Finite Group Databasebeta

← #10691247

Subgroups — #10691247

Order 768.

762 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 113 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#82710C211yesmaximal, characteristic
2#74005C211yesmaximal, characteristic, complemented
3C3×#1519C211yesmaximal, characteristic, complemented
4#71675C211yesmaximal, characteristic, complemented
5C2×#4468C211yesmaximal, characteristic, complemented
6#67676C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9203—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#937—21no—
11C22×(C2×D3)⋊C4C2211yescharacteristic
12C2×D3×C22⋊C4—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15#5223C412yescomplemented
16#5220—21no—
17#5212C2211yescharacteristic
18#5212C2211yescharacteristic
19C2×#830C2211yescharacteristic
20#4992C412yescomplemented
21#4989—22no—
22#4961—22no—
23#4942—21no—
24#4668—22no—
25C3×#327—22no—
26#4537—22no—
27#4468—22no—
28#2617—31no—
29#2315—31no—
30#1519—31no—
31#1519D311yescharacteristic, complemented
32C2×#323—31no—
33#1289—31no—
34#1289—31no—
35C24×D3D411yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4C2×C411yescharacteristic, complemented
38C23×C3×C4C2311yescharacteristic
39C22×#67—22no—
40C23×Dic3D411yescharacteristic
41C22×C4×D3—22no—
42C22×C3×C8—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×#121—21no—
46C3×#121—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—22no—
54#937—42no—
55#937—42no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4C2×C412yes—
63C2×(C2×D3)⋊C4D412yes—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×Dic3⋊C4—22no—
67C22×C3⋊C8—21no—
68D3×C22⋊C4—42no—
69D3×C22⋊C4—42no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95D412yescomplemented
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95C2×C412yescomplemented
76C3⋊#95D412yescomplemented
77#830—21no—
78#830—21no—
79C2×#121C2×D311yescharacteristic
80C2×#120—61no—
81C22×C22⋊C4—31no—
82C22×C22⋊C4—61no—
83C2×C22⋊C8—31no—
84C2×C22⋊C8C2×D311yescharacteristic
85#397—32no—
86#394—62no—
87#392—32no—
88#386—61no—
89#385—62no—
90#385—62no—
91#380—31no—
92#380—31no—
93#380—61no—
94#380C2×D311yescharacteristic
95#327—22no—
96#327—62no—
97#323—62no—
98C24×C3—21no—
99C24×C3C2×D411yescharacteristic
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—81no—
105C23×D3—81no—
106C23×D3C22⋊C412yes—
107C23×D3D812yes—
108C23×D3SD1612yes—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—81no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—81no—
121C22×C3×C4C22×C411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4#4111yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C2×#67—44no—
128C2×#67—82no—
129C2×#67—82no—
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—22no—
133C22×Dic3—22no—
134C22×Dic3—81no—
135C22×Dic3C22⋊C412yes—
136C2×C4×D3—44no—
137C2×C4×D3—82no—
138C2×C4×D3—82no—
139C2×C3×C8—41no—
140C2×C3×C8—81no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—81no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148#217—41no—
149#217—41no—
150#217—41no—
151#217—41no—
152#217—82no—
153(C2×D3)⋊C4—22no—
154(C2×D3)⋊C4—22no—
155(C2×D3)⋊C4—22no—
156(C2×D3)⋊C4—22no—
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—44no—
159C4⋊Dic3—81no—
160C4⋊Dic3—81no—
161Dic3⋊C4—82no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—81no—
164C25—31no—
165C22×D4—21no—
166C22×D4—62no—
167C22×D4C4×D311yescharacteristic, complemented
168C23×C4—31no—
169C23×C4—62no—
170C23×C4C22×D311yescharacteristic
171C22×C8—21no—
172C22×C8—61no—
173#121—21no—
174#121—21no—
175#121—21no—
176#121—21no—
177#120—122no—
178#120—122no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4C4×D311yescharacteristic, complemented
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—32no—
187C2×C22⋊C4—32no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—62no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—122no—
198C2×C22⋊C4—122no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203#95—32no—
204#95—32no—
205#95—62no—
206#95—62no—
207#95#6712yescomplemented
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3C2×C22⋊C411yescharacteristic
216C23×C3#12011yescharacteristic
217C23×C3#12311yescharacteristic
218C23×C3C2×D811yescharacteristic
219C23×C3C2×SD1611yescharacteristic
220C23×C3AGL(1,ℤ/8)11yescharacteristic
221C23×C3#13711yescharacteristic
222C22×D3—22no—
223C22×D3—22no—
224C22×D3—22no—
225C22×D3—22no—
226C22×D3—22no—
227C22×D3—22no—
228C22×D3—42no—
229C22×D3—42no—
230C22×D3—42no—
231C22×D3—42no—
232C22×D3—82no—
233C22×D3—82no—
234C22×D3—81no—
235C22×D3—81no—
236C22×D3—82no—
237C22×D3—82no—
238C22×D3#10214yes—
