Finite Group Databasebeta

← #10690776

Subgroups — #10690776

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4719C211yesmaximal, characteristic, complemented
2#73999C211yesmaximal, characteristic
3C2×#4470C211yesmaximal, characteristic
4#69402C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9118—31nomaximal, Sylow
9C2×#930C2211yescharacteristic
10C22×C4×Dic3C2211yescharacteristic
11C2×#886—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15C2×C3⋊#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2211yescharacteristic
18C2×#830C2211yescharacteristic
19C2×#815—21no—
20C2×Dic3⋊C8—21no—
21#4988C411yescharacteristic, complemented
22#4988C411yescharacteristic, complemented
23#4984C411yescharacteristic, complemented
24#4984C411yescharacteristic, complemented
25#4958—21no—
26#4719—22no—
27#4719—22no—
28#4670—21no—
29#4670—21no—
30#4470—22no—
31#4451—22no—
32C2×#432—31no—
33#2854—31no—
34C2×#327—31no—
35C2×#327—31no—
36#1434—31no—
37#1294—31no—
38#1294D311yescharacteristic, complemented
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C23×Dic3—21no—
43C23×Dic3D411yescharacteristic
44C23×Dic3D411yescharacteristic
45C22×D12C2×C411yescharacteristic
46C22×C3×C8—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C4⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4—21no—
50#930—21no—
51#930—21no—
52#930—21no—
53#930—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic
62C2×C4×Dic3—21no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3—21no—
66C2×C4×Dic3C2×C411yescharacteristic
67C2×C4×Dic3C2×C411yescharacteristic
68C2×C4×Dic3D411yescharacteristic
69C2×C4×Dic3D411yescharacteristic
70C22×C3⋊C8—21no—
71#886—42no—
72#886—42no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yes—
75C3×#95—21no—
76C3×#95—21no—
77C3×#95—21no—
78C3⋊#95—24no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—42no—
84#830—21no—
85#830D412yes—
86#815—24no—
87Dic3⋊C8—24no—
88C2×#123—61no—
89C2×#121—31no—
90C22×C42—31no—
91#432—62no—
92#432—62no—
93C2×C4⋊C8—61no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8C2×D311yescharacteristic
96C2×#97—61no—
97#390—31no—
98#390—31no—
99#389—31no—
100#389—31no—
101#388—61no—
102#388—61no—
103#386—61no—
104#380C2×D311yescharacteristic
105C2×#95—31no—
106C2×#95—61no—
107C2×#95—61no—
108C2×#95C2×D311yescharacteristic
109#327—62no—
110#327—62no—
111C24×C3C2×D411yescharacteristic
112C23×D3—21no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
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—41no—
131C2×#67—41no—
132C2×#67—41no—
133C2×#67—41no—
134C2×#67—41no—
135C2×#67—41no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3—21no—
140C22×Dic3—21no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—21no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3C22⋊C411yescharacteristic
155C22×Dic3C22⋊C411yescharacteristic
156C22×Dic3C22⋊C411yescharacteristic
157C22×Dic3C22⋊C411yescharacteristic
158C2×D12—21no—
159C2×D12—21no—
160C2×D12—41no—
161C2×D12—41no—
162C2×C3×C8—22no—
163C2×C3×C8—41no—
164C2×C3×C8—41no—
165C3×C4⋊C4—41no—
166C3×C4⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169(C2×D3)⋊C4—42no—
170(C2×D3)⋊C4—42no—
171(C2×D3)⋊C4—42no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—42no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4⋊Dic3—42no—
178C4⋊Dic3—42no—
179C4×Dic3—21no—
180C4×Dic3—21no—
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3—21no—
185C4×Dic3—21no—
186C4×Dic3—21no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—41no—
189C2×C3⋊C8—41no—
190C22×D4—31no—
191C22×D4—61no—
192C23×C4—21no—
193C23×C4—31no—
194C23×C4—31no—
195C23×C4—61no—
196C23×C4C22×D311yescharacteristic
197C22×C8—21no—
198C22×C8—61no—
199#123—122no—
200#123—122no—
201#121—61no—
202#121—61no—
203#121—61no—
204#121—61no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—61no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4C4×D311yescharacteristic
209C2×C4⋊C4C4×D311yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—61no—
216C2×C42—31no—
217C2×C42—31no—
218C2×C42—31no—
219C2×C42—31no—
220C2×C42—61no—
221C2×C42—61no—
222C2×C42—61no—
223C2×C42—61no—
224C4⋊C8—64no—
225C22⋊C8—21no—
226C22⋊C8—32no—
227C22⋊C8—61no—
228C22⋊C8D1212yes—
229#97—64no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—61no—
234#95—61no—
235#95—64no—
236#95—61no—
237#95—61no—
238#95—122no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3C2×C22⋊C411yescharacteristic
244C23×C3#12111yescharacteristic
245C23×C3#12111yescharacteristic
246C23×C3AGL(1,ℤ/8)11yescharacteristic
247C23×C3AGL(1,ℤ/8)11yescharacteristic
248C23×C3AGL(1,ℤ/8)11yescharacteristic
249C23×C3#13711yescharacteristic
250C22×D3—41no—
251C22×D3—41no—
252C22×D3—41no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4—81no—
277C2×C3×C4—81no—
