Finite Group Databasebeta

← #10698049

Subgroups — #10698049

Order 768.

726 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 101 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#82699C211yesmaximal, characteristic, complemented
2#82625C211yesmaximal, characteristic
3#73999C211yesmaximal, characteristic, complemented
4C3×#1519C211yesmaximal, characteristic, complemented
5#69944C211yesmaximal, characteristic, complemented
6#69910C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#10321—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#935—21no—
11C2×#930C2211yescharacteristic
12C22×C4×Dic3C2211yescharacteristic
13C2×#886—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16#5221—21no—
17#5220—21no—
18#5217—21no—
19#5212C2211yescharacteristic
20C2×#830C2211yescharacteristic
21#5038—22no—
22#5036—21no—
23#5033—21no—
24#4988—21no—
25#4984—21no—
26#4972—21no—
27#4967—21no—
28#4958—21no—
29#4944—22no—
30#4670—22no—
31C3×#327—22no—
32#4550—22no—
33#4470—22no—
34#3186—31no—
35#2854—31no—
36#2854—31no—
37#1817—31no—
38#1519—31no—
39#1519D311yescharacteristic, complemented
40#1289—31no—
41C22×C3×D4—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C23×Dic3D411yescharacteristic
46C23×Dic3D411yescharacteristic
47C22×D12D411yescharacteristic
48C22×C3×C8—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—21no—
57C2×#217—22no—
58C2×#217—22no—
59C2×#217—21no—
60#935—42no—
61#935—42no—
62C2×#213—22no—
63#930—21no—
64#930—21no—
65#930—21no—
66#930—21no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×Dic3⋊C4—22no—
74C2×C4×Dic3—21no—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—21no—
77C2×C4×Dic3—21no—
78C2×C4×Dic3—21no—
79C2×C4×Dic3—21no—
80C22×C3⋊C8D412yes—
81#886—42no—
82#886—42no—
83C3×C22⋊C8—21no—
84C3×C22⋊C8—21no—
85C3×#95D412yes—
86#832—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95D412yes—
90#830—21no—
91#830—21no—
92#809—22no—
93#807—22no—
94C2×#123—61no—
95C2×#123—61no—
96C2×#121—31no—
97C2×#121C2×D311yescharacteristic
98C22×C42—31no—
99#480—61no—
100#480—61no—
101#466—62no—
102#447—61no—
103#447—61no—
104C2×C22⋊C8—31no—
105C2×C22⋊C8C2×D311yescharacteristic
106#390—61no—
107#390—61no—
108#389—61no—
109#389—61no—
110#388—62no—
111#388—62no—
112#386—61no—
113#386—61no—
114#380—31no—
115#380C2×D311yescharacteristic
116#336—62no—
117#327—22no—
118#327—62no—
119C24×C3—21no—
120C24×C3C2×D411yescharacteristic
121C23×D3—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—42no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—42no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—41no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—22no—
157C22×Dic3—42no—
158C22×Dic3D812yescomplemented
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—42no—
162C2×C3×C8—42no—
163C2×C3×C8—41no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—42no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—41no—
170C3×C22⋊C4—41no—
171#217—44no—
172#217—44no—
173#217—41no—
174#217—41no—
175#213—82no—
176(C2×D3)⋊C4—44no—
177(C2×D3)⋊C4—44no—
178(C2×D3)⋊C4—41no—
179(C2×D3)⋊C4—41no—
180C4⋊Dic3—21no—
181C4⋊Dic3—21no—
182C4⋊Dic3—44no—
183C4⋊Dic3—42no—
184Dic3⋊C4—44no—
185C4×Dic3—41no—
186C4×Dic3—41no—
187C4×Dic3—41no—
188C4×Dic3—41no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—42no—
191C22×D4—21no—
192C22×D4—31no—
193C22×D4—61no—
194C22×D4D1211yescharacteristic
195C23×C4—31no—
196C23×C4—31no—
197C23×C4C22×D311yescharacteristic
198C2×D8—62no—
199C22×C8—21no—
200C22×C8—32no—
201#123—122no—
202#123—122no—
203#123—122no—
204#123—122no—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209#121—61no—
210#121—61no—
211#121—61no—
212#121—61no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—31no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4—62no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4D1211yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—62no—
222C2×C22⋊C4—62no—
223C2×C22⋊C4—62no—
224C2×C22⋊C4—62no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C42—61no—
228C2×C42—61no—
229C2×C42—61no—
230C2×C42—61no—
231C2×C42—61no—
232C2×C42—61no—
233#107—62no—
234#102—62no—
235#102—62no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—61no—
239C22⋊C8—61no—
240#95—32no—
241#95—62no—
242#95—62no—
243#95#6712yes—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3#12011yescharacteristic
252C23×C3#12011yescharacteristic
253C23×C3#12111yescharacteristic
254C23×C3C2×D811yescharacteristic
255C23×C3#13511yescharacteristic
