Finite Group Databasebeta

← #10693097

Subgroups — #10693097

Order 768.

1184 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 1184 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#84044C211yesmaximal, characteristic, complemented
3#72130C212yesmaximal, complemented
4#72130C211yesmaximal, characteristic, complemented
5C3⋊#1304C212yesmaximal, complemented
6#9497—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C23×C3×D4C2211yescharacteristic
9C2×C3×#120—21no—
10C2×C3×#120C412yescomplemented
11C2×#926—21no—
12C3×#392—22no—
13C3×#386—21no—
14C3×#386C412yescomplemented
15C3×#379C2211yescharacteristic
16#5247—21no—
17#5247C2212yes—
18#5247C2212yes—
19#5221—24no—
20C2×#834—22no—
21C2×#834—24no—
22#2403D311yescharacteristic
23#2309—31no—
24#1781—32no—
25#1781—31no—
26#1304—32no—
27C25×C3C2×C411yescharacteristic
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
42C23×C3×C4D411yescharacteristic
43C3×#120—44no—
44C3×#120—44no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4C2×C412yes—
50C2×#217—22no—
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—22no—
54C2×#217—22no—
55C2×#217—22no—
56C2×#217—22no—
57C2×#217D414yescomplemented
58#926—44no—
59C2×C4⋊Dic3—22no—
60C2×C4×Dic3—22no—
61C3×#95—22no—
62#834—48no—
63#834—44no—
64C23×D4C2×D311yescharacteristic
65C23×D4C2×D311yescharacteristic
66C2×#120—21no—
67C2×#120Dic312yescomplemented
68C2×#117—61no—
69C2×#99—64no—
70C2×#99—62no—
71#392—22no—
72#390—64no—
73#386—21no—
74#386Dic312yescomplemented
75#379—32no—
76#379—32no—
77#379—61no—
78#379C2×D311yescharacteristic
79C24×C3—21no—
80C24×C3—21no—
81C24×C3—22no—
82C24×C3—21no—
83C24×C3—42no—
84C24×C3—42no—
85C24×C3—42no—
86C24×C3—42no—
87C24×C3—42no—
88C24×C3—42no—
89C24×C3—81no—
90C24×C3—81no—
91C24×C3—81no—
92C24×C3C22⋊C412yes—
93C24×C3C22×C411yescharacteristic
94C24×C3C2×D412yes—
95C24×C3C2×D411yescharacteristic
96C24×C3C2×D411yescharacteristic
97C24×C3#4112yes—
98C2×C3×D4—28no—
99C2×C3×D4—24no—
100C2×C3×D4—24no—
101C2×C3×D4—24no—
102C2×C3×D4—24no—
103C2×C3×D4—24no—
104C2×C3×D4—42no—
105C2×C3×D4—42no—
106C2×C3×D4—44no—
107C2×C3×D4—44no—
108C2×C3×D4—48no—
109C2×C3×D4—42no—
110C2×C3×D4—44no—
111C2×C3×D4—42no—
112C2×C3×D4—44no—
113C2×C3×D4—42no—
114C2×C3×D4—44no—
115C2×C3×D4—41no—
116C2×C3×D4—42no—
117C2×C3×D4—41no—
118C2×C3×D4—44no—
119C2×C3×D4—44no—
120C2×C3×D4—82no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—24no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—42no—
128C22×C3×C4C22⋊C412yes—
129C22×Dic3—24no—
130C22×Dic3—42no—
131C3×C22⋊C4—44no—
132C3×C22⋊C4—42no—
133C3×C22⋊C4—44no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—81no—
136#217—44no—
137#217—44no—
138#217—44no—
139#217—44no—
140#217—82no—
141#217—82no—
142#217—82no—
143#217—82no—
144C4⋊Dic3—44no—
145C4×Dic3—44no—
146C25C2×Dic311yescharacteristic
147C25#6711yescharacteristic
148C25C22×D311yescharacteristic
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—22no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4C2×Dic312yes—
159C22×D4#6712yes—
160C23×C4C2×Dic311yescharacteristic
161C23×C4#6711yescharacteristic
162#120—44no—
163#120—44no—
164#117—124no—
165C2×C4⋊C4—62no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—34no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—62no—
175C2×C22⋊C4—62no—
176C2×C22⋊C4—62no—
177C2×C22⋊C4—62no—
178C2×C22⋊C4C2×Dic312yes—
179C2×C42—62no—
180#99—128no—
181#99—124no—
182#95—22no—
183C23×C3—22no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—24no—
188C23×C3—22no—
189C23×C3—24no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—22no—
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—42no—
211C23×C3—41no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—81no—
215C23×C3—81no—
216C23×C3—82no—
217C23×C3—82no—
218C23×C3—88no—
219C23×C3—88no—
