Finite Group Databasebeta

← #10898401

Subgroups — #10898401

Order 768.

674 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2319C211yesmaximal, characteristic
2#73408C211yesmaximal, characteristic, complemented
3#71612C211yesmaximal, characteristic
4#71606C211yesmaximal, characteristic
5#69653C211yesmaximal, characteristic
6#69651C211yesmaximal, characteristic
7#69646C211yesmaximal, characteristic
8#69645C211yesmaximal, characteristic
9#69635C211yesmaximal, characteristic, complemented
10#69624C211yesmaximal, characteristic, complemented
11#69619C211yesmaximal, characteristic, complemented
12#69616C211yesmaximal, characteristic, complemented
13#69465C211yesmaximal, characteristic, complemented
14#69460C211yesmaximal, characteristic, complemented
15#69436C211yesmaximal, characteristic, complemented
16#29441—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C22×#214C2211yescharacteristic
19C3×#390C2211yescharacteristic
20C3×#390C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#388—21no—
24C3×#388—21no—
25C3×#381—21no—
26C3×#380—21no—
27C3×C4×C4⋊C4C2211yescharacteristic
28C3×#376C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#810C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2211yescharacteristic
35C2×#808C2211yescharacteristic
36C2×#807C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4929C2211yescharacteristic
39#4929C2211yescharacteristic
40#4929C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46C2×#803C2211yescharacteristic
47C2×C4×C3⋊C8C2211yescharacteristic
48C3⋊#401—21no—
49#4658—21no—
50#4658—21no—
51#4658—21no—
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2645—31no—
61#2319D311yescharacteristic
62#2088—31no—
63#2068—31no—
64#2067—31no—
65#2055—31no—
66#1968—31no—
67#1942—31no—
68#1936—31no—
69#1935—31no—
70#1928—31no—
71#1920—31no—
72#1877—31no—
73#1862—31no—
74#1768—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×Dic6C2311yescharacteristic
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#214—22no—
95C2×#214—22no—
96C2×#214—41no—
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C22×C3⋊C8C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95D411yescharacteristic
111C3×#95D411yescharacteristic
112C3×#95D411yescharacteristic
113C3×#95D411yescharacteristic
114#832—22no—
115#832—22no—
116#832—41no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126#810—22no—
127#810—22no—
128#810—41no—
129#808—22no—
130#807—22no—
131#807—41no—
132C4⋊(C3⋊C8)—22no—
133C4⋊(C3⋊C8)—22no—
134#803—41no—
135C4×C3⋊C8—41no—
136C22×SD16—31no—
137C2×C4×D4C2×D311yescharacteristic
138C2×#107—31no—
139C2×#107—31no—
140C2×#106—31no—
141C2×C4⋊C8—31no—
142C2×C4⋊C8—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#97—31no—
150C2×C4×C8—31no—
151#401—61no—
152#400—61no—
153#400—61no—
154#400—61no—
155#391—31no—
156#391—31no—
157#391—31no—
158#390C2×D311yescharacteristic
159#390C2×D311yescharacteristic
160#389—31no—
161#389—31no—
162#389—31no—
163#389C2×D311yescharacteristic
164#389C2×D311yescharacteristic
165#388—21no—
166#388—21no—
167#381—21no—
168#380—21no—
169C4×C4⋊C4C2×D311yescharacteristic
170#376C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336—31no—
178#336—31no—
179C24×C3—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—81no—
191C22×C3×C4—81no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—81no—
215C3×C4⋊C4—81no—
216C3×C4⋊C4#4112yes—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—81no—
219C3×C22⋊C4—81no—
220C3×C22⋊C4—81no—
221C3×C22⋊C4—81no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—81no—
229#214—44no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236Dic3⋊C4—81no—
237Dic3⋊C4—81no—
238Dic3⋊C4—81no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8#4112yes—
250C2×C3⋊C8#4112yes—
251C22×Q8—31no—
252C22×D4C22×D311yescharacteristic
253C23×C4—21no—
254C2×SD16—62no—
255C2×SD16—62no—
256C2×SD16—121no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—31no—
260C22×C8—31no—
261C4×D4—22no—
262C4×D4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4—61no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4C22×D311yescharacteristic
272C2×C4⋊C4C22×D311yescharacteristic
273C2×C4⋊C4C22×D311yescharacteristic
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C42—21no—
280C2×C42C22×D311yescharacteristic
281C2×C42C22×D311yescharacteristic
282C2×C42C22×D311yescharacteristic
283#107—62no—
284#107—121no—
285#106—62no—
