Finite Group Databasebeta

← #10867938

Subgroups — #10867938

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76205C211yesmaximal, characteristic
2#73793C211yesmaximal, characteristic
3C3×#1921C211yesmaximal, characteristic, complemented
4#70635C211yesmaximal, characteristic
5#70615C211yesmaximal, characteristic, complemented
6#69619C211yesmaximal, characteristic, complemented
7C4×#810C211yesmaximal, characteristic, complemented
8#68914C211yesmaximal, characteristic, complemented
9#68913C211yesmaximal, characteristic, complemented
10#68874C211yesmaximal, characteristic
11#68872C211yesmaximal, characteristic
12#68825C211yesmaximal, characteristic
13#68820C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68657C211yesmaximal, characteristic
16#27036—31nomaximal, Sylow
17C2×#897C2211yescharacteristic
18C2×D3×C4⋊C4C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C8C2211yescharacteristic
24C2×(C2×D3)⋊C8C2211yescharacteristic
25C2×#818C2211yescharacteristic
26C2×#816C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29#4989—21no—
30#4987C2211yescharacteristic
31#4987C2211yescharacteristic
32#4975C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34#4973C2211yescharacteristic
35#4972—21no—
36#4972C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4940—21no—
42C4×(C2×D3)⋊C4C2211yescharacteristic
43#4933—21no—
44#4928C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4676—21no—
47#4666—21no—
48#4658—21no—
49#4654—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2497—31no—
61#2360—31no—
62#2072—31no—
63#2068—31no—
64#2036—31no—
65#2036—31no—
66#1947—31no—
67#1942—31no—
68#1923—31no—
69#1921—31no—
70#1921D311yescharacteristic, complemented
71#1863—31no—
72#1852—31no—
73#1824—31no—
74C4×#103—31no—
75C22×C4×D3—21no—
76C22×C4×D3C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
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×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4D411yescharacteristic
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107#897—22no—
108#897—22no—
109D3×C4⋊C4—22no—
110D3×C4⋊C4—41no—
111C4×Dic6—22no—
112C4×Dic6—41no—
113C3×#107—22no—
114C3×#95—21no—
115C3×#95—21no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95D411yescharacteristic
124C3⋊#95D411yescharacteristic
125(C2×D3)⋊C8—22no—
126(C2×D3)⋊C8—41no—
127#818—22no—
128#816—22no—
129#816—41no—
130Dic3⋊C8—22no—
131#810—22no—
132#810—22no—
133#807—22no—
134#807—41no—
135C4×C3⋊C8—41no—
136C2×C22⋊Q8—31no—
137C2×C4×Q8—31no—
138C22×C4⋊C4—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#107—31no—
142C2×#107C2×D311yescharacteristic
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#103—31no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×C4×C8—31no—
151#400—61no—
152#396—61no—
153#391—31no—
154#389—31no—
155#389—31no—
156#389—61no—
157#389C2×D311yescharacteristic
158#388—61no—
159#387—61no—
160#387—61no—
161#385—61no—
162#384—31no—
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4C2×D311yescharacteristic
167C4×C22⋊C4—31no—
168C4×C22⋊C4—61no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C23×D3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—21no—
197C2×C4×D3—21no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—42no—
201C2×C4×D3—42no—
202C2×C4×D3—81no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6#4112yes—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
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—21no—
222C3×C4⋊C4—81no—
223C3×C4⋊C4#4112yes—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—81no—
227(C2×D3)⋊C4—24no—
228(C2×D3)⋊C4—41no—
229(C2×D3)⋊C4—81no—
230(C2×D3)⋊C4—81no—
231(C2×D3)⋊C4Q1614yes—
232C4⋊Dic3—24no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3#4112yes—
239Dic3⋊C4—42no—
240Dic3⋊C4—42no—
241Dic3⋊C4—42no—
242Dic3⋊C4—81no—
243Dic3⋊C4—81no—
244Dic3⋊C4—81no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—42no—
248C4×Dic3—81no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8#4112yes—
255C22×Q8—31no—
256C23×C4—31no—
257C23×C4—61no—
258C22×C8—31no—
259C22×C8—31no—
260C22×C8C22×D311yescharacteristic
261C22×C8C22×D311yescharacteristic
262C22⋊Q8—62no—
263C22⋊Q8—62no—
264C4×Q8—62no—
265C4×Q8—121no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—121no—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C22⋊C4—31no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C42—21no—
289C2×C42—31no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42C22×D311yescharacteristic
293#107—22no—
294#107—62no—