256C23×C3AGL(1,ℤ/8)11yescharacteristic
257C23×C3AGL(1,ℤ/8)11yescharacteristic
258C22×D3—42no—
259C22×D3—42no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C3×D4—42no—
264C3×D4—81no—
265C3×D4—81no—
266C3×D4—81no—
267C3×D4—81no—
268C3×D4—81no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4#12011yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12111yescharacteristic
284C2×C3×C4#12711yescharacteristic
285#67—81no—
286#67—81no—
287#67—81no—
288#67—81no—
289C2×Dic3—22no—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—41no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309D12—42no—
310D12—81no—
311C3×C8—81no—
312C3⋊C8—42no—
313C24—21no—
314C24—61no—
315C24D3×D411yescharacteristic
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—42no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—61no—
326C2×D4—61no—
327C2×D4—122no—
328C2×D4—121no—
329C2×D4—121no—
330C2×D4—121no—
331C2×D4—121no—
332C2×D4—121no—
333C2×D4—121no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—32no—
337C22×C4—42no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—62no—
348C22×C4—61no—
349C22×C4—122no—
350C22×C4C2×D1211yescharacteristic
351C22×C4#23511yescharacteristic
352C22×C4D3×D411yescharacteristic
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4C2×#6711yescharacteristic
356D8—242no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—62no—
360C2×C8—122no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—42no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—122no—
367C4⋊C4—124no—
368C4⋊C4—124no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—124no—
374C22⋊C4—124no—
375C22⋊C4—124no—
376C22⋊C4—124no—
377C22⋊C4—121no—
378C22⋊C4—121no—
379C22⋊C4—121no—
380C22⋊C4—121no—
381C42—121no—
382C42—121no—
383C42—121no—
384C42—121no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—82no—
396C22×C3—82no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—81no—
400C22×C3#39211yescharacteristic
401C22×C3#44711yescharacteristic
402C22×C3#44711yescharacteristic
403C22×C3#44911yescharacteristic
404C22×C3#44911yescharacteristic
405C22×C3#46611yescharacteristic
406C22×C3#46911yescharacteristic
407C2×D3—82no—
408C2×D3—82no—
409C2×D3—81no—
410C2×D3—81no—
411C2×D3—81no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—82no—
416C3×C4—81no—
417C3×C4#45812yes—
418Dic3—42no—
419Dic3—42no—
420Dic3—82no—
421Dic3—81no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—122no—
430C23—122no—
431C23—121no—
432C23—121no—
433C23—121no—
434C23#88211yescharacteristic
435C23#88411yescharacteristic
436C23D3×D811yescharacteristic
437C23#91111yescharacteristic
438C23#91411yescharacteristic
439C23#91611yescharacteristic
440C23#93711yescharacteristic
441D4—42no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—81no—
446D4—81no—
447D4—122no—
448D4—241no—
449D4—241no—
450D4—241no—
451D4—241no—
452D4—241no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—122no—
485C2×C4#88211yescharacteristic
486C2×C4#88411yescharacteristic
487C2×C4#93011yescharacteristic
488C2×C4#94011yescharacteristic
489C8—81no—
490C8—122no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—161no—
498C2×C3#199911yescharacteristic
499C2×C3#201011yescharacteristic
500C2×C3#202211yescharacteristic
501D3—161no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—82no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22—242no—
518C22—242no—
519C22—241no—
520C22—241no—
521C22—241no—
522C22#477311yescharacteristic
523C22#477511yescharacteristic
524C22#478111yescharacteristic
525C22#478211yescharacteristic
526C22#496011yescharacteristic
527C22#515611yescharacteristic
528C22#517211yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—82no—
533C4—81no—
534C4—122no—
535C4—122no—
536C4—242no—
537C4—241no—
538C4#474212yes—
539C3#1032111yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2—161no—
547C2—481no—
548C2#7003111yescharacteristic
549C2#7020411yescharacteristic
550C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#392
prime radical#82625C2
hypercenter#1519D3
hypocenterC3#10321
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
1. #82699C2semidirect
3. #73999C2semidirect
4. C3×#1519C2semidirect
5. #69944C2semidirect
6. #69910C2semidirect
7. #67625C2semidirect
39. #1519D3semidirect
158. C22×Dic3D8semidirect
539. C3#10321semidirect

Maximal subgroups

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

subgroupindex
#826992
#826252
#739992
C3×#15192
#699442
#699102
#676252
#103213