Finite Group Databasebeta

← #10703012

Subgroups — #10703012

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#825C211yesmaximal, characteristic
2C2×#5110C211yesmaximal, characteristic
3C2×#5002C211yesmaximal, characteristic
4C3×#1729C211yesmaximal, characteristic, complemented
5C2×#4558C211yesmaximal, characteristic, complemented
6C2×#4550C211yesmaximal, characteristic, complemented
7C2×#4532C211yesmaximal, characteristic, complemented
8#11191—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C23×C3×C8C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C22×C4×Dic3C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C23×C3⋊C8C2211yescharacteristic
16C2×#825—22no—
17C2×#825—22no—
18C2×#825—22no—
19#5110—22no—
20#5110—22no—
21C2×(C2×D3)⋊C8—22no—
22#5002—22no—
23#5002—22no—
24C2×#810C412yescomplemented
25C2×#809C412yescomplemented
26C3×#336—22no—
27#4558—22no—
28#4550—22no—
29#4532—22no—
30C22×#104—31no—
31C2×#416—31no—
32C2×#408—31no—
33C2×#341—31no—
34C2×#339—31no—
35#1729D311yescharacteristic, complemented
36C2×#336—31no—
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C23×Dic3—21no—
40C2×#235—21no—
41C2×#235—21no—
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235D412yes—
51C2×#235D412yes—
52C22×D12C2×C411yescharacteristic
53C22×C4×D3—21no—
54C22×Dic6C2×C411yescharacteristic
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—
62C22×C3×C8—21no—
63C22×C3×C8—21no—
64C22×C3×C8—21no—
65C22×C3×C8—21no—
66C22×C3×C8—21no—
67C22×C3×C8—21no—
68C22×C3×C8D412yes—
69C3×#117—21no—
70C3×#117—21no—
71C3×#117—21no—
72C3×#117—21no—
73C2×C3×C4⋊C4C2×C412yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×C4×Dic3—21no—
77C2×C4×Dic3—21no—
78C2×C4×Dic3—21no—
79C2×C4×Dic3—21no—
80C2×C4×Dic3—21no—
81C2×C4×Dic3D412yes—
82C2×#208—21no—
83C2×#208—21no—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208D412yes—
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⋊C8C2×C412yes—
96#825—28no—
97(C2×D3)⋊C8—28no—
98#810—24no—
99#809—24no—
100C22×#41—31no—
101C22×M16—31no—
102C22×M16C2×D311yescharacteristic
103C23×C8—31no—
104C23×C8C2×D311yescharacteristic
105C2×#117C2×D311yescharacteristic
106C22×C42—31no—
107C2×#104—62no—
108C2×#104—62no—
109C2×#104—62no—
110#416—62no—
111#416—62no—
112C2×#103—32no—
113C2×#102—32no—
114#408—62no—
115#408—62no—
116C2×C22⋊C8—62no—
117#341—62no—
118#339—62no—
119#336—22no—
120#336—62no—
121C24×C3#4111yescharacteristic
122C23×D3—21no—
123C22×C3×C4—22no—
124C22×C3×C4C22×C411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×#67—22no—
139C2×#67—22no—
140C2×#67—41no—
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—41no—
144C22×Dic3—24no—
145C22×Dic3—21no—
146C22×Dic3—22no—
147C22×Dic3—42no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150#235—22no—
151#235—22no—
152#235—22no—
153#235—22no—
154#235—22no—
155#235—22no—
156#235—22no—
157#235—22no—
158#235—41no—
159#235—41no—
160#235—41no—
161#235—41no—
162#235—41no—
163#235—41no—
164#235—41no—
165#235—41no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12C22⋊C412yes—
172C2×C4×D3—22no—
173C2×C4×D3—22no—
174C2×C4×D3—41no—
175C2×C4×D3—41no—
176C2×C4×D3—41no—
177C2×C4×D3—41no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6C22⋊C412yes—
184C3×M16—22no—
185C3×M16—22no—
186C3×M16—41no—
187C3×M16—41no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—41no—
193C2×C3×C8—41no—
194C2×C3×C8—41no—
195C2×C3×C8—41no—
196C2×C3×C8—41no—
197C2×C3×C8—41no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C22⋊C4—41no—
201C3×C22⋊C4—41no—
202C3×C42—41no—
203C3×C42—41no—
204C4×Dic3—22no—
205C4×Dic3—22no—
206C4×Dic3—41no—
207C4×Dic3—41no—
208#208—22no—
209#208—22no—
210#208—41no—
211#208—41no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—41no—
217C2×C3⋊C8—41no—
218C2×C3⋊C8—41no—
219C2×C3⋊C8—41no—
220C2×C3⋊C8—41no—
221C2×C3⋊C8—41no—
222C2×#41—32no—
223C2×#41—32no—
224C2×#41—61no—
225C2×#41—61no—
226C2×#41—61no—
227C2×#41—61no—
228C2×#41—61no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C22×Q8—31no—
