Finite Group Databasebeta

← #10898342

Subgroups — #10898342

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76295C211yesmaximal, characteristic, complemented
2#76252C211yesmaximal, characteristic, complemented
3C3×#2065C211yesmaximal, characteristic, complemented
4#70569C211yesmaximal, characteristic, complemented
5#70559C211yesmaximal, characteristic, complemented
6#70483C211yesmaximal, characteristic, complemented
7#70442C211yesmaximal, characteristic, complemented
8#70190C211yesmaximal, characteristic
9#70188C211yesmaximal, characteristic, complemented
10#70172C211yesmaximal, characteristic
11#70170C211yesmaximal, characteristic, complemented
12#69650C211yesmaximal, characteristic
13#69622C211yesmaximal, characteristic, complemented
14#68856C211yesmaximal, characteristic, complemented
15#68815C211yesmaximal, characteristic, complemented
16#29437—31nomaximal, Sylow
17C2×D4×Dic3C2211yescharacteristic
18C22×#213C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#389C2211yescharacteristic
25#5221C2211yescharacteristic
26C2×#832C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×#815C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4994—21no—
33#4992—21no—
34#4988C2211yescharacteristic
35#4988C2211yescharacteristic
36#4984C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4972C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4945—21no—
43#4944—21no—
44#4943—21no—
45#4942—21no—
46#4939C2211yescharacteristic
47#4932C2211yescharacteristic
48#4928C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4675—21no—
51#4674—21no—
52C3×#340C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2644—31no—
61#2319—31no—
62#2086—31no—
63#2065—31no—
64#2065D311yescharacteristic, complemented
65#2054—31no—
66#1970—31no—
67#1941—31no—
68#1936—31no—
69#1936—31no—
70#1926—31no—
71#1918—31no—
72#1879—31no—
73#1864—31no—
74#1768—31no—
75C22×C3×D4C2311yescharacteristic
76C23×Dic3—21no—
77C22×D12C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90D4×Dic3—22no—
91D4×Dic3—22no—
92C2×#213—22no—
93C2×#213—22no—
94C2×#213—41no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3C2311yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C3×#106—41no—
114C3×C4⋊C8—22no—
115C3×#102—22no—
116C3×#102—41no—
117C3×#95—21no—
118#832—22no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95D411yescharacteristic
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128#821—22no—
129#821—22no—
130#817—22no—
131#815—41no—
132C8×Dic3—41no—
133#809—41no—
134#807—22no—
135C4⋊(C3⋊C8)—22no—
136C22×D8—31no—
137C2×C4×D4—31no—
138C2×#107—31no—
139C2×#106—31no—
140C2×#106C2×D311yescharacteristic
141C2×C4⋊C8—31no—
142C2×C4⋊C8C2×D311yescharacteristic
143C2×#102—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102C2×D311yescharacteristic
148C2×#102C2×D311yescharacteristic
149C2×#97—31no—
150C2×C4×C8—31no—
151#397—61no—
152#397—61no—
153#397—61no—
154#396—61no—
155#390—31no—
156#390—31no—
157#390—31no—
158#390—31no—
159#390C2×D311yescharacteristic
160#389—31no—
161#389—31no—
162#389—31no—
163#389—31no—
164#389C2×D311yescharacteristic
165#388—61no—
166#388—61no—
167#381—61no—
168#380—61no—
169C4×C4⋊C4—31no—
170#376—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
179C24×C3—21no—
180C23×D3—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4#4112yes—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C22×Dic3—21no—
194C22×Dic3—24no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—81no—
198C22×Dic3—81no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8#4112yes—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—81no—
222C3×C22⋊C4—81no—
223C3×C22⋊C4—81no—
224C3×C42—21no—
225C3×C42—21no—
226#217—24no—
227#217—81no—
228#217—81no—
229#213—44no—
230(C2×D3)⋊C4—81no—
231(C2×D3)⋊C4—81no—
232(C2×D3)⋊C4—81no—
233(C2×D3)⋊C4—81no—
234(C2×D3)⋊C4—81no—
235(C2×D3)⋊C4—81no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—81no—
242C4⋊Dic3#4112yes—
243Dic3⋊C4—81no—
244Dic3⋊C4—81no—
245Dic3⋊C4—81no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—81no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8#4112yes—
257C22×D4—31no—
258C22×D4C22×D311yescharacteristic
259C23×C4—61no—
260C2×D8—62no—
261C2×D8—62no—
262C2×D8—121no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267C4×D4—62no—
268C4×D4—62no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C42—31no—
292C2×C42—31no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#106—41no—
297#106—62no—
298C4⋊C8—22no—
299C4⋊C8—62no—
300#102—22no—
301#102—41no—
302#102—62no—
303#102—62no—
304#102—62no—
305#102—121no—
306#97—121no—
307C4×C8—121no—
308#95—21no—
309#95—31no—
310#95—31no—
311#95—31no—
