Finite Group Databasebeta

← #10692923

Subgroups — #10692923

Order 768.

1034 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 97 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#82653C211yesmaximal, characteristic
2#74119C211yesmaximal, characteristic, complemented
3C3×#1510C211yesmaximal, characteristic, complemented
4#71674C211yesmaximal, characteristic, complemented
5#69988C211yesmaximal, characteristic, complemented
6#67682C211yesmaximal, characteristic, complemented
7#67628C211yesmaximal, characteristic, complemented
8#9463—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#940C412yescomplemented
12C2×#937—21no—
13C2×#937—21no—
14C2×D3×C22⋊C4—21no—
15C3×#407C2211yescharacteristic
16C3×#381C2211yescharacteristic
17#5211C2211yescharacteristic
18C4×#217C2211yescharacteristic
19#5207C2211yescharacteristic
20C2×#823—22no—
21#4988C412yescomplemented
22#4960—22no—
23#4955—21no—
24#4665—22no—
25#4665—22no—
26C3×#327—22no—
27#4537—22no—
28#2433—31no—
29#2308—31no—
30#1783—31no—
31#1510—31no—
32#1510D311yescharacteristic, complemented
33#1284—31no—
34#1284—31no—
35C24×D3D411yescharacteristic
36C24×D3D411yescharacteristic
37C22×C3×D4—21no—
38C22×C3×D4C2×C411yescharacteristic, complemented
39C23×C3×C4C2311yescharacteristic
40C22×#67—22no—
41C22×#67—22no—
42C22×D12—21no—
43C22×D12—21no—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—44no—
47C22×D12C2×C412yes—
48C22×D12D412yes—
49C22×C4×D3—22no—
50C2×C3×M16—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×#217—21no—
59C2×#217—21no—
60#940—44no—
61#937—44no—
62#937—44no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4×Dic3C2×C412yescomplemented
69C2×#208—21no—
70D3×C22⋊C4—44no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#95—21no—
74C3×#95D412yescomplemented
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95D412yescomplemented
78#830—21no—
79#830—21no—
80#823—44no—
81C23×D4—31no—
82C2×#127—32no—
83C2×#121C2×D311yescharacteristic
84C2×#120—61no—
85C2×#120—61no—
86C22×C22⋊C4—61no—
87C2×#100—62no—
88#407—31no—
89#407C2×D311yescharacteristic
90#392—62no—
91#390—32no—
92#386—62no—
93#386—62no—
94#381—31no—
95#381C2×D311yescharacteristic
96#379—61no—
97C4×C22⋊C4—31no—
98#327—22no—
99#327—62no—
100C24×C3—21no—
101C24×C3C2×D411yescharacteristic
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—24no—
105C23×D3—21no—
106C23×D3—21no—
107C23×D3—42no—
108C23×D3—42no—
109C23×D3—42no—
110C23×D3—81no—
111C23×D3—81no—
112C23×D3C22⋊C412yes—
113C23×D3C22⋊C412yes—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—42no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C22×C3×C4—21no—
122C22×C3×C4—22no—
123C22×C3×C4—42no—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C2×#67—44no—
131C2×#67—44no—
132C2×#67—44no—
133C2×#67—44no—
134C2×#67—82no—
135C2×#67—82no—
136C22×Dic3—22no—
137C22×Dic3—22no—
138C2×D12—24no—
139C2×D12—24no—
140C2×D12—22no—
141C2×D12—22no—
142C2×D12—22no—
143C2×D12—22no—
144C2×D12—42no—
145C2×D12—42no—
146C2×D12—42no—
147C2×D12—42no—
148C2×D12—42no—
149C2×D12—42no—
150C2×D12—44no—
151C2×D12—44no—
152C2×C4×D3—44no—
153C2×C4×D3—82no—
154C2×C4×D3—82no—
155C3×M16—42no—
156C2×C3×C8—41no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C22⋊C4—42no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162#217—24no—
