Finite Group Databasebeta

← #10691757

Subgroups — #10691757

Order 768.

648 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 107 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#82625C211yesmaximal, characteristic
2#73338C211yesmaximal, characteristic, complemented
3C2×C3×#327C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic, complemented
5#69944C211yesmaximal, characteristic, complemented
6#67677C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9277—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×C4×#67—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C2×C3×#95C2211yescharacteristic
14C2×#834—21no—
15C2×#834—21no—
16#5217C412yescomplemented
17#5213—21no—
18#5212C2211yescharacteristic
19C4×#217C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#5038—22no—
22#5036—21no—
23#5033—21no—
24#4967C412yescomplemented
25#4669—21no—
26#4666—21no—
27#4665—21no—
28#4654—21no—
29C3×#327—21no—
30C3×#327—21no—
31#4470—22no—
32#3186—31no—
33#2364—31no—
34#1964—31no—
35#1519—31no—
36C2×#327D311yescharacteristic, complemented
37#1294—31no—
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4C2×C411yescharacteristic, complemented
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C22×C3×C8—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
52C2×C3×C22⋊C4—21no—
53C2×#217—21no—
54C2×#217D411yescharacteristic
55C2×#217D411yescharacteristic
56C2×#213—22no—
57#930—21no—
58#930—21no—
59#930C2×C412yes—
60#930D412yes—
61C4×#67—42no—
62C4×#67—42no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×Dic3⋊C4—21no—
70C2×C4×Dic3—21no—
71C2×C4×Dic3—21no—
72C22×C3⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×#95—21no—
76C3×#95D411yescharacteristic
77C3×#95D411yescharacteristic
78#834—42no—
79#834—42no—
80#832—22no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85#830—21no—
86#830C2×C412yescomplemented
87#809—22no—
88#807—22no—
89C2×#121—31no—
90C2×#121C2×D311yescharacteristic
91C2×C4×D4—61no—
92#480—32no—
93#480—61no—
94#466—62no—
95#447—32no—
96#447—61no—
97C2×#99—61no—
98C2×#99—61no—
99C2×C22⋊C8—31no—
100C2×C22⋊C8C2×D311yescharacteristic
101#388—61no—
102#387—61no—
103#386—61no—
104#386—61no—
105#380—31no—
106C4×C22⋊C4—31no—
107C4×C22⋊C4—61no—
108C2×#95C2×D311yescharacteristic
109#327—21no—
110#327—21no—
111#327—62no—
112C24×C3—21no—
113C24×C3C2×D411yescharacteristic
114C23×D3—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—81no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—42no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—81no—
131C22×C3×C4—81no—
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×#67—22no—
139C2×#67—22no—
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—41no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—41no—
148C2×D12—21no—
149C2×D12—81no—
150C2×D12C22⋊C412yes—
151C2×C4×D3—42no—
152C2×C4×D3—42no—
153C2×C4×D3—81no—
154C2×C3×C8—41no—
155C2×C3×C8—81no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160#217—21no—
161#217—21no—
162#217—21no—
163#217—41no—
164#217—41no—
165#217D812yescomplemented
166#213—82no—
167(C2×D3)⋊C4—22no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—42no—
170(C2×D3)⋊C4—42no—
171(C2×D3)⋊C4—81no—
172C4⋊Dic3—21no—
173C4⋊Dic3—81no—
174C4⋊Dic3C22⋊C412yes—
175Dic3⋊C4—42no—
176C4×Dic3—42no—
177C4×Dic3—81no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—81no—
180C22×D4—21no—
181C22×D4—31no—
182C22×D4—61no—
183C22×D4C4×D311yescharacteristic, complemented
184C23×C4—21no—
185C23×C4—61no—
186C23×C4C22×D311yescharacteristic
187C2×D8—62no—
188C22×C8—21no—
189C22×C8—61no—
190#121—21no—
191#121—21no—
192#121—21no—
193#121—21no—
194#121—32no—
195#121—32no—
196#121—61no—
197#121—61no—
198C4×D4—122no—
199C4×D4—122no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4C4×D311yescharacteristic, complemented
204C2×C22⋊C4—21no—
205C2×C22⋊C4—31no—
206C2×C22⋊C4—31no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C42—61no—
213C2×C42—61no—
214#107—62no—
215#102—62no—
216#102—62no—
217#99—122no—
218#99—122no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—32no—
222C22⋊C8—61no—
223#95—21no—
224#95—61no—
225#95—61no—
226#95—61no—
227#95—61no—
228#95#6711yescharacteristic
229#95#6711yescharacteristic
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—81no—
236C23×C3—81no—
237C23×C3C4×D411yescharacteristic
238C23×C3#12011yescharacteristic
239C23×C3#12111yescharacteristic
240C23×C3C2×D811yescharacteristic
241C23×C3#13511yescharacteristic
242C23×C3AGL(1,ℤ/8)11yescharacteristic
243C23×C3AGL(1,ℤ/8)11yescharacteristic
244C22×D3—22no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3—41no—
250C22×D3—41no—
251C3×D4—42no—
252C3×D4—81no—
253C3×D4—81no—