239C3×D4—81no—
240C3×D4—81no—
241C3×D4—81no—
242C3×D4—81no—
243C3×D4—81no—
244C3×D4—81no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4C4×D411yescharacteristic
258C2×C3×C4C4×D411yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12111yescharacteristic
261#67—88no—
262C2×Dic3—22no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—81no—
272C4×D3—88no—
273C3×C8—81no—
274C3⋊C8—81no—
275C24—21no—
276C24—32no—
277C24—32no—
278C24—32no—
279C24—61no—
280C24—61no—
281C24—61no—
282C24—61no—
283C24—241no—
284C24—241no—
285C24D3×D411yescharacteristic
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—81no—
295C2×D4—124no—
296C2×D4—242no—
297C2×D4—242no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—32no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—81no—
306C22×C4—124no—
307C22×C4—242no—
308C22×C4—242no—
309C22×C4—241no—
310C22×C4C2×C4×D311yescharacteristic
311C22×C4#23511yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4#23711yescharacteristic
315C22×C4C2×#6711yescharacteristic
316C2×C8—41no—
317C2×C8—81no—
318C2×C8—121no—
319C2×C8—241no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—81no—
323C4⋊C4—242no—
324C4⋊C4—241no—
325C4⋊C4—241no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—62no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4—124no—
335C22⋊C4—124no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C22⋊C4—121no—
340C22⋊C4—242no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—82no—
352C22×C3—82no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3#38611yescharacteristic
357C22×C3C2×#10211yescharacteristic
358C22×C3#41711yescharacteristic
359C22×C3#44711yescharacteristic
360C22×C3#44811yescharacteristic
361C22×C3#48011yescharacteristic
362C22×C3#48111yescharacteristic
363C2×D3—24no—
364C2×D3—44no—
365C2×D3—44no—
366C2×D3—42no—
367C2×D3—42no—
368C2×D3—44no—
369C2×D3—82no—
370C2×D3—82no—
371C2×D3—82no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—82no—
377C3×C4—81no—
378Dic3—42no—
379Dic3—82no—
380Dic3—82no—
381C23—34no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—62no—
390C23—62no—
391C23—62no—
392C23—62no—
393C23—62no—
394C23—62no—
395C23—122no—
396C23—122no—
397C23—122no—
398C23—122no—
399C23—241no—
400C23—241no—
401C23—242no—
402C23—242no—
403C23—242no—
404C23—242no—
405C23D3×C22⋊C411yescharacteristic
406C23#88311yescharacteristic
407C23D3×D811yescharacteristic
408C23#91111yescharacteristic
409C23D3×SD1611yescharacteristic
410C23#91511yescharacteristic
411C23#93711yescharacteristic
412D4—81no—
413D4—81no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—248no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—62no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—248no—
441C2×C4—241no—
442C2×C4#67⋊C411yescharacteristic
443C2×C4#88411yescharacteristic
444C2×C4C4×#6711yescharacteristic
445C2×C4#93811yescharacteristic
446C8—81no—
447C8—241no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—161no—
455C2×C3#187411yescharacteristic
456C2×C3#191111yescharacteristic
457C2×C3#191311yescharacteristic
458D3—44no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—64no—
470C22—82no—
471C22—82no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—124no—
476C22—124no—
477C22—124no—
478C22—122no—
479C22—122no—
480C22—242no—
481C22—242no—
482C22—242no—
483C22D3×#10211yescharacteristic
484C22#477111yescharacteristic
485C22#477411yescharacteristic
486C22#477711yescharacteristic
487C22#495611yescharacteristic
488C22#515611yescharacteristic
489C22#517011yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—82no—
495C4—81no—
496C4—122no—
497C4—242no—
498C4—242no—
499C3#920311yescharacteristic, complemented, Sylow
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—21no—
506C2—124no—
507C2—161no—
508C2#7003411yescharacteristic
509C2#7020011yescharacteristic
510C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#386
prime radical#82710C2
cosocleC23×C3×C4C23
hypercenter#1519D3
hypocenterC3#9203
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #74005C2semidirect
3. C3×#1519C2semidirect
4. #71675C2semidirect
5. C2×#4468C2semidirect
6. #67676C2semidirect
7. #67625C2semidirect
15. #5223C4semidirect
20. #4992C4semidirect
31. #1519D3semidirect
37. C22×C3×D4C2×C4semidirect
48. C2×C3×C4⋊C4C2×C4semidirect
72. C3×#95D4semidirect
75. C3⋊#95C2×C4semidirect
76. C3⋊#95D4semidirect
167. C22×D4C4×D3semidirect
183. C2×C4⋊C4C4×D3semidirect
207. #95#67semidirect
499. C3#9203semidirect

Maximal subgroups

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

subgroupindex
#827102
#740052
C3×#15192
#716752
C2×#44682
#676762
#676252
#92033