Finite Group Databasebeta

← #10695664

Subgroups — #10695664

Order 768.

1018 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 103 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#84049C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic, complemented
3#78612C211yesmaximal, characteristic, complemented
4C3×#1786C211yesmaximal, characteristic, complemented
5#69529C211yesmaximal, characteristic, complemented
6#69509C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9954—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×#953—22no—
11C2×#940—21no—
12C22×#214—21no—
13C2×#930C2211yescharacteristic
14C2×#924C2211yescharacteristic
15C2×C3×#102—22no—
16C2×C3×C22⋊C8C2211yescharacteristic
17C3×#380C2211yescharacteristic
18#5223—22no—
19#5221—21no—
20#5218—22no—
21#5218—22no—
22C2×#832—21no—
23C4×#217C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5114—22no—
26#4738—21no—
27#4735—21no—
28#4731—21no—
29#4721—21no—
30#4527—22no—
31#4451—22no—
32#3187—31no—
33C2×#450—31no—
34#2433—31no—
35#1996—31no—
36#1786D311yescharacteristic, complemented
37#1519—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—44no—
46C22×C3×D4D412yes—
47C23×C3×C4C2311yescharacteristic
48C22×#67—21no—
49C22×#67—24no—
50C22×D12D411yescharacteristic
51C22×Dic6D411yescharacteristic, complemented
52C22×C3×C8D412yescomplemented
53C2×C3×C4⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55#953—44no—
56#953—44no—
57C2×#217—24no—
58C2×#217—21no—
59C2×#217D412yes—
60#940—44no—
61C2×#214—24no—
62C2×#214—44no—
63#930—21no—
64#930—21no—
65#930—21no—
66#930—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic, complemented
70#924—21no—
71#924—21no—
72#924—21no—
73#924—21no—
74C2×Dic3⋊C4—21no—
75C2×C4×Dic3—21no—
76C22×C3⋊C8—21no—
77C2×#204—22no—
78C3×#102—44no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×#95—21no—
82#832—24no—
83C3⋊#95—21no—
84#830—21no—
85#830D412yescomplemented
86#821—22no—
87#817—22no—
88#816—22no—
89C23×D4C2×D311yescharacteristic
90C22×SD16—61no—
91C2×#127—61no—
92C2×C22⋊Q8—31no—
93C2×#121—31no—
94C2×#120—62no—
95#483—61no—
96#481—61no—
97#465—62no—
98#450—61no—
99#450—62no—
100#450—62no—
101#448—61no—
102C2×#102—22no—
103C2×#102—61no—
104C2×C22⋊C8—31no—
105C2×C22⋊C8C2×D311yescharacteristic
106#392—62no—
107#390—61no—
108#380C2×D311yescharacteristic
109C4×C22⋊C4—31no—
110#342—62no—
111#327—62no—
112C24×C3—22no—
113C24×C3—44no—
114C24×C3—41no—
115C24×C3—41no—
116C24×C3—42no—
117C24×C3—42no—
118C24×C3—82no—
119C24×C3C2×D411yescharacteristic
120C23×D3—21no—
121C2×C3×D4—24no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4—42no—
129C2×C3×D4—44no—
130C2×C3×D4—44no—
131C2×C3×D4—44no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C22×C3×C4—42no—
135C22×C3×C4—41no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C2×#67—48no—
143C2×#67—48no—
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—42no—
147C2×#67—42no—
148C2×#67—84no—
149C22×Dic3—22no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—42no—
156C2×Dic6—21no—
157C2×Dic6—21no—
158C2×Dic6—42no—
159C2×C3×C8—22no—
160C2×C3×C8—42no—
161C3×C4⋊C4—42no—
162C3×C22⋊C4—42no—
163#217—22no—
164#217—48no—
165#217—41no—
166#217—41no—
167#217SD1614yescomplemented
168#214—48no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—41no—
171C4⋊Dic3—21no—
172C4⋊Dic3—21no—
173C4⋊Dic3—21no—
174C4⋊Dic3—21no—
175Dic3⋊C4—41no—
176Dic3⋊C4—41no—
177C4×Dic3—42no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—42no—
180#204—82no—
181C25—21no—
182C22×Q8—31no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—31no—
189C22×D4—44no—
190C22×D4—61no—
191C22×D4—64no—
192C22×D4D1212yes—
193C23×C4C22×D311yescharacteristic
194C2×SD16—62no—
195C2×SD16—64no—
196C2×SD16—124no—
197C22×C8—61no—
198C22×C8#6712yescomplemented
199#127—124no—
200C22⋊Q8—61no—
201C22⋊Q8—61no—
202C22⋊Q8—61no—
203C22⋊Q8—61no—
204#121—61no—
205#121—61no—
206#121—61no—
207#121—61no—
208#120—124no—
209#120—124no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—31no—
213C2×C4⋊C4—61no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—32no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—61no—
218C2×C22⋊C4—61no—
219C2×C42—61no—
220#106—62no—
221#103—62no—
222#102—44no—
223#102—62no—
224#102—64no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—32no—
228C22⋊C8—61no—
229#95—21no—
230#95—61no—
231C23×C3—24no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—44no—
235C23×C3—44no—
236C23×C3—44no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—44no—
240C23×C3—82no—
241C23×C3—82no—
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×C3C2×SD1612yes—
251C23×C3AGL(1,ℤ/8)12yes—
252C22×D3—42no—