220C23×C3—88no—
221C23×C3C2×C22⋊C411yescharacteristic
222C23×C3C4×D412yes—
223C23×C3#12011yescharacteristic
224C23×C3#12112yes—
225C23×C3#12311yescharacteristic
226C3×D4—48no—
227C3×D4—48no—
228C3×D4—48no—
229C3×D4—44no—
230C3×D4—44no—
231C3×D4—88no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—24no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4#9914yescomplemented
246C2×Dic3—44no—
247C2×Dic3—44no—
248C2×Dic3—44no—
249C2×Dic3—82no—
250C2×Dic3—82no—
251C2×Dic3—82no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—21no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—81no—
263C24—81no—
264C24—81no—
265C24#21712yes—
266C24D3×D412yes—
267C24#23712yes—
268C24C22×Dic311yescharacteristic
269C24C2×#6711yescharacteristic
270C24C2×#6711yescharacteristic
271C2×D4—28no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—48no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—42no—
287C2×D4—44no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—82no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—64no—
302C22×C4—122no—
303C22×C4#21712yes—
304C4⋊C4—124no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—81no—
310C22⋊C4—124no—
311C22⋊C4—124no—
312C22⋊C4—124no—
313C22⋊C4—124no—
314C22⋊C4—242no—
315C22⋊C4—242no—
316C22⋊C4—242no—
317C22⋊C4—242no—
318C42—124no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—21no—
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—42no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—84no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3—84no—
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—24no—
355C3×C4—42no—
356C3×C4—82no—
357Dic3—84no—
358Dic3—162no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—24no—
364C23—22no—
365C23—24no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—42no—
390C23—81no—
391C23—81no—
392C23—82no—
393C23—82no—
394C23—88no—
395C23—88no—
396C23—88no—
397C23D4×Dic312yes—
398C23#93511yescharacteristic
399C23#93711yescharacteristic
400C23#93912yes—
401C23C2×#21711yescharacteristic
402D4—48no—
403D4—48no—
404D4—48no—
405D4—44no—
406D4—44no—
407D4—88no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—124no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4—242no—
425C2×C4—242no—
426C2×C4—242no—
427C2×C4#83414yescomplemented
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—42no—
431C2×C3—42no—
432C2×C3—41no—
433C2×C3—82no—
434C2×C3—82no—
435C2×C3#188911yescharacteristic
436C2×C3#189811yescharacteristic
437C2×C3#191311yescharacteristic
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—22no—
444C22—21no—
445C22—22no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—82no—
460C22—84no—
461C22—82no—
462C22—82no—
463C22—84no—
464C22—84no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—81no—
469C22—81no—
470C22#503212yes—
471C22C2×#83412yes—
472C22#522011yescharacteristic
473C4—24no—
474C4—42no—
475C4—82no—
476C4—244no—
477C4—482no—
478C3#949711yescharacteristic, complemented, Sylow
479C2—22no—
480C2—22no—
481C2—42no—
482C2—42no—
483C2—41no—
484C2—82no—
485C2—82no—
486C2#7166911yescharacteristic
487C2#7167011yescharacteristic
488C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC24×C3C22×C4
FrattiniC25C22×D3
Fitting, prime radicalC3×#2403C2
socleC22×C3#386
cosocleC25×C3C23
hypercenter#2403D3
hypocenterC3#9497
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. #84044C2semidirect
3. #72130C2semidirect
4. #72130C2semidirect
5. C3⋊#1304C2semidirect
10. C2×C3×#120C4semidirect
14. C3×#386C4semidirect
57. C2×#217D4semidirect
67. C2×#120Dic3semidirect
74. #386Dic3semidirect
245. C2×C3×C4#99semidirect
427. C2×C4#834semidirect
478. C3#9497semidirect

Maximal subgroups

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

subgroupindex
C3×#24032
#840442
#721302
C3⋊#13042
#94973