Finite Group Databasebeta

← #10695611

Subgroups — #10695611

Order 768.

930 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×C3×#447C211yesmaximal, characteristic
2#84050C211yesmaximal, characteristic, complemented
3#82630C211yesmaximal, characteristic, complemented
4#72128C211yesmaximal, characteristic, complemented
5#71686C211yesmaximal, characteristic, complemented
6#69946C211yesmaximal, characteristic, complemented
7#67678C211yesmaximal, characteristic, complemented
8#9947—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C22×C3×D8—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#936—21no—
13C2×#924C2211yescharacteristic
14C3×#447—22no—
15C3×#447—22no—
16C2×C3×#102—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5247—22no—
19#5222—22no—
20#5221—21no—
21C2×#834—22no—
22#5218—21no—
23C2×#832—22no—
24#5212C2211yescharacteristic
25C4×#217C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#5041—22no—
28#5040—21no—
29#5034—21no—
30#4971—21no—
31#4467—22no—
32#3187—31no—
33C2×#447D311yescharacteristic
34#2435—31no—
35#1964—31no—
36#1786—31no—
37#1520—31no—
38#1282—31no—
39C25×C3—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4—21no—
46C22×C3×D4—44no—
47C22×C3×D4D411yescharacteristic
48C22×C3×D4D412yes—
49C23×C3×C4C2311yescharacteristic
50C22×Dic6D411yescharacteristic
51C2×C3×D8—24no—
52C2×C3×D8—44no—
53C22×C3×C8—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×#217—24no—
61C2×#217—21no—
62C2×#217—21no—
63C2×#217—24no—
64C2×#217D412yes—
65#936—44no—
66C2×#214—22no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69#924—21no—
70#924—21no—
71#924—21no—
72#924—21no—
73C2×Dic3⋊C4—21no—
74C2×C4×Dic3—21no—
75C22×C3⋊C8D412yes—
76C3×#102—24no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8D412yes—
79#834—44no—
80#832—22no—
81#832—44no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#830—21no—
86#810—22no—
87#808—22no—
88C23×D4C2×D311yescharacteristic
89C22×D8—21no—
90C2×#124—61no—
91C2×C22⋊Q8—31no—
92C2×#121C2×D311yescharacteristic
93#483—61no—
94#481—61no—
95#465—62no—
96#450—61no—
97#448—61no—
98#447—22no—
99#447—22no—
100C2×#102—21no—
101C2×#102—62no—
102C2×#99—62no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8C2×D311yescharacteristic
105#394—62no—
106#390—61no—
107#380—31no—
108#379—62no—
109C4×C22⋊C4—31no—
110#328—62no—
111C24×C3—22no—
112C24×C3—21no—
113C24×C3—44no—
114C24×C3—41no—
115C24×C3—41no—
116C24×C3—42no—
117C24×C3—42no—
118C24×C3—82no—
119C24×C3C2×D411yescharacteristic
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—22no—
127C2×C3×D4—42no—
128C2×C3×D4—42no—
129C2×C3×D4—42no—
130C2×C3×D4—42no—
131C2×C3×D4—44no—
132C2×C3×D4—44no—
133C2×C3×D4—44no—
134C2×C3×D4—41no—
135C2×C3×D4—41no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C22×C3×C4—21no—
141C22×C3×C4—42no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×Dic3—22no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—41no—
152C2×Dic6—21no—
153C2×Dic6—21no—
154C2×Dic6—42no—
155C3×D8—48no—
156C2×C3×C8—22no—
157C2×C3×C8—42no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162#217—22no—
163#217—22no—
164#217—48no—
165#217—41no—
166#217—41no—
167#217—42no—
168#217—84no—
169#214—82no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172C4⋊Dic3—42no—
173Dic3⋊C4—41no—
174Dic3⋊C4—41no—
175C4×Dic3—42no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—42no—
178C25—21no—
179C22×Q8—31no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—44no—
187C22×D4#6711yescharacteristic
188C22×D4#6712yes—
189C23×C4C22×D311yescharacteristic
190C2×SD16—62no—
191C2×D8—24no—
192C2×D8—44no—
193C22×C8—21no—
194C22×C8—32no—
195#124—124no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198C22⋊Q8—61no—
199C22⋊Q8—61no—
200#121—21no—
201#121—21no—
202#121—21no—
203#121—21no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—61no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—32no—
210C2×C22⋊C4—64no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—64no—
214C2×C42—61no—
215#106—62no—
216#103—62no—
217#102—24no—
218#102—62no—
219#102—124no—
220#99—124no—
221C22⋊C8—21no—
222C22⋊C8—61no—
223C22⋊C8—61no—
224C22⋊C8#6712yes—
225#95—61no—
226#95—61no—
227C23×C3—24no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—44no—
231C23×C3—44no—
232C23×C3—44no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—44no—
236C23×C3—41no—
237C23×C3—42no—
238C23×C3—82no—
239C23×C3—82no—
