Finite Group Databasebeta

← #10823183

Subgroups — #10823183

Order 768.

1410 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#942C211yesmaximal, characteristic
2C3×#2310C211yesmaximal, characteristic, complemented
3C22×#835C211yesmaximal, characteristic, complemented
4#82270C211yesmaximal, characteristic
5C2×#5002C211yesmaximal, characteristic
6C2×#4999C211yesmaximal, characteristic, complemented
7#71581C211yesmaximal, characteristic, complemented
8#69681C214yesmaximal, complemented
9#69680C214yesmaximal, complemented
10#23545—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×#216C2212yes—
16C2×#942—24no—
17C2×#942—24no—
18C2×#942—24no—
19C22×#215C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×#208C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23C3×#391—22no—
24C3×#391C2214yes—
25C3×#381—22no—
26C2×#835—24no—
27C2×#835C2214yes—
28#5002—24no—
29#4999—24no—
30#4987—22no—
31C2×#810C2212yes—
32C2×#809C2212yes—
33#4958—22no—
34#4939C2212yes—
35#4929C2212yes—
36C2×C4⋊(C3⋊C8)C2214yes—
37#4529C2214yes—
38C22×#137—31no—
39C2×#424—31no—
40C2×#416—31no—
41C22×#103—31no—
42#2311—31no—
43#2310D311yescharacteristic, complemented
44#2056—34no—
45#2055—34no—
46#1736—31no—
47C22×C3×Q8—21no—
48C22×C3×Q8—22no—
49C22×C3×Q8—22no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8D414yes—
54C22×C3×Q8C2312yes—
55C23×C3×C4—21no—
56C23×C3×C4C2311yescharacteristic
57C22×#67—21no—
58C23×Dic3—21no—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235—22no—
62C2×#235—22no—
63C2×#235—22no—
64C2×#235D414yes—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6C2311yescharacteristic
68C3×#117—22no—
69C3×#117—22no—
70C3×#117—22no—
71C3×#117—22no—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42—21no—
77C2×C3×C42C2312yes—
78C2×#216—24no—
79C2×#216—24no—
80C2×#216—24no—
81#942—232no—
82C2×#215—24no—
83C2×#215—24no—
84C2×#215—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90C2×#208—22no—
91C2×#208—22no—
92C2×#208D414yes—
93C22×C3⋊C8—22no—
94C22×C3⋊C8—22no—
95C22×C3⋊C8—22no—
96C22×C3⋊C8C2312yes—
97C22×C3⋊C8C2312yes—
98C3×#95—28no—
99#835—28no—
100C3⋊#95—24no—
101#810—24no—
102#809—24no—
103C4⋊(C3⋊C8)—28no—
104C22×#41—31no—
105C23×Q8C2×D311yescharacteristic
106C2×#137—64no—
107C2×#137—64no—
108C2×#137—64no—
109C22×Q16—32no—
110C22×SD16—32no—
111C22×M16—31no—
112C23×C8—31no—
113C2×#117C2×D311yescharacteristic
114C2×#117C2×D311yescharacteristic
115C22×C4⋊C4—31no—
116#424—64no—
117C2×C4⋊C8—34no—
118#416—64no—
119C2×#103—34no—
120C2×#103—32no—
121C2×#103—64no—
122C2×#102—32no—
123#391—22no—
124#391—32no—
125#391C2×D314yes—
126#390—32no—
127#386—62no—
128#381—22no—
129#380—62no—
130#340—34no—
131C24×C3C2×D411yescharacteristic
132C23×D3—21no—
133C2×C3×Q8—24no—
134C2×C3×Q8—22no—
135C2×C3×Q8—28no—
136C2×C3×Q8—28no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—42no—
142C2×C3×Q8—42no—
143C2×C3×Q8—44no—
144C2×C3×Q8—44no—
145C2×C3×Q8C2×D414yes—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D414yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4#4112yes—
157C22×C3×C4#4112yes—
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C2×#67—24no—
161C2×#67—42no—
162C2×#67—42no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3—42no—
166C22×Dic3—82no—
167#235—28no—
168#235—28no—
169#235—44no—
170#235—44no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12C2×D412yes—
177C2×C4×D3—24no—
178C2×C4×D3—42no—
179C2×C4×D3—42no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6C2×D412yes—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—42no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—42no—
194#216—216no—
195#215—216no—
196(C2×D3)⋊C4—84no—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199C4⋊Dic3—42no—
200#208—28no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—44no—
207C2×C3⋊C8C2×D414yes—
208C2×#41—34no—
209C2×#41—62no—
210C2×#41—62no—
211C2×#41—62no—
212C2×#41—62no—
213C2×#41—62no—
214C22×Q8—22no—
215C22×Q8—21no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—31no—
221C22×Q8D1214yes—
222C22×Q8C22×D312yes—
223C22×D4—31no—
224C22×D4—61no—
225C23×C4—21no—
226C23×C4—61no—
227C23×C4—61no—
228C23×C4C22×D311yescharacteristic
229#137—632no—
230C2×Q16—64no—
231C2×Q16—64no—
232C2×Q16—64no—
233C2×SD16—64no—
234C2×SD16—64no—
235C2×SD16—64no—
236C2×M16—34no—
237C2×M16—62no—
238C2×M16—62no—
239C22×C8—32no—
240C22×C8—32no—
241C22×C8—62no—
242C22×C8—62no—
243C22×C8—62no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4#6712yes—
253C2×C4⋊C4C22×D312yes—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—64no—
256C2×C22⋊C4#6712yes—
257C2×C42—21no—
258C2×C42C22×D312yes—
259C4⋊C8—68no—
260#103—64no—
261#103—68no—
262#102—64no—
263#95—28no—
264#95—64no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3#12112yes—
