Finite Group Databasebeta

← #10702210

Subgroups — #10702210

Order 768.

1006 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#823C211yesmaximal, characteristic
2C2×#5001C211yesmaximal, characteristic
3C2×#4999C211yesmaximal, characteristic, complemented
4C22×#809C211yesmaximal, characteristic
5C3×#1736C211yesmaximal, characteristic, complemented
6C2×#4558C211yesmaximal, characteristic, complemented
7C2×#4530C211yesmaximal, characteristic, complemented
8#11061—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×#208C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C23×C3⋊C8C2211yescharacteristic
16C2×#823—22no—
17C2×#823—22no—
18C2×#823—22no—
19#5001—22no—
20#5001C414yescomplemented
21#4999—22no—
22#4999—22no—
23C2×#809—22no—
24C2×#809—22no—
25C2×#809—22no—
26C2×#809—22no—
27C2×C4⋊(C3⋊C8)—22no—
28C3×#340—22no—
29#4558—22no—
30#4530—22no—
31C2×#424—31no—
32C2×#418—31no—
33C22×#102—31no—
34C22×#100—31no—
35C2×#343—31no—
36C2×#341—31no—
37#1736D311yescharacteristic, complemented
38C24×D3—21no—
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—21no—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12—21no—
51C22×D12—21no—
52C22×D12C2×C412yes—
53C22×D12D412yes—
54C22×D12Q812yes—
55C2×C3×M16—21no—
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16D412yes—
61C22×C3×C8—21no—
62C3×#117—21no—
63C3×#117—21no—
64C3×#117C2×C412yes—
65C3×#117D412yes—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4D412yes—
72C2×C3×C4⋊C4D412yes—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42—21no—
75C2×#208—21no—
76C2×#208—21no—
77C2×#208—21no—
78C2×#208—21no—
79C2×#208—21no—
80C2×#208—21no—
81C2×#208—21no—
82C2×#208—21no—
83C2×#208—21no—
84C2×#208—21no—
85C2×#208C2×C412yes—
86C2×#208D412yes—
87C22×C3⋊C8—21no—
88C22×C3⋊C8—21no—
89C22×C3⋊C8—21no—
90C22×C3⋊C8—21no—
91C22×C3⋊C8—21no—
92C22×C3⋊C8—21no—
93C22×C3⋊C8—21no—
94C22×C3⋊C8—21no—
95C22×C3⋊C8D412yes—
96#823—28no—
97#809—28no—
98#809—28no—
99C4⋊(C3⋊C8)—28no—
100C23×D4—31no—
101C22×M16—31no—
102C22×M16—31no—
103C22×M16C2×D311yescharacteristic
104C23×C8—31no—
105C2×#117C2×D311yescharacteristic
106C22×C4⋊C4C2×D311yescharacteristic
107#424—62no—
108#424—62no—
109C2×C4⋊C8—62no—
110#418—34no—
111#418—62no—
112C2×#102—62no—
113C2×#102—62no—
114C2×#102—62no—
115C2×#102—62no—
116C2×#100—62no—
117C2×#100—62no—
118C2×#100—62no—
119#343—62no—
120#341—62no—
121#340—22no—
122C24×C3#4111yescharacteristic
123C23×D3—22no—
124C23×D3—22no—
125C23×D3—22no—
126C23×D3—41no—
127C23×D3—41no—
128C23×D3—41no—
129C23×D3—41no—
130C23×D3—82no—
131C23×D3—82no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—82no—
137C22×C3×C4C22×C411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×Q811yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C22×C3×C4#4111yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C2×D12—22no—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—41no—
167C2×D12—41no—
168C2×D12—41no—
169C2×D12—41no—
170C2×D12—41no—
171C2×D12—41no—
172C2×D12—41no—
173C2×D12—41no—
174C2×D12—41no—
175C2×D12—41no—
176C2×D12—41no—
177C2×D12—41no—
178C2×D12C4⋊C414yes—
179C3×M16—22no—
180C3×M16—22no—
181C3×M16—41no—
182C3×M16—41no—
183C2×C3×C8—22no—
184C2×C3×C8—41no—
185C2×C3×C8—41no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—41no—
190C3×C4⋊C4—41no—
191C3×C4⋊C4C22⋊C414yes—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—22no—
194C3×C42—22no—
195C3×C42—22no—
196#208—22no—
197#208—22no—
198#208—22no—
199#208—22no—
200#208—41no—
201#208—41no—
202#208—41no—
203#208—41no—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—41no—
210C2×C3⋊C8—41no—
211C2×C3⋊C8—41no—
212C2×C3⋊C8—41no—
213C2×C3⋊C8—41no—
214C2×C3⋊C8—41no—
215C2×C3⋊C8—41no—
216C2×C3⋊C8—41no—
217C25—61no—
218C22×D4—32no—
219C22×D4—32no—
220C22×D4—32no—
221C22×D4—61no—
222C22×D4—61no—
223C22×D4—61no—
224C22×D4—61no—
225C22×D4—61no—
226C22×D4—61no—
227C22×D4—61no—
228C22×D4—61no—
229C22×D4—61no—
230C22×D4—61no—
231C22×D4—61no—
232C23×C4—21no—
233C23×C4C22×D311yescharacteristic