278C2×C3×C4C4×D411yescharacteristic
279C2×C3×C4C4×D411yescharacteristic
280C2×C3×C4#12011yescharacteristic
281C2×C3×C4#12311yescharacteristic
282#67—81no—
283#67—81no—
284#67—81no—
285#67—81no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—81no—
310C2×Dic3—81no—
311C2×Dic3—81no—
312C2×Dic3—81no—
313C2×Dic3#10412yes—
314D12—81no—
315D12—81no—
316C3×C8—42no—
317C3⋊C8—42no—
318C24—61no—
319C24D3×D411yescharacteristic
320C2×D4—61no—
321C2×D4—61no—
322C2×D4—121no—
323C2×D4—121no—
324C2×D4—121no—
325C2×D4—121no—
326C2×D4—121no—
327C2×D4—121no—
328C2×D4—121no—
329C2×D4—121no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—62no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—121no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4—121no—
360C22×C4—121no—
361C22×C4—121no—
362C22×C4—121no—
363C22×C4C2×C4×D311yescharacteristic
364C22×C4C2×D1211yescharacteristic
365C22×C4#23511yescharacteristic
366C22×C4D3×D411yescharacteristic
367C22×C4D3×D411yescharacteristic
368C22×C4#23911yescharacteristic
369C2×C8—22no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—62no—
373C2×C8—121no—
374C2×C8—121no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—122no—
380C4⋊C4—122no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—122no—
384C22⋊C4—122no—
385C22⋊C4—122no—
386C22⋊C4—122no—
387C22⋊C4—121no—
388C22⋊C4—121no—
389C42—61no—
390C42—61no—
391C42—61no—
392C42—61no—
393C42—61no—
394C42—61no—
395C42—61no—
396C42—61no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3#38611yescharacteristic
410C22×C3#41011yescharacteristic
411C22×C3#41011yescharacteristic
412C22×C3#41711yescharacteristic
413C22×C3#41811yescharacteristic
414C22×C3C2×#10411yescharacteristic
415C22×C3#42111yescharacteristic
416C22×C3#46811yescharacteristic
417C22×C3#46811yescharacteristic
418C22×C3#46911yescharacteristic
419C22×C3#46911yescharacteristic
420C2×D3—81no—
421C2×D3—81no—
422C2×D3—81no—
423C2×D3—81no—
424C2×D3—81no—
425C2×D3—81no—
426C2×D3—81no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—82no—
432C3×C4—81no—
433C3×C4—81no—
434Dic3—22no—
435Dic3—41no—
436Dic3—41no—
437Dic3—41no—
438Dic3—82no—
439Dic3—81no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—121no—
445C23—121no—
446C23—121no—
447C23—121no—
448C23—121no—
449C23—121no—
450C23—121no—
451C23D3×C22⋊C411yescharacteristic
452C23#88411yescharacteristic
453C23#89511yescharacteristic
454C23#91111yescharacteristic
455C23#91411yescharacteristic
456C23#91411yescharacteristic
457C23#91811yescharacteristic
458D4—241no—
459D4—241no—
460D4—241no—
461D4—241no—
462D4—241no—
463D4—241no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—32no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—61no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4—121no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—241no—
510C2×C4—241no—
511C2×C4—241no—
512C2×C4—241no—
513C2×C4C4×D1211yescharacteristic
514C2×C4#88211yescharacteristic
515C2×C4#89311yescharacteristic
516C2×C4#89411yescharacteristic
517C8—42no—
518C8—122no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3#150511yescharacteristic
524C2×C3#150911yescharacteristic
525C2×C3#151211yescharacteristic
526C2×C3#151411yescharacteristic
527C2×C3#187111yescharacteristic
528C2×C3#189511yescharacteristic
529C2×C3#189911yescharacteristic
530D3—161no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—241no—
544C22—241no—
545C22—241no—
546C22—241no—
547C22—241no—
548C22—241no—
549C22—241no—
550C22#466511yescharacteristic
551C22#474011yescharacteristic
552C22#474011yescharacteristic
553C22#477011yescharacteristic
554C22#478211yescharacteristic
555C22#480311yescharacteristic
556C22#481211yescharacteristic
557C22D3×#10411yescharacteristic
558C22#482111yescharacteristic
559C22#486511yescharacteristic
560C22#488611yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—62no—
566C4—82no—
567C4—81no—
568C4—81no—
569C4—121no—
570C4—121no—
571C4—121no—
572C4—242no—
573C4—241no—
574C3#911811yescharacteristic, complemented, Sylow
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—481no—
580C2#6827811yescharacteristic
581C2#6828111yescharacteristic
582C2#6831111yescharacteristic
583C2#6831211yescharacteristic
584C2#6875511yescharacteristic
585C2#6888511yescharacteristic
586C2#6895411yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9118
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
1. C2×#4719C2semidirect
4. #69402C2semidirect
6. C3×#1294C2semidirect
7. #67624C2semidirect
21. #4988C4semidirect
22. #4988C4semidirect
23. #4984C4semidirect
24. #4984C4semidirect
38. #1294D3semidirect
574. C3#9118semidirect

Maximal subgroups

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

subgroupindex
C2×#47192
#739992
C2×#44702
#694022
C2×#44512
C3×#12942
#676242
#91183