235C22×D4—31no—
236C22×D4—61no—
237C23×C4—61no—
238C23×C4—61no—
239C23×C4C22×D311yescharacteristic
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—32no—
246C2×M16—61no—
247C2×M16—61no—
248C2×M16—61no—
249C2×M16—61no—
250C2×M16—61no—
251C2×M16D1212yes—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—32no—
260C22×C8—61no—
261C22×C8—61no—
262C22×C8—61no—
263C22×C8—61no—
264C22×C8—61no—
265C22×C8—61no—
266C22×C8—61no—
267C22×C8#6712yes—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272C2×C4⋊C4C4×D312yes—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42—32no—
276C2×C42—61no—
277C2×C42—61no—
278C2×C42—61no—
279C2×C42—61no—
280C2×C42—61no—
281#104—68no—
282#103—64no—
283#102—64no—
284C22⋊C8—68no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3C4×D411yescharacteristic
288C23×C3#12111yescharacteristic
289C23×C3#12311yescharacteristic
290C23×C3#12311yescharacteristic
291C23×C3#12411yescharacteristic
292C23×C3#12611yescharacteristic
293C23×C3#12611yescharacteristic
294C22×D3—22no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—42no—
298C22×D3—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4C2×C22⋊C411yescharacteristic
303C2×C3×C4#11711yescharacteristic
304C2×C3×C4#11711yescharacteristic
305C2×C3×C4C4×D411yescharacteristic
306C2×C3×C4C4×D411yescharacteristic
307C2×C3×C4C4×D411yescharacteristic
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4#12111yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12311yescharacteristic
318C2×C3×C4#12311yescharacteristic
319C2×C3×C4#12311yescharacteristic
320C2×C3×C4#12311yescharacteristic
321C2×C3×C4#12311yescharacteristic
322C2×C3×C4#12311yescharacteristic
323C2×C3×C4#12311yescharacteristic
324C2×C3×C4#12311yescharacteristic
325C2×C3×C4#12411yescharacteristic
326C2×C3×C4#12411yescharacteristic
327C2×C3×C4#12411yescharacteristic
328C2×C3×C4#12611yescharacteristic
329C2×C3×C4#12611yescharacteristic
330#67—42no—
331#67—42no—
332#67—42no—
333#67—42no—
334C2×Dic3—24no—
335C2×Dic3—22no—
336C2×Dic3—44no—
337C2×Dic3—44no—
338C2×Dic3—42no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342D12—22no—
343D12—22no—
344D12—41no—
345D12—41no—
346C4×D3—42no—
347C4×D3—42no—
348C4×D3—42no—
349C4×D3—42no—
350Dic6—22no—
351Dic6—22no—
352Dic6—41no—
353Dic6—41no—
354C3×C8—42no—
355C3×C8—42no—
356C3⋊C8—42no—
357C3⋊C8—42no—
358C24—61no—
359C24#23511yescharacteristic
360#41—62no—
361#41—62no—
362#41—62no—
363#41—62no—
364#41—62no—
365#41—62no—
366#41—62no—
367#41—62no—
368#41—121no—
369#41—121no—
370#41—121no—
371#41—121no—
372#41—121no—
373#41—121no—
374#41—121no—
375#41—121no—
376C2×Q8—32no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×D4—32no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—121no—
391C2×D4—121no—
392C2×D4—121no—
393C2×D4—121no—
394C22×C4—22no—
395C22×C4—64no—
396C22×C4—62no—
397C22×C4—61no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—122no—
401C22×C4—121no—
402C22×C4—121no—
403C22×C4—121no—
404C22×C4—121no—
405C22×C4—121no—
406C22×C4—121no—
407C22×C4C2×C4×D311yescharacteristic
408C22×C4C2×D1211yescharacteristic
409C22×C4#23511yescharacteristic
410C22×C4#23511yescharacteristic
411C22×C4#23511yescharacteristic
412C22×C4D3×D411yescharacteristic
413C22×C4D3×D411yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4#23711yescharacteristic
417C22×C4#23711yescharacteristic
418C22×C4#23911yescharacteristic
419C22×C4#23911yescharacteristic
420C22×C4C2×#6711yescharacteristic
421M16—22no—
422M16—22no—
423M16—41no—
424M16—41no—
425M16—62no—
426M16—62no—
427M16—121no—
428M16—121no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—62no—
440C2×C8—62no—
441C2×C8—62no—
442C2×C8—62no—
443C2×C8—121no—
444C2×C8—121no—
445C2×C8—121no—
446C2×C8—121no—
447C2×C8—121no—
448C2×C8—121no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C42—41no—
454C42—41no—
455C42—62no—
456C42—62no—
457C42—121no—
458C42—121no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3#38611yescharacteristic