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16—21no—
239C2×M16—32no—
240C2×M16—32no—
241C2×M16—61no—
242C2×M16—61no—
243C2×M16—61no—
244C2×M16—61no—
245C2×M16—61no—
246C2×M16—61no—
247C2×M16—61no—
248C2×M16—61no—
249C2×M16—61no—
250C2×M16—61no—
251C2×M16D1212yes—
252C22×C8—21no—
253C22×C8—32no—
254C22×C8—61no—
255C22×C8—61no—
256C22×C8—61no—
257C22×C8—61no—
258C22×C8—61no—
259C22×C8—61no—
260C22×C8—61no—
261C22×C8—61no—
262#117—21no—
263#117—21no—
264#117C4×D312yes—
265#117D1212yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4D1212yes—
272C2×C4⋊C4#6712yes—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C4⋊C8—68no—
276#102—68no—
277#102—68no—
278#100—68no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3C4×D411yescharacteristic
282C23×C3#12111yescharacteristic
283C23×C3C22⋊Q811yescharacteristic
284C23×C3#12311yescharacteristic
285C23×C3#12311yescharacteristic
286C23×C3#12311yescharacteristic
287C23×C3#12611yescharacteristic
288C22×D3—24no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C22×D3—84no—
296C22×D3—84no—
297C22×D3—82no—
298C22×D3—82no—
299C22×D3—82no—
300C22×D3—82no—
301C22×D3—84no—
302C22×D3—84no—
303C22×D3—82no—
304C22×D3—82no—
305C2×C3×C4—24no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—82no—
310C2×C3×C4—82no—
311C2×C3×C4—82no—
312C2×C3×C4C2×C22⋊C411yescharacteristic
313C2×C3×C4C2×C4⋊C411yescharacteristic
314C2×C3×C4#11711yescharacteristic
315C2×C3×C4C4×D411yescharacteristic
316C2×C3×C4C4×D411yescharacteristic
317C2×C3×C4C4×D411yescharacteristic
318C2×C3×C4#12011yescharacteristic
319C2×C3×C4#12011yescharacteristic
320C2×C3×C4#12011yescharacteristic
321C2×C3×C4#12111yescharacteristic
322C2×C3×C4#12111yescharacteristic
323C2×C3×C4#12111yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4#12111yescharacteristic
327C2×C3×C4#12111yescharacteristic
328C2×C3×C4C22⋊Q811yescharacteristic
329C2×C3×C4C22⋊Q811yescharacteristic
330C2×C3×C4C22⋊Q811yescharacteristic
331C2×C3×C4C22⋊Q811yescharacteristic
332C2×C3×C4C22⋊Q811yescharacteristic
333C2×C3×C4#12311yescharacteristic
334C2×C3×C4#12311yescharacteristic
335C2×C3×C4#12411yescharacteristic
336C2×C3×C4#12411yescharacteristic
337C2×C3×C4#12411yescharacteristic
338C2×C3×C4#12611yescharacteristic
339C2×C3×C4#12711yescharacteristic
340D12—24no—
341D12—24no—
342D12—42no—
343D12—42no—
344D12—42no—
345D12—42no—
346D12—42no—
347D12—42no—
348C3×C8—42no—
349C3⋊C8—42no—
350C3⋊C8—42no—
351C3⋊C8—42no—
352C24—62no—
353C24—62no—
354C24—62no—
355C24—121no—
356C24—121no—
357C24—121no—
358C24—121no—
359C24—242no—
360C24—242no—
361C24#23711yescharacteristic
362C2×D4—34no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—121no—
379C2×D4—121no—
380C2×D4—121no—
381C2×D4—121no—
382C2×D4—121no—
383C2×D4—121no—
384C2×D4—121no—
385C2×D4—121no—
386C2×D4—121no—
387C2×D4—121no—
388C2×D4—121no—
389C2×D4—121no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—82no—
395C22×C4C2×C4×D311yescharacteristic
396C22×C4C2×D1211yescharacteristic
397C22×C4C2×D1211yescharacteristic
398C22×C4C2×D1211yescharacteristic
399C22×C4#23511yescharacteristic
400C22×C4#23511yescharacteristic
401C22×C4D3×D411yescharacteristic
402C22×C4D3×D411yescharacteristic
403C22×C4D3×D411yescharacteristic
404C22×C4#23711yescharacteristic
405C22×C4#23711yescharacteristic
406C22×C4D3×Q811yescharacteristic
407C22×C4#23911yescharacteristic
408C22×C4C2×#6711yescharacteristic
409M16—22no—
410M16—22no—
411M16—41no—
412M16—41no—
413M16—62no—
414M16—62no—
415M16—62no—
416M16—62no—
417M16—121no—
418M16—121no—
419M16—121no—
420M16—121no—
421C2×C8—22no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—62no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8—62no—
429C2×C8—121no—
430C2×C8—121no—
431C2×C8—121no—
432C2×C8—121no—
433C2×C8—121no—
434C2×C8—121no—
435C2×C8—121no—
436C2×C8—121no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4(C2×D3)⋊C414yes—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C42—22no—
446C42—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3#38511yescharacteristic