253C22×D3—42no—
254C22×D3—41no—
255C22×D3—42no—
256C3×D4—24no—
257C3×D4—24no—
258C3×D4—42no—
259C3×D4—44no—
260C3×D4—42no—
261C3×D4—44no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—81no—
269C2×C3×C4—81no—
270C2×C3×C4—81no—
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12011yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12111yescharacteristic
275#67—816no—
276#67—82no—
277#67—81no—
278#67—81no—
279C2×Dic3—24no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—42no—
286C2×Dic3—81no—
287C2×Dic3—81no—
288D12—42no—
289D12—81no—
290Dic6—42no—
291Dic6—42no—
292C3×C8—42no—
293C3⋊C8—42no—
294C24—22no—
295C24—44no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24—61no—
301C24—82no—
302C24D3×D411yescharacteristic
303C2×Q8—61no—
304C2×Q8—61no—
305C2×Q8—122no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—122no—
322C2×D4—128no—
323C2×D4—128no—
324C2×D4—121no—
325C2×D4—122no—
326C2×D4—122no—
327C2×D4—121no—
328C2×D4—244no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—62no—
332C22×C4—61no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4C2×D1211yescharacteristic
336C22×C4#23511yescharacteristic
337C22×C4D3×D411yescharacteristic
338C22×C4D3×D411yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4C2×#6711yescharacteristic
341SD16—128no—
342SD16—242no—
343C2×C8—22no—
344C2×C8—42no—
345C2×C8—62no—
346C2×C8—122no—
347C4⋊C4—42no—
348C4⋊C4—61no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—121no—
353C4⋊C4—121no—
354C22⋊C4—34no—
355C22⋊C4—42no—
356C22⋊C4—62no—
357C22⋊C4—128no—
358C22⋊C4—121no—
359C22⋊C4—121no—
360C22⋊C4—121no—
361C22⋊C4—121no—
362C42—122no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—48no—
372C22×C3—44no—
373C22×C3—84no—
374C22×C3—84no—
375C22×C3—88no—
376C22×C3—84no—
377C22×C3—82no—
378C22×C3—82no—
379C22×C3#39211yescharacteristic
380C22×C3#45012yes—
381C22×C3#46012yes—
382C22×C3#49211yescharacteristic
383C22×C3#49611yescharacteristic
384C2×D3—84no—
385C2×D3—81no—
386C2×D3—81no—
387C2×D3—81no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—161no—
392C3×C4B(4,𝔽2)12yes—
393Dic3—44no—
394Dic3—81no—
395Dic3—81no—
396Dic3—81no—
397C23—24no—
398C23—41no—
399C23—41no—
400C23—44no—
401C23—44no—
402C23—42no—
403C23—44no—
404C23—42no—
405C23—44no—
406C23—82no—
407C23—82no—
408C23—88no—
409C23—88no—
410C23—81no—
411C23—81no—
412C23—82no—
413C23—122no—
414C23—122no—
415C23—121no—
416C23—122no—
417C23#88211yescharacteristic
418C23#88411yescharacteristic
419C23#91112yes—
420C23D3×SD1612yes—
421C23#94011yescharacteristic
422Q8—122no—
423Q8—122no—
424D4—24no—
425D4—24no—
426D4—42no—
427D4—44no—
428D4—42no—
429D4—44no—
430D4—122no—
431D4—241no—
432D4—2416no—
433D4—241no—
434D4—241no—
435D4—242no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—64no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—122no—
447C2×C4—122no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—122no—
452C2×C4—241no—
453C2×C4—241no—
454C2×C4#88211yescharacteristic
455C2×C4#88411yescharacteristic
456C2×C4#93011yescharacteristic
457C2×C4#93711yescharacteristic
458C8—42no—
459C8—122no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—22no—
463C2×C3—21no—
464C2×C3—84no—
465C2×C3#200811yescharacteristic
466C2×C3#201411yescharacteristic
467C2×C3#202111yescharacteristic
468D3—161no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—42no—
476C22—42no—
477C22—48no—
478C22—44no—
479C22—84no—
480C22—84no—
481C22—84no—
482C22—88no—
483C22—82no—
484C22—82no—
485C22—244no—
486C22—241no—
487C22—241no—
488C22—241no—
489C22#476212yes—
490C22#477612yes—
491C22#496011yescharacteristic
492C22#515411yescharacteristic
493C22#517511yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—124no—
498C4—161no—
499C4—241no—
500C4—241no—
501C4—241no—
502C4#503212yes—
503C3#995411yescharacteristic, complemented, Sylow
504C2—22no—
505C2—21no—
506C2—21no—
507C2—22no—
508C2—84no—
509C2—481no—
510C2#6950011yescharacteristic
511C2#7021011yescharacteristic
512C2#7033111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1786C2
socleC22×C3#392
prime radical#84049C2
hypercenter#1786D3
hypocenterC3#9954
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. C2×#5218C2semidirect
3. #78612C2semidirect
4. C3×#1786C2semidirect
5. #69529C2semidirect
6. #69509C2semidirect
7. #67677C2semidirect
36. #1786D3semidirect
51. C22×Dic6D4semidirect
52. C22×C3×C8D4semidirect
69. C2×C4⋊Dic3D4semidirect
85. #830D4semidirect
167. #217SD16semidirect
198. C22×C8#67semidirect
503. C3#9954semidirect

Maximal subgroups

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

subgroupindex
#840492
C2×#52182
#786122
C3×#17862
#695292
#695092
#676772
#99543