312#95—31no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318C23×C3—22no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3C22×D411yescharacteristic
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—41no—
336C3×D4—41no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—84no—
380D12—41no—
381D12—41no—
382D12—41no—
383D12—41no—
384C3×C8—22no—
385C3×C8—41no—
386C3⋊C8—22no—
387C3⋊C8—41no—
388C24—21no—
389C24—61no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4#23912yes—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—61no—
410C22×C4—64no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—241no—
414C22×C4—241no—
415C22×C4D3×D411yescharacteristic
416C22×C4D3×D411yescharacteristic
417C22×C4C23×D311yescharacteristic
418D8—124no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—32no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8#23712yes—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—32no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—81no—
444C4⋊C4—241no—
445C4⋊C4—241no—
446C4⋊C4—241no—
447C4⋊C4—241no—
448C22⋊C4—64no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C22⋊C4—241no—
452C22⋊C4—241no—
453C22⋊C4—241no—
454C22⋊C4—241no—
455C22⋊C4—241no—
456C22⋊C4—241no—
457C22⋊C4—241no—
458C22⋊C4—241no—
459C42—21no—
460C42—21no—
461C42—61no—
462C42—61no—
463C42—61no—
464C42—61no—
465C42—241no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3#52511yescharacteristic
474C22×C3#53211yescharacteristic
475C22×C3#53711yescharacteristic
476C22×C3#57611yescharacteristic
477C22×C3#57611yescharacteristic
478C22×C3#57711yescharacteristic
479C22×C3#57711yescharacteristic
480C2×D3—81no—
481C2×D3—81no—
482C2×D3—81no—
483C2×D3—81no—
484C2×D3—81no—
485C2×D3—81no—
486C2×D3—81no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4#55111yescharacteristic
493C3×C4#55211yescharacteristic
494C3×C4#55311yescharacteristic
495C3×C4#55611yescharacteristic
496Dic3—41no—
497Dic3—42no—
498Dic3—41no—
499Dic3—41no—
500Dic3—41no—
501Dic3—81no—
502Dic3—81no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—121no—
511C23—121no—
512C23—121no—
513C23—121no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23C2×D3×D411yescharacteristic
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
522D4—121no—
523D4—121no—
524D4—121no—
525D4—121no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—62no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—244no—
563C2×C4C2×#23711yescharacteristic
564C2×C4#100411yescharacteristic
565C2×C4C2×#23911yescharacteristic
566C2×C4D3×#4111yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4#101011yescharacteristic
569C8—22no—
570C8—41no—
571C8—62no—
572C8—121no—
573C2×C3—82no—
574C2×C3#260111yescharacteristic
575C2×C3#311811yescharacteristic
576C2×C3#311911yescharacteristic
577C2×C3#317311yescharacteristic
578C2×C3#317611yescharacteristic
579C2×C3#325711yescharacteristic
580C2×C3#325711yescharacteristic
581D3—161no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—241no—
590C22—241no—
591C22—241no—
592C22—241no—
593C22—241no—
594C22—241no—
595C22—241no—
596C22#559611yescharacteristic
597C22#560811yescharacteristic
598C22#562711yescharacteristic
599C22#576211yescharacteristic
600C22#577011yescharacteristic
601C22#577411yescharacteristic
602C22#577411yescharacteristic
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—121no—
609C4—121no—
610C4—121no—
611C4—122no—
612C4—121no—
613C4—241no—
614C4—241no—
615C4#560111yescharacteristic
616C4#563711yescharacteristic
617C4#564211yescharacteristic
618C4#570211yescharacteristic
619C3#2943711yescharacteristic, complemented, Sylow
620C2—82no—
621C2—481no—
622C2#7732711yescharacteristic
623C2#8040311yescharacteristic
624C2#8051511yescharacteristic
625C2#8063411yescharacteristic
626C2#8085011yescharacteristic
627C2#8104811yescharacteristic
628C2#8114411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2065C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2065D3
hypocenterC3#29437
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2065

Complements

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

NH ≅ G/Ntype
1. #76295C2semidirect
2. #76252C2semidirect
3. C3×#2065C2semidirect
4. #70569C2semidirect
5. #70559C2semidirect
6. #70483C2semidirect
7. #70442C2semidirect
9. #70188C2semidirect
11. #70170C2semidirect
13. #69622C2semidirect
14. #68856C2semidirect
15. #68815C2semidirect
64. #2065D3semidirect
619. C3#29437semidirect

Maximal subgroups

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

subgroupindex
#762952
#762522
C3×#20652
#705692
#705592
#704832
#704422
#701902
#701882
#701722
#701702
#696502
#696222
#688562
#688152
#294373