240C23×C3—81no—
241C23×C3—81no—
242C23×C3—88no—
243C23×C3—88no—
244C23×C3—82no—
245C23×C3—81no—
246C23×C3—81no—
247C23×C3#12011yescharacteristic
248C23×C3#12111yescharacteristic
249C23×C3#12711yescharacteristic
250C23×C3#13512yes—
251C23×C3AGL(1,ℤ/8)12yes—
252C3×D4—24no—
253C3×D4—24no—
254C3×D4—42no—
255C3×D4—44no—
256C3×D4—42no—
257C3×D4—42no—
258C3×D4—44no—
259C3×D4—42no—
260C3×D4—82no—
261C3×D4—81no—
262C3×D4—81no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12011yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12111yescharacteristic
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—42no—
282C2×Dic3—81no—
283C2×Dic3—81no—
284C2×Dic3—81no—
285Dic6—42no—
286Dic6—81no—
287C3×C8—42no—
288C3⋊C8—42no—
289C24—22no—
290C24—21no—
291C24—44no—
292C24—41no—
293C24—41no—
294C24—42no—
295C24—42no—
296C24—82no—
297C24D3×D411yescharacteristic
298C2×Q8—61no—
299C2×Q8—61no—
300C2×Q8—122no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—21no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C22×C4—21no—
322C22×C4—42no—
323C22×C4—62no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—121no—
327C22×C4D3×D411yescharacteristic
328C22×C4D3×D411yescharacteristic
329C22×C4#23711yescharacteristic
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332C22×C4C2×#6711yescharacteristic
333SD16—242no—
334D8—48no—
335C2×C8—22no—
336C2×C8—42no—
337C2×C8—62no—
338C2×C8—122no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—61no—
342C4⋊C4—61no—
343C4⋊C4—122no—
344C4⋊C4—121no—
345C4⋊C4—121no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—62no—
349C22⋊C4—62no—
350C22⋊C4—128no—
351C22⋊C4—121no—
352C22⋊C4—121no—
353C22⋊C4—122no—
354C22⋊C4—244no—
355C42—122no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—48no—
365C22×C3—44no—
366C22×C3—84no—
367C22×C3—84no—
368C22×C3—88no—
369C22×C3—84no—
370C22×C3—82no—
371C22×C3—82no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—162no—
376C22×C3#39211yescharacteristic
377C22×C3#44912yes—
378C22×C3#46312yes—
379C22×C3#49511yescharacteristic
380C22×C3#49611yescharacteristic
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—81no—
385C3×C4B(4,𝔽2)12yes—
386Dic3—82no—
387Dic3—81no—
388Dic3—81no—
389Dic3—161no—
390C23—24no—
391C23—41no—
392C23—41no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—42no—
397C23—44no—
398C23—42no—
399C23—41no—
400C23—42no—
401C23—82no—
402C23—82no—
403C23—81no—
404C23—81no—
405C23—88no—
406C23—88no—
407C23—82no—
408C23—81no—
409C23—81no—
410C23#91112yes—
411C23#91212yes—
412C23#93711yescharacteristic
413C23#93911yescharacteristic
414C23#94011yescharacteristic
415Q8—122no—
416Q8—241no—
417D4—24no—
418D4—24no—
419D4—42no—
420D4—44no—
421D4—42no—
422D4—42no—
423D4—44no—
424D4—42no—
425D4—81no—
426D4—81no—
427D4—82no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—241no—
444C2×C4—241no—
445C2×C4—241no—
446C2×C4#93711yescharacteristic
447C2×C4#93811yescharacteristic
448C2×C4#93911yescharacteristic
449C2×C4#95311yescharacteristic
450C8—42no—
451C8—122no—
452C2×C3—22no—
453C2×C3—21no—
454C2×C3—22no—
455C2×C3—21no—
456C2×C3—84no—
457C2×C3—161no—
458C2×C3#200811yescharacteristic
459C2×C3#201211yescharacteristic
460C2×C3#202111yescharacteristic
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—42no—
468C22—42no—
469C22—48no—
470C22—44no—
471C22—84no—
472C22—84no—
473C22—88no—
474C22—84no—
475C22—82no—
476C22—82no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—162no—
481C22#515312yes—
482C22#515411yescharacteristic
483C22#515511yescharacteristic
484C22#515812yes—
485C22#522311yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—81no—
490C4—242no—
491C4—241no—
492C4—241no—
493C4—481no—
494C4#503212yes—
495C3#994711yescharacteristic, complemented, Sylow
496C2—22no—
497C2—21no—
498C2—22no—
499C2—21no—
500C2—84no—
501C2—161no—
502C2#7167111yescharacteristic
503C2#7204511yescharacteristic
504C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC2×C3×#447C2
socleC22×C3#392
hypercenterC2×#447D3
hypocenterC3#9947
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#447

Complements

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

NH ≅ G/Ntype
2. #84050C2semidirect
3. #82630C2semidirect
4. #72128C2semidirect
5. #71686C2semidirect
6. #69946C2semidirect
7. #67678C2semidirect
495. C3#9947semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4472
#840502
#826302
#721282
#716862
#699462
#676782
#99473