268C23×C3#12112yes—
269C23×C3#12112yes—
270C23×C3C22×D411yescharacteristic
271C22×D3—22no—
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—42no—
275C22×D3—42no—
276C3×Q8—24no—
277C3×Q8—24no—
278C3×Q8—42no—
279C3×Q8—42no—
280C3×Q8—48no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—48no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4#12014yes—
292C2×C3×C4#12112yes—
293C2×C3×C4#12112yes—
294C2×C3×C4#12112yes—
295C2×C3×C4#12314yes—
296C2×C3×C4#12314yes—
297C2×C3×C4#12414yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304#67—48no—
305C2×Dic3—22no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—44no—
310C2×Dic3—84no—
311D12—22no—
312D12—22no—
313D12—41no—
314D12—41no—
315C4×D3—48no—
316Dic6—22no—
317Dic6—22no—
318Dic6—41no—
319Dic6—41no—
320C3⋊C8—24no—
321C3⋊C8—42no—
322C24—61no—
323C24C2×#6711yescharacteristic
324#41—68no—
325#41—68no—
326#41—124no—
327#41—124no—
328C2×Q8—24no—
329C2×Q8—22no—
330C2×Q8—28no—
331C2×Q8—28no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—32no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8C2×D1214yes—
347C2×D4—32no—
348C2×D4—64no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—61no—
353C2×D4—61no—
354C2×D4—122no—
355C2×D4—122no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—61no—
362C22×C4—64no—
363C22×C4—62no—
364C22×C4—122no—
365C22×C4—122no—
366C22×C4—122no—
367C22×C4—242no—
368C22×C4C2×D1212yes—
369C22×C4#23512yes—
370C22×C4D3×D412yes—
371C22×C4D3×D412yes—
372C22×C4#23712yes—
373C22×C4#23712yes—
374C22×C4C2×#6714yes—
375C22×C4C2×#6711yescharacteristic
376C22×C4C23×D311yescharacteristic
377Q16—616no—
378SD16—616no—
379M16—68no—
380C2×C8—34no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—62no—
385C2×C8—62no—
386C2×C8—124no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—122no—
393C22⋊C4—24no—
394C22⋊C4—42no—
395C22⋊C4—244no—
396C42—22no—
397C42—22no—
398C42—42no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3#39414yes—
404C22×C3C2×#12111yescharacteristic
405C22×C3C2×#12111yescharacteristic
406C22×C3C2×#12111yescharacteristic
407C22×C3#57712yes—
408C22×C3#57812yes—
409C2×D3—42no—
410C2×D3—44no—
411C2×D3—81no—
412C2×D3—81no—
413C2×D3—82no—
414C3×C4—44no—
415C3×C4—44no—
416C3×C4—82no—
417C3×C4#39414yes—
418C3×C4C2×#12011yescharacteristic
419C3×C4C2×#12311yescharacteristic
420C3×C4C2×#12311yescharacteristic
421C3×C4C2×#12411yescharacteristic
422Dic3—42no—
423Dic3—82no—
424C23—22no—
425C23—22no—
426C23—62no—
427C23—121no—
428C23—121no—
429C23—122no—
430C23—122no—
431C23#93012yes—
432C23#93812yes—
433C23#93912yes—
434C23C22×#6711yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—48no—
438Q8—42no—
439Q8—42no—
440Q8—62no—
441Q8—62no—
442Q8—121no—
443Q8—121no—
444D4—62no—
445D4—62no—
446D4—128no—
447D4—121no—
448D4—121no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—62no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—128no—
463C2×C4—122no—
464C2×C4—124no—
465C2×C4—244no—
466C2×C4#88214yes—
467C2×C4#88314yes—
468C2×C4#88514yes—
469C2×C4#88614yes—
470C2×C4#93012yes—
471C2×C4#93812yes—
472C2×C4#93912yes—
473C2×C4C22×D1211yescharacteristic
474C2×C4C2×#23511yescharacteristic
475C2×C4C2×D3×D411yescharacteristic
476C2×C4C2×D3×D411yescharacteristic
477C2×C4C2×#23711yescharacteristic
478C2×C4C2×#23711yescharacteristic
479C8—64no—
480C8—122no—
481C2×C3—24no—
482C2×C3C2×#39411yescharacteristic
483C2×C3#304311yescharacteristic
484C2×C3#304411yescharacteristic
485C2×C3#304612yes—
486C2×C3#306312yes—
487D3—82no—
488C22—22no—
489C22—22no—
490C22—24no—
491C22—24no—
492C22—122no—
493C22—124no—
494C22—241no—
495C22—241no—
496C22—242no—
497C22#496114yes—
498C22C2×#93011yescharacteristic
499C22C2×#93811yescharacteristic
500C22C2×#93911yescharacteristic
501C22#583712yes—
502C22#583812yes—
503C4—44no—
504C4—44no—
505C4—82no—
506C4—122no—
507C4—242no—
508C4#496114yes—
509C4C2×#88211yescharacteristic
510C4C2×#88311yescharacteristic
511C4C2×#88511yescharacteristic
512C4C2×#88611yescharacteristic
513C3#2354511yescharacteristic, complemented, Sylow
514C2—24no—
515C2—242no—
516C2C2×#496111yescharacteristic
517C2#8255111yescharacteristic
518C2#8255411yescharacteristic
519C2#8340012yes—
520C2#8340612yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2310C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#2310D3
hypocenterC3#23545
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C3×#2310C2semidirect
3. C22×#835C2semidirect
6. C2×#4999C2semidirect
7. #71581C2semidirect
8. #69681C2semidirect
9. #69680C2semidirect
43. #2310D3semidirect
513. C3#23545semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
C3×#23102
C22×#8352
#822702
C2×#50022
C2×#49992
#715812
#696812
#696802
#235453