254C3×D4—81no—
255C3×D4—81no—
256C3×D4—81no—
257C2×C3×C4—22no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—82no—
273C2×C3×C4—82no—
274C2×C3×C4C2×C22⋊C411yescharacteristic
275C2×C3×C4C4×D411yescharacteristic
276C2×C3×C4#12111yescharacteristic
277C2×C3×C4#12311yescharacteristic
278#67—42no—
279#67—42no—
280#67—42no—
281#67—42no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—41no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—81no—
292C2×Dic3—81no—
293C2×Dic3—81no—
294D12—42no—
295D12—81no—
296C4×D3—84no—
297C3×C8—81no—
298C3⋊C8—42no—
299C24—21no—
300C24—61no—
301C24D3×D411yescharacteristic
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—32no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—62no—
312C2×D4—62no—
313C2×D4—61no—
314C2×D4—81no—
315C2×D4—121no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—121no—
319C2×D4—241no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—61no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—122no—
332C22×C4—122no—
333C22×C4—121no—
334C22×C4—241no—
335C22×C4C2×C4×D311yescharacteristic
336C22×C4#23511yescharacteristic
337C22×C4D3×D411yescharacteristic
338C22×C4D3×D411yescharacteristic
339C22×C4#23711yescharacteristic
340C22×C4C2×#6711yescharacteristic
341D8—242no—
342C2×C8—41no—
343C2×C8—62no—
344C2×C8—81no—
345C2×C8—241no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—32no—
349C4⋊C4—61no—
350C4⋊C4—122no—
351C4⋊C4—241no—
352C22⋊C4—32no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—61no—
356C22⋊C4—62no—
357C22⋊C4—62no—
358C22⋊C4—61no—
359C22⋊C4—61no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—241no—
365C42—122no—
366C42—241no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—81no—
378C22×C3—81no—
379C22×C3—81no—
380C22×C3—161no—
381C22×C3—161no—
382C22×C3#38611yescharacteristic
383C22×C3C4×D811yescharacteristic
384C22×C3#44211yescharacteristic
385C22×C3#44711yescharacteristic
386C22×C3#44911yescharacteristic
387C22×C3#45911yescharacteristic
388C22×C3#46311yescharacteristic
389C2×D3—42no—
390C2×D3—42no—
391C2×D3—42no—
392C2×D3—81no—
393C2×D3—81no—
394C2×D3—81no—
395C2×D3—81no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—42no—
401C3×C4—81no—
402C3×C4—81no—
403Dic3—42no—
404Dic3—42no—
405Dic3—81no—
406Dic3—161no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—62no—
413C23—81no—
414C23—81no—
415C23—121no—
416C23—121no—
417C23—121no—
418C23—121no—
419C23—121no—
420C23—121no—
421C23#67⋊C411yescharacteristic
422C23#88411yescharacteristic
423C23D3×D811yescharacteristic
424C23#91111yescharacteristic
425C23#91411yescharacteristic
426C23#91611yescharacteristic
427C23#93711yescharacteristic
428D4—42no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—122no—
435D4—122no—
436D4—122no—
437D4—122no—
438D4—122no—
439D4—241no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—244no—
467C2×C4—241no—
468C2×C4—241no—
469C2×C4—241no—
470C2×C4D3×C22⋊C411yescharacteristic
471C2×C4#88311yescharacteristic
472C2×C4C4×#6711yescharacteristic
473C2×C4#93811yescharacteristic
474C8—81no—
475C8—122no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—161no—
483C2×C3#187911yescharacteristic
484C2×C3#190611yescharacteristic
485C2×C3#191311yescharacteristic
486D3—82no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—122no—
501C22—122no—
502C22—122no—
503C22—161no—
504C22—161no—
505C22—241no—
506C22—241no—
507C22—241no—
508C22—241no—
509C22#475611yescharacteristic
510C22#478511yescharacteristic
511C22#478611yescharacteristic
512C22#478711yescharacteristic
513C22#495611yescharacteristic
514C22#515611yescharacteristic
515C22#517211yescharacteristic
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—81no—
522C4—81no—
523C4—122no—
524C4—122no—
525C4—241no—
526C4—481no—
527C3#927711yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—161no—
535C2—242no—
536C2#7003411yescharacteristic
537C2#7020811yescharacteristic
538C2#7039111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC22×C3#386
prime radical#82625C2
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#9277
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. #73338C2semidirect
3. C2×C3×#327C2semidirect
4. #71686C2semidirect
5. #69944C2semidirect
6. #67677C2semidirect
7. #67612C2semidirect
16. #5217C4semidirect
24. #4967C4semidirect
36. C2×#327D3semidirect
40. C22×C3×D4C2×C4semidirect
51. C2×C3×C4⋊C4C2×C4semidirect
86. #830C2×C4semidirect
165. #217D8semidirect
183. C22×D4C4×D3semidirect
203. C2×C4⋊C4C4×D3semidirect
527. C3#9277semidirect

Maximal subgroups

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

subgroupindex
#826252
#733382
C2×C3×#3272
#716862
#699442
#676772
#676122
#92773