163#217—42no—
164(C2×D3)⋊C4—44no—
165(C2×D3)⋊C4—44no—
166(C2×D3)⋊C4—44no—
167(C2×D3)⋊C4—82no—
168(C2×D3)⋊C4—82no—
169C4×Dic3—42no—
170#208—42no—
171C2×C3⋊C8—41no—
172C25—31no—
173C25—31no—
174C22×D4—21no—
175C22×D4—32no—
176C22×D4—32no—
177C22×D4—62no—
178C22×D4—62no—
179C22×D4—61no—
180C22×D4—61no—
181C22×D4—61no—
182C22×D4—61no—
183C22×D4—124no—
184C22×D4C4×D311yescharacteristic, complemented
185C23×C4—62no—
186C23×C4C22×D311yescharacteristic
187C2×M16—21no—
188C2×M16—61no—
189#127—124no—
190#121—21no—
191#121—21no—
192#121—21no—
193#121—21no—
194#120—124no—
195#120—124no—
196C2×C4⋊C4C4×D311yescharacteristic, complemented
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—124no—
207C2×C42—32no—
208#100—124no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#95—21no—
214#95—32no—
215#95—61no—
216#95—61no—
217#95#6712yescomplemented
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—41no—
225C23×C3—42no—
226C23×C3C2×C22⋊C411yescharacteristic
227C23×C3#12011yescharacteristic
228C23×C3#12311yescharacteristic
229C22×D3—22no—
230C22×D3—22no—
231C22×D3—22no—
232C22×D3—24no—
233C22×D3—22no—
234C22×D3—22no—
235C22×D3—22no—
236C22×D3—22no—
237C22×D3—42no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3—42no—
247C22×D3—42no—
248C22×D3—88no—
249C22×D3—81no—
250C22×D3—81no—
251C22×D3—81no—
252C22×D3—81no—
253C22×D3—88no—
254C22×D3#9914yes—
255C3×D4—81no—
256C3×D4—81no—
257C3×D4—82no—
258C3×D4—82no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
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—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4C4×D411yescharacteristic
272C2×C3×C4C4×D411yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12111yescharacteristic
275#67—88no—
276#67—88no—
277C2×Dic3—24no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283D12—44no—
284D12—44no—
285D12—44no—
286D12—44no—
287C4×D3—88no—
288C3×C8—81no—
289C3⋊C8—81no—
290C24—21no—
291C24—32no—
292C24—32no—
293C24—64no—
294C24—61no—
295C24—61no—
296C24—61no—
297C24—61no—
298C24—122no—
299C24—122no—
300C24—122no—
301C24—241no—
302C24—241no—
303C24D3×D411yescharacteristic
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—64no—
312C2×D4—64no—
313C2×D4—62no—
314C2×D4—62no—
315C2×D4—62no—
316C2×D4—62no—
317C2×D4—122no—
318C2×D4—122no—
319C2×D4—124no—
320C2×D4—122no—
321C2×D4—122no—
322C2×D4—124no—
323C2×D4—122no—
324C2×D4—122no—
325C2×D4—124no—
326C2×D4—124no—
327C2×D4—124no—
328C2×D4—124no—
329C2×D4—242no—
330C2×D4—242no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—42no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—124no—
337C22×C4—242no—
338C22×C4—242no—
339C22×C4C2×C4×D311yescharacteristic
340C22×C4#23511yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4D3×D411yescharacteristic
343C22×C4#23711yescharacteristic
344C22×C4C2×#6711yescharacteristic
345M16—42no—
346M16—122no—
347C2×C8—41no—
348C2×C8—121no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—64no—
355C22⋊C4—124no—
356C22⋊C4—124no—
357C22⋊C4—124no—
358C22⋊C4—122no—
359C22⋊C4—242no—
360C22⋊C4—242no—
361C42—122no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—44no—
370C22×C3—82no—