468C22×C3#38811yescharacteristic
469C22×C3#38811yescharacteristic
470C22×C3#39411yescharacteristic
471C22×C3#39711yescharacteristic
472C22×C3#39711yescharacteristic
473C22×C3#39711yescharacteristic
474C22×C3#44312yes—
475C22×C3#44412yes—
476C2×D3—44no—
477C2×D3—42no—
478C2×D3—82no—
479C2×D3—81no—
480C2×D3—81no—
481C3×C4—82no—
482C3×C4#38111yescharacteristic
483C3×C4#38611yescharacteristic
484C3×C4#38811yescharacteristic
485C3×C4#38811yescharacteristic
486C3×C4#39411yescharacteristic
487C3×C4#39411yescharacteristic
488C3×C4#39411yescharacteristic
489C3×C4#39711yescharacteristic
490Dic3—44no—
491Dic3—42no—
492C23—22no—
493C23—22no—
494C23—62no—
495C23—122no—
496C23—122no—
497C23—121no—
498C23—121no—
499C23#87511yescharacteristic
500C23#87611yescharacteristic
501C23#88311yescharacteristic
502C23#88411yescharacteristic
503C23#89411yescharacteristic
504C23#89811yescharacteristic
505C23C4×#6711yescharacteristic
506Q8—62no—
507Q8—62no—
508Q8—121no—
509Q8—121no—
510D4—62no—
511D4—62no—
512D4—122no—
513D4—122no—
514D4—122no—
515D4—122no—
516D4—121no—
517D4—121no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—64no—
522C2×C4—62no—
523C2×C4—124no—
524C2×C4—124no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4#87211yescharacteristic
534C2×C4C4×D1211yescharacteristic
535C2×C4D3×C22⋊C411yescharacteristic
536C2×C4#67⋊C411yescharacteristic
537C2×C4#88211yescharacteristic
538C2×C4#88311yescharacteristic
539C2×C4#88311yescharacteristic
540C2×C4#88411yescharacteristic
541C2×C4#88411yescharacteristic
542C2×C4#88511yescharacteristic
543C2×C4#88511yescharacteristic
544C2×C4#88611yescharacteristic
545C2×C4#89211yescharacteristic
546C2×C4#89311yescharacteristic
547C2×C4#89411yescharacteristic
548C2×C4#89411yescharacteristic
549C2×C4#89411yescharacteristic
550C2×C4#89411yescharacteristic
551C2×C4#89511yescharacteristic
552C2×C4#89511yescharacteristic
553C2×C4#89811yescharacteristic
554C2×C4#89811yescharacteristic
555C2×C4#92911yescharacteristic
556C2×C4#93011yescharacteristic
557C2×C4#93711yescharacteristic
558C2×C4#93811yescharacteristic
559C2×C4#93911yescharacteristic
560C2×C4#94711yescharacteristic
561C8—42no—
562C8—42no—
563C8—122no—
564C8—122no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3#144011yescharacteristic
570C2×C3#187612yes—
571C2×C3#192912yes—
572C2×C3#193711yescharacteristic
573C2×C3#193811yescharacteristic
574D3—82no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—122no—
584C22—124no—
585C22—242no—
586C22—241no—
587C22—241no—
588C22#467211yescharacteristic
589C22#467411yescharacteristic
590C22#467511yescharacteristic
591C22#467511yescharacteristic
592C22#494011yescharacteristic
593C22#495611yescharacteristic
594C22#499011yescharacteristic
595C22#514412yes—
596C22#514512yes—
597C4—82no—
598C4—124no—
599C4—122no—
600C4#466411yescharacteristic
601C4#466511yescharacteristic
602C4#466911yescharacteristic
603C4#467011yescharacteristic
604C4#495911yescharacteristic
605C4#496111yescharacteristic
606C4#499111yescharacteristic
607C4#499211yescharacteristic
608C3#1119111yescharacteristic, complemented, Sylow
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—242no—
614C2#6942911yescharacteristic
615C2#6990012yes—
616C2#7001112yes—
617C2#7053511yescharacteristic
618C2#7053611yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1729C2
socleC23×C3C4×D4
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1729D3
hypocenterC3#11191
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C23×C8 ◁ #1729

Complements

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

NH ≅ G/Ntype
4. C3×#1729C2semidirect
5. C2×#4558C2semidirect
6. C2×#4550C2semidirect
7. C2×#4532C2semidirect
24. C2×#810C4semidirect
25. C2×#809C4semidirect
35. #1729D3semidirect
608. C3#11191semidirect

Maximal subgroups

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

subgroupindex
C22×#8252
C2×#51102
C2×#50022
C3×#17292
C2×#45582
C2×#45502
C2×#45322
#111913