295#107—62no—
296#107—121no—
297C4⋊C8—62no—
298#103—62no—
299#103—62no—
300#103—62no—
301#103—121no—
302C22⋊C8—62no—
303C22⋊C8—121no—
304C4×C8—121no—
305#95—21no—
306#95—21no—
307#95—31no—
308#95—31no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316C23×C3C22×D411yescharacteristic
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3C2×Q1612yes—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—22no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4C2×Q1612yes—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C42−1+411yescharacteristic
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3C2×Q1612yes—
369C4×D3—41no—
370C4×D3—41no—
371C4×D3—41no—
372C4×D3—41no—
373C4×D3—84no—
374Dic6—22no—
375Dic6—22no—
376Dic6—41no—
377Dic6—41no—
378C3×C8—41no—
379C3×C8—41no—
380C3⋊C8—22no—
381C3⋊C8—41no—
382C24—31no—
383C2×Q8—32no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—31no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4—241no—
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4C23×D311yescharacteristic
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—32no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—32no—
432C4⋊C4—64no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—81no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4—122no—
442C4⋊C4—241no—
443C4⋊C4—241no—
444C4⋊C4—241no—
445C4⋊C4#23512yes—
446C22⋊C4—34no—
447C22⋊C4—64no—
448C22⋊C4—121no—
449C22⋊C4—241no—
450C22⋊C4—241no—
451C42—21no—
452C42—21no—
453C42—61no—
454C42—61no—
455C42—81no—
456C42—122no—
457C42—241no—
458C22×C3#52511yescharacteristic
459C22×C3#54611yescharacteristic
460C22×C3#54711yescharacteristic
461C22×C3C22×Q1611yescharacteristic
462C22×C3#57511yescharacteristic
463C22×C3#57611yescharacteristic
464C22×C3#57711yescharacteristic
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—41no—
471C2×D3—41no—
472C3×C4—22no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—81no—
476C3×C4—81no—
477C3×C4—81no—
478C3×C4#52911yescharacteristic
479C3×C4#53911yescharacteristic
480C3×C4#55311yescharacteristic
481C3×C4#55611yescharacteristic
482Dic3—22no—
483Dic3—41no—
484Dic3—42no—
485Dic3—41no—
486Dic3—42no—
487Dic3—81no—
488Dic3—81no—
489C23—32no—
490C23—61no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23—61no—
496C23C2×D3×D411yescharacteristic
497Q8—62no—
498Q8—62no—
499Q8—121no—
500Q8—121no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—32no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—62no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—244no—
544C2×C4D3×Q1612yes—
545C2×C4C2×#23511yescharacteristic
546C2×C4#100711yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4#100911yescharacteristic
551C8—41no—
552C8—41no—
553C8—62no—
554C8—121no—
555C2×C3#246611yescharacteristic
556C2×C3#310411yescharacteristic
557C2×C3#311911yescharacteristic
558C2×C3#329911yescharacteristic
559C2×C3#331911yescharacteristic
560C2×C3#333811yescharacteristic
561C2×C3#335111yescharacteristic
562D3—42no—
563C22—62no—
564C22—62no—
565C22—62no—
566C22—121no—
567C22—121no—
568C22—121no—
569C22—121no—
570C22#554711yescharacteristic
571C22#563111yescharacteristic
572C22#565411yescharacteristic
573C22C2×D3×Q1611yescharacteristic
574C22#576811yescharacteristic
575C22#577411yescharacteristic
576C22#577511yescharacteristic
577C4—22no—
578C4—42no—
579C4—41no—
580C4—62no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—121no—
585C4—121no—
586C4—122no—
587C4—122no—
588C4—241no—
589C4—241no—
590C4#553011yescharacteristic
591C4#556811yescharacteristic
592C4#569211yescharacteristic
593C4#570611yescharacteristic
594C3#2703611yescharacteristic, complemented, Sylow
595C2—122no—
596C2#7533011yescharacteristic
597C2#7972211yescharacteristic
598C2#7987311yescharacteristic
599C2#8112511yescharacteristic
600C2#8120011yescharacteristic
601C2#8140211yescharacteristic
602C2#8149011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1921C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1921D3
hypocenterC3#27036
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
3. C3×#1921C2semidirect
5. #70615C2semidirect
6. #69619C2semidirect
7. C4×#810C2semidirect
8. #68914C2semidirect
9. #68913C2semidirect
13. #68820C2semidirect
14. #68683C2semidirect
70. #1921D3semidirect
594. C3#27036semidirect

Maximal subgroups

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

subgroupindex
#762052
#737932
C3×#19212
#706352
#706152
#696192
C4×#8102
#689142
#689132
#688742
#688722
#688252
#688202
#686832
#686572
#270363