456C22×C3#38611yescharacteristic
457C22×C3#38811yescharacteristic
458C22×C3#39411yescharacteristic
459C22×C3#39611yescharacteristic
460C22×C3#39711yescharacteristic
461C22×C3#39911yescharacteristic
462C22×C3#47112yes—
463C22×C3#47212yes—
464C2×D3—44no—
465C2×D3—84no—
466C2×D3—84no—
467C2×D3—84no—
468C2×D3—84no—
469C2×D3—84no—
470C2×D3—82no—
471C2×D3—82no—
472C2×D3—84no—
473C3×C4—44no—
474C3×C4—82no—
475C3×C4#38011yescharacteristic
476C3×C4#38511yescharacteristic
477C3×C4#38811yescharacteristic
478C3×C4#39011yescharacteristic
479C3×C4#39211yescharacteristic
480C3×C4#39311yescharacteristic
481C3×C4#39411yescharacteristic
482C3×C4#39611yescharacteristic
483C23—22no—
484C23—22no—
485C23—64no—
486C23—122no—
487C23—122no—
488C23—122no—
489C23—122no—
490C23—122no—
491C23—122no—
492C23—244no—
493C23—244no—
494C23—242no—
495C23—242no—
496C23—244no—
497C23—244no—
498C23—242no—
499C23—242no—
500C23—242no—
501C23—242no—
502C23#67⋊C411yescharacteristic
503C23#88311yescharacteristic
504C23#88611yescharacteristic
505C23#88611yescharacteristic
506C23#89711yescharacteristic
507C23#89811yescharacteristic
508C23#93911yescharacteristic
509D4—64no—
510D4—64no—
511D4—122no—
512D4—122no—
513D4—122no—
514D4—122no—
515D4—122no—
516D4—122no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4C4×D1211yescharacteristic
525C2×C4C4×D1211yescharacteristic
526C2×C4#87411yescharacteristic
527C2×C4#87511yescharacteristic
528C2×C4#67⋊C411yescharacteristic
529C2×C4#88211yescharacteristic
530C2×C4#88211yescharacteristic
531C2×C4#88211yescharacteristic
532C2×C4#88311yescharacteristic
533C2×C4#88511yescharacteristic
534C2×C4#88511yescharacteristic
535C2×C4#88611yescharacteristic
536C2×C4D3×C4⋊C411yescharacteristic
537C2×C4#89211yescharacteristic
538C2×C4#89511yescharacteristic
539C2×C4#89511yescharacteristic
540C2×C4#89511yescharacteristic
541C2×C4#89611yescharacteristic
542C2×C4#89611yescharacteristic
543C2×C4#89711yescharacteristic
544C2×C4#89711yescharacteristic
545C2×C4#89811yescharacteristic
546C2×C4C2×(C2×D3)⋊C411yescharacteristic
547C2×C4#93011yescharacteristic
548C2×C4#93011yescharacteristic
549C2×C4#93811yescharacteristic
550C2×C4#93911yescharacteristic
551C2×C4#94611yescharacteristic
552C8—42no—
553C8—122no—
554C8—122no—
555C8—122no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3#143911yescharacteristic
561C2×C3#190712yes—
562C2×C3#204511yescharacteristic
563C2×C3#204611yescharacteristic
564C2×C3#206312yes—
565D3—84no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—124no—
575C22—244no—
576C22—244no—
577C22—244no—
578C22—244no—
579C22—244no—
580C22—244no—
581C22—242no—
582C22—242no—
583C22#466811yescharacteristic
584C22#467011yescharacteristic
585C22#467411yescharacteristic
586C22#467711yescharacteristic
587C22#495811yescharacteristic
588C22#496111yescharacteristic
589C22#499411yescharacteristic
590C22#519912yes—
591C22#520312yes—
592C4—44no—
593C4—82no—
594C4#466811yescharacteristic
595C4#467011yescharacteristic
596C4#467111yescharacteristic
597C4#467311yescharacteristic
598C4#493911yescharacteristic
599C4#496111yescharacteristic
600C4#498711yescharacteristic
601C4#499411yescharacteristic
602C3#1106111yescharacteristic, complemented, Sylow
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—244no—
608C2#6942711yescharacteristic
609C2#7030312yes—
610C2#7031911yescharacteristic
611C2#7033911yescharacteristic
612C2#7067612yes—
subgroupG/H
centerC23#939
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1736C2
socleC23×C3#121
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenter#1736D3
hypocenterC3#11061
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
3. C2×#4999C2semidirect
5. C3×#1736C2semidirect
6. C2×#4558C2semidirect
7. C2×#4530C2semidirect
20. #5001C4semidirect
37. #1736D3semidirect
602. C3#11061semidirect

Maximal subgroups

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

subgroupindex
C22×#8232
C2×#50012
C2×#49992
C22×#8092
C3×#17362
C2×#45582
C2×#45302
#110613