371C22×C3—82no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3#38611yescharacteristic
376C22×C3C2×#9912yes—
377C22×C3#45312yes—
378C2×D3—24no—
379C2×D3—44no—
380C2×D3—44no—
381C2×D3—42no—
382C2×D3—42no—
383C2×D3—44no—
384C2×D3—44no—
385C2×D3—42no—
386C2×D3—42no—
387C2×D3—42no—
388C2×D3—82no—
389C2×D3—84no—
390C2×D3—82no—
391C2×D3—82no—
392C2×D3—84no—
393C2×D3—82no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—82no—
399C3×C4—81no—
400Dic3—44no—
401Dic3—82no—
402C23—22no—
403C23—22no—
404C23—34no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—41no—
410C23—42no—
411C23—62no—
412C23—62no—
413C23—64no—
414C23—62no—
415C23—62no—
416C23—62no—
417C23—62no—
418C23—62no—
419C23—122no—
420C23—122no—
421C23—122no—
422C23—122no—
423C23—122no—
424C23—122no—
425C23—121no—
426C23—121no—
427C23—122no—
428C23—122no—
429C23—122no—
430C23—241no—
431C23—241no—
432C23—241no—
433C23—241no—
434C23—248no—
435C23—248no—
436C23D3×C22⋊C411yescharacteristic
437C23#88311yescharacteristic
438C23#93711yescharacteristic
439D4—81no—
440D4—81no—
441D4—82no—
442D4—82no—
443D4—124no—
444D4—124no—
445D4—124no—
446D4—124no—
447D4—248no—
448D4—248no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—64no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—248no—
468C2×C4#67⋊C411yescharacteristic
469C2×C4#88411yescharacteristic
470C2×C4C4×#6711yescharacteristic
471C2×C4#93811yescharacteristic
472C8—81no—
473C8—241no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—41no—
477C2×C3—161no—
478C2×C3#188711yescharacteristic
479C2×C3#189811yescharacteristic
480C2×C3#191111yescharacteristic
481D3—44no—
482D3—82no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—44no—
491C22—64no—
492C22—82no—
493C22—82no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—124no—
498C22—124no—
499C22—124no—
500C22—124no—
501C22—122no—
502C22—122no—
503C22—122no—
504C22—122no—
505C22—122no—
506C22—242no—
507C22—242no—
508C22—242no—
509C22—242no—
510C22—244no—
511C22—244no—
512C22D3×#9912yes—
513C22#495611yescharacteristic
514C22#519812yes—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—82no—
520C4—81no—
521C4—124no—
522C4—242no—
523C3#946311yescharacteristic, complemented, Sylow
524C2—22no—
525C2—22no—
526C2—41no—
527C2—124no—
528C2—161no—
529C2—242no—
530C2#7003611yescharacteristic
531C2#7032311yescharacteristic
532C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1510C2
socleC22×C3#386
prime radical#82653C2
cosocleC23×C3×C4C23
hypercenter#1510D3
hypocenterC3#9463
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1510

Complements

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

NH ≅ G/Ntype
2. #74119C2semidirect
3. C3×#1510C2semidirect
4. #71674C2semidirect
5. #69988C2semidirect
6. #67682C2semidirect
7. #67628C2semidirect
11. C2×#940C4semidirect
21. #4988C4semidirect
32. #1510D3semidirect
38. C22×C3×D4C2×C4semidirect
55. C2×C3×C4⋊C4C2×C4semidirect
68. C2×C4×Dic3C2×C4semidirect
74. C3×#95D4semidirect
77. C3⋊#95D4semidirect
184. C22×D4C4×D3semidirect
196. C2×C4⋊C4C4×D3semidirect
217. #95#67semidirect
523. C3#9463semidirect

Maximal subgroups

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

subgroupindex
#826532
#741192
C3×#15102
#716742
#699882
#676822
#676282
#94633