacteristic
93C24×C3C2×D412yes—
94C24×C3C2×D411yescharacteristic
95C24×C3C2×D411yescharacteristic
96C24×C3#4112yes—
97C2×C3×D4—28no—
98C2×C3×D4—24no—
99C2×C3×D4—24no—
100C2×C3×D4—24no—
101C2×C3×D4—24no—
102C2×C3×D4—24no—
103C2×C3×D4—42no—
104C2×C3×D4—42no—
105C2×C3×D4—42no—
106C2×C3×D4—44no—
107C2×C3×D4—44no—
108C2×C3×D4—44no—
109C2×C3×D4—48no—
110C2×C3×D4—42no—
111C2×C3×D4—44no—
112C2×C3×D4—42no—
113C2×C3×D4—44no—
114C2×C3×D4—41no—
115C2×C3×D4—42no—
116C2×C3×D4—41no—
117C2×C3×D4—44no—
118C2×C3×D4—44no—
119C2×C3×D4—82no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—42no—
126C22×C3×C4—42no—
127C22×C3×C4C22⋊C412yes—
128C22×Dic3—24no—
129C22×Dic3—44no—
130C22×Dic3—42no—
131C22×Dic3—82no—
132C22×Dic3—82no—
133C3×C22⋊C4—44no—
134C3×C22⋊C4—44no—
135C3×C22⋊C4—42no—
136C3×C22⋊C4—42no—
137C3×C22⋊C4—81no—
138#217—44no—
139#217—44no—
140#217—44no—
141#217—44no—
142#217—44no—
143#217—82no—
144#217—82no—
145C25C2×Dic311yescharacteristic, complemented
146C25#6711yescharacteristic
147C25C22×D311yescharacteristic
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4C2×Dic312yes—
158C22×D4#6712yes—
159C23×C4—62no—
160C23×C4C2×Dic311yescharacteristic, complemented
161C23×C4#6711yescharacteristic
162#120—44no—
163#120—44no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—34no—
169C2×C22⋊C4—62no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—62no—
175C2×C22⋊C4—124no—
176C2×C22⋊C4C2×Dic312yescomplemented
177#99—128no—
178#99—124no—
179#95—22no—
180#95—64no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—24no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3—24no—
191C23×C3—22no—
192C23×C3—24no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—42no—
197C23×C3—42no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3—41no—
201C23×C3—42no—
202C23×C3—42no—
203C23×C3—42no—
204C23×C3—42no—
205C23×C3—42no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3—81no—
213C23×C3—81no—
214C23×C3—82no—
215C23×C3—82no—
216C23×C3—88no—
217C23×C3—88no—
218C23×C3—88no—
219C23×C3#9914yescomplemented
220C23×C3C2×C22⋊C411yescharacteristic
221C23×C3C4×D412yes—
222C23×C3#12011yescharacteristic
223C23×C3#12112yes—
224C23×C3#12311yescharacteristic
225C3×D4—48no—
226C3×D4—48no—
227C3×D4—48no—
228C3×D4—44no—
229C3×D4—44no—
230C3×D4—88no—
231C2×C3×C4—22no—
232C2×C3×C4—24no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×Dic3—44no—
245C2×Dic3—44no—
246C2×Dic3—44no—
247C2×Dic3—88no—
248C2×Dic3—82no—
249C2×Dic3—82no—
250C2×Dic3—82no—
251C24—21no—
252C24—24no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—81no—
262C24—81no—
263C24—81no—
264C24#21712yes—
265C24D3×D412yes—
266C24#23712yes—
267C24C22×Dic311yescharacteristic
268C24C2×#6711yescharacteristic
269C24C2×#6711yescharacteristic
270C2×D4—28no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—42no—
284C2×D4—44no—
285C2×D4—42no—
286C2×D4—44no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—82no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—64no—
301C22×C4—124no—
302C22×C4—122no—
303C22×C4—242no—
304C22×C4—242no—
305C22×C4#21712yes—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—81no—
311C22⋊C4—124no—
312C22⋊C4—124no—
313C22⋊C4—124no—
314C22⋊C4—124no—
315C22⋊C4—124no—
316C22⋊C4—242no—
317C22⋊C4—242no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—21no—
327C22×C3—24no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—21no—
332C22×C3—44no—
333C22×C3—44no—
334C22×C3—44no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—44no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—84no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3—82no—
345C22×C3—84no—
346C22×C3—84no—
347C22×C3—84no—
348C22×C3—84no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3#38611yescharacteristic
352C22×C3C2×#9912yes—
353C22×C3B(4,𝔽2)12yes—
354C3×C4—42no—
355C3×C4—42no—
356C3×C4—82no—
357Dic3—84no—
358Dic3—162no—
359C23—22no—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—22no—
370C23—24no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—41no—
390C23—81no—
391C23—81no—
392C23—82no—
393C23—82no—
394C23—88no—
395C23—88no—
396C23—88no—
397C23#83414yescomplemented
398C23D4×Dic312yes—
399C23#93511yescharacteristic
400C23#93711yescharacteristic
401C23#93912yes—
402C23C2×#21711yescharacteristic
403D4—48no—
404D4—48no—
405D4—48no—
406D4—44no—
407D4—44no—
408D4—88no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4—124no—
425C2×C4—248no—
426C2×C4—242no—
427C2×C4—242no—
428C2×C4—242no—
429C2×C3—22no—
430C2×C3—22no—
431C2×C3—42no—
432C2×C3—42no—
433C2×C3—41no—
434C2×C3—44no—
435C2×C3—82no—
436C2×C3#188911yescharacteristic
437C2×C3#189811yescharacteristic
438C2×C3#191311yescharacteristic
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—24no—
452C22—22no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—82no—
463C22—82no—
464C22—84no—
465C22—82no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—81no—
471C22—81no—
472C22#503212yes—
473C22C2×#83412yes—
474C22#522011yescharacteristic
475C4—42no—
476C4—42no—
477C4—82no—
478C4—244no—
479C4—482no—
480C3#949511yescharacteristic, complemented, Sylow
481C2—22no—
482C2—22no—
483C2—42no—
484C2—42no—
485C2—41no—
486C2—44no—
487C2—82no—
488C2#7166911yescharacteristic
489C2#7167011yescharacteristic
490C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC24×C3C22×C4
FrattiniC25C22×D3
Fitting, prime radicalC3×#2403C2
socleC22×C3#386
cosocleC25×C3C23
hypercenter#2403D3
hypocenterC3#9495
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #2403

Complements

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

NH ≅ G/Ntype
2. #84043C2semidirect
3. #72130C2semidirect
4. #72130C2semidirect
5. C3⋊#1304C2semidirect
10. C2×C3×#120C4semidirect
14. C3×#386C4semidirect
27. C25×C3C2×C4semidirect
41. C23×C3×C4C2×C4semidirect
50. C2×C3×C22⋊C4C2×C4semidirect
58. C2×#217D4semidirect
66. C2×#120Dic3semidirect
73. #386Dic3semidirect
145. C25C2×Dic3semidirect
160. C23×C4C2×Dic3semidirect
176. C2×C22⋊C4C2×Dic3semidirect
219. C23×C3#99semidirect
397. C23#834semidirect
480. C3#9495semidirect

Maximal subgroups

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

subgroupindex
C3×#24032
#840432
#721302
C3⋊#13042
#94953