286C4⋊C8—62no—
287C4⋊C8—62no—
288#103—62no—
289#103—62no—
290#103—121no—
291#102—62no—
292#102—62no—
293#102—121no—
294#97—121no—
295C4×C8—121no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—61no—
300#95—61no—
301#95—61no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95#6711yescharacteristic
309#95#6711yescharacteristic
310#95#6711yescharacteristic
311#95#6711yescharacteristic
312C23×C3—22no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3C22×D411yescharacteristic
320C3×D4—22no—
321C3×D4—22no—
322C3×D4—41no—
323C3×D4—41no—
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—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
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—84no—
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—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×Dic3—41no—
369C2×Dic3—41no—
370Dic6—41no—
371Dic6—41no—
372Dic6—41no—
373Dic6—41no—
374C3⋊C8—22no—
375C3⋊C8—22no—
376C3⋊C8—41no—
377C3⋊C8—41no—
378C24—21no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4#23512yes—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4C2×#6711yescharacteristic
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6711yescharacteristic
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C23×D311yescharacteristic
408SD16—124no—
409C2×C8—32no—
410C2×C8—32no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—61no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—241no—
435C4⋊C4—241no—
436C4⋊C4—241no—
437C4⋊C4#23512yes—
438C22⋊C4—24no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—81no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3#52511yescharacteristic
458C22×C3#53211yescharacteristic
459C22×C3#53711yescharacteristic
460C22×C3#57611yescharacteristic
461C22×C3#57711yescharacteristic
462C22×C3#57711yescharacteristic
463C22×C3#57811yescharacteristic
464C3×C4—41no—
465C3×C4—42no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—81no—
472C3×C4—81no—
473C3×C4#55111yescharacteristic
474C3×C4#55211yescharacteristic
475C3×C4#55311yescharacteristic
476C3×C4#55611yescharacteristic
477Dic3—81no—
478Dic3—81no—
479Dic3—81no—
480Dic3—81no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23C22×#6711yescharacteristic
489Q8—121no—
490Q8—121no—
491Q8—121no—
492Q8—121no—
493D4—22no—
494D4—22no—
495D4—41no—
496D4—41no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—84no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
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×C4C2×#23511yescharacteristic
538C2×C4C2×#23511yescharacteristic
539C2×C4D3×#4111yescharacteristic
540C2×C4D3×#4111yescharacteristic
541C2×C4#100911yescharacteristic
542C2×C4#101011yescharacteristic
543C8—62no—
544C8—62no—
545C8—121no—
546C8—121no—
547C2×C3—82no—
548C2×C3#260111yescharacteristic
549C2×C3#311811yescharacteristic
550C2×C3#312011yescharacteristic
551C2×C3#317411yescharacteristic
552C2×C3#317511yescharacteristic
553C2×C3#325711yescharacteristic
554C2×C3#325811yescharacteristic
555C22—42no—
556C22—42no—
557C22—42no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22#579111yescharacteristic
563C22#583211yescharacteristic
564C22#583311yescharacteristic
565C22#583511yescharacteristic
566C22#583611yescharacteristic
567C22#583611yescharacteristic
568C22#583811yescharacteristic
569C4—41no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—241no—
579C4—241no—
580C4—241no—
581C4—241no—
582C4#552711yescharacteristic
583C4#553011yescharacteristic
584C4#553111yescharacteristic
585C4#553311yescharacteristic
586C3#2944111yescharacteristic, complemented, Sylow
587C2—82no—
588C2#8225911yescharacteristic
589C2#8254111yescharacteristic
590C2#8257511yescharacteristic
591C2#8343111yescharacteristic
592C2#8343711yescharacteristic
593C2#8345711yescharacteristic
594C2#8346811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2319C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2319D3
hypocenterC3#29441
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #73408C2semidirect
9. #69635C2semidirect
10. #69624C2semidirect
11. #69619C2semidirect
12. #69616C2semidirect
13. #69465C2semidirect
14. #69460C2semidirect
15. #69436C2semidirect
586. C3#29441semidirect

Maximal subgroups

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

subgroupindex
C3×#23192
#734082
#716122
#716062
#696532
#696512
#696462
#696452
#696352
#696242
#696192
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