Finite Group Databasebeta

← #10876140

Subgroups — #10876140

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2297C211yesmaximal, characteristic
2#82062C211yesmaximal, characteristic, complemented
3#72135C212yesmaximal
4#71576C212yesmaximal, complemented
5#71531C211yesmaximal, characteristic, complemented
6#69684C211yesmaximal, characteristic, complemented
7#69670C211yesmaximal, characteristic, complemented
8#69666C212yesmaximal, complemented
9#69656C211yesmaximal, characteristic, complemented
10#69640C211yesmaximal, characteristic, complemented
11#69636C212yesmaximal, complemented
12#27584—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C2×#926—21no—
17C3×#391C2211yescharacteristic
18C3×#391C2211yescharacteristic
19C3×#390C2211yescharacteristic
20C3×#390C2211yescharacteristic
21C3×#386—21no—
22C3×#386—21no—
23C3×C4×C22⋊C4—21no—
24C3×C4×C22⋊C4—21no—
25C2×#835C2212yes—
26C2×#835C2212yes—
27C2×#832C2212yes—
28C2×#832C2212yes—
29#5212—21no—
30#5212C2212yes—
31C2×#830C2212yes—
32C2×#808C2212yes—
33C2×#807C2212yes—
34#4936—21no—
35#4935C2211yescharacteristic
36#4935C2211yescharacteristic
37C4×C4⋊Dic3—21no—
38C4×C4⋊Dic3C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4524C2212yes—
42#4524C2211yescharacteristic
43#4524C2211yescharacteristic
44#2297D311yescharacteristic
45#2285—31no—
46#2095—31no—
47#2094—31no—
48#2072—31no—
49#2071—31no—
50#1927—32no—
51#1925—32no—
52#1854—32no—
53#1817—31no—
54#1788—32no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41—24no—
58C2×C3×#41—42no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C3×#117—42no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42—24no—
75C2×C3×C42—21no—
76C2×C3×C42D411yescharacteristic
77C2×C3×C42D411yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81#926—44no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3C2312yes—
87C2×C4⋊Dic3C2312yes—
88C2×C4×Dic3—22no—
89C22×C3⋊C8C2312yes—
90C22×C3⋊C8C2312yes—
91C3×#95—21no—
92C3×#95—21no—
93C3×#95—21no—
94C3×#95—21no—
95#835—24no—
96#835—42no—
97#832—24no—
98#832—42no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#830—24no—
102#808—42no—
103#807—42no—
104C22×#41C2×D311yescharacteristic
105C2×#117—61no—
106C2×#117C2×D311yescharacteristic
107C22×C42C2×D311yescharacteristic
108C2×#107—32no—
109C2×#106—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×#102—32no—
113C2×#102—32no—
114C2×C22⋊C8—32no—
115#391C2×D311yescharacteristic
116#391C2×D311yescharacteristic
117#390C2×D311yescharacteristic
118#390C2×D311yescharacteristic
119#386—21no—
120#386—21no—
121#384—31no—
122#384—31no—
123#382—61no—
124#380—32no—
125#380—61no—
126C4×C4⋊C4—32no—
127C4×C4⋊C4—61no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130#340—32no—
131#340—32no—
132#336—31no—
133#336—31no—
134#336—32no—
135C24×C3—21no—
136C24×C3C2×D411yescharacteristic
137C3×#41—48no—
138C3×#41—48no—
139C2×C3×Q8—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8#4112yes—
143C2×C3×D4—24no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4#4112yes—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—24no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—41no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4#4114yes—
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—22no—
173C22×Dic3—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C22⋊C4—42no—
177C3×C22⋊C4—42no—
178C3×C22⋊C4—42no—
179C3×C22⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—42no—
186#217—44no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—42no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4⋊Dic3—42no—
195C4×Dic3—44no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—24no—
205C2×#41—42no—
206C22×Q8C22×D311yescharacteristic
207C22×D4—21no—
208C22×D4C22×D311yescharacteristic
209C23×C4—21no—
210C23×C4#6711yescharacteristic
211C23×C4#6711yescharacteristic
212C23×C4C22×D311yescharacteristic
213C22×C8—32no—
214C22×C8—32no—
215#117—42no—
216#117—124no—
217C2×C4⋊C4—32no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—62no—
231C2×C42—24no—
232C2×C42—21no—
233C2×C42—62no—
234C2×C42#6711yescharacteristic
235C2×C42#6711yescharacteristic
236C2×C42C22×D311yescharacteristic
237C2×C42C22×D311yescharacteristic
238#107—122no—
239#106—122no—
240#103—64no—
241#103—122no—
242#102—64no—
243#102—122no—
244C22⋊C8—64no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—62no—
250#95—62no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—22no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3#12312yes—
265C23×C3C22×D411yescharacteristic
266C3×Q8—24no—
267C3×Q8—42no—
268C3×D4—24no—
269C3×D4—48no—
270C3×D4—42no—
271C2×C3×C4—28no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—44no—
287C2×C3×C4—48no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4#12312yes—
297C2×C3×C4#13512yes—
298C2×C3×C4AGL(1,ℤ/8)12yes—
299C2×C3×C4#13712yes—
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4112yes—
303C2×C3×C4H(5,𝔽2)12yes—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C3⋊C8—42no—
311C3⋊C8—42no—
312C24—21no—
313C24C2×#6711yescharacteristic
314#41—48no—
315#41—48no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8#23512yes—
320C2×D4—24no—
321C2×D4—21no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4#23512yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—62no—
343C22×C4—62no—
344C22×C4#23714yes—
345C22×C4C2×#6711yescharacteristic
346C22×C4C2×#6711yescharacteristic
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350C22×C4C23×D311yescharacteristic
351C2×C8—62no—
352C2×C8—62no—
353C2×C8—62no—
354C2×C8—62no—
355C2×C8—62no—
356C2×C8—62no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—62no—
360C4⋊C4—62no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—122no—
364C4⋊C4—122no—
365C4⋊C4—122no—
366C4⋊C4—122no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—124no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—42no—
378C42—124no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3C2×#12311yescharacteristic
394C22×C3#52511yescharacteristic
395C22×C3#53511yescharacteristic
396C22×C3C2×#13511yescharacteristic
397C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
398C22×C3C2×#13711yescharacteristic
399C22×C3#57511yescharacteristic
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—41no—
405C3×C4—42no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4#55112yes—
409C3×C4#55112yes—
410Dic3—82no—
411Dic3—82no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23#93512yes—
426C23C22×#6711yescharacteristic
427Q8—24no—
428Q8—42no—
429D4—24no—
430D4—48no—
431D4—42no—
432C2×C4—28no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4#93512yes—
464C2×C4#94912yes—
465C2×C4#95012yes—
466C2×C4#95112yes—
467C2×C4C2×#23511yescharacteristic
468C2×C4C2×#23511yescharacteristic
469C2×C4C2×#23712yes—
470C2×C4#100412yes—
471C8—122no—
472C8—122no—
473C2×C3—42no—
474C2×C3—82no—
475C2×C3#248211yescharacteristic
476C2×C3#309111yescharacteristic
477C2×C3#309211yescharacteristic
478C2×C3#310611yescharacteristic
479C2×C3#311611yescharacteristic
480C2×C3#321011yescharacteristic
481C2×C3#321411yescharacteristic
482C22—22no—
483C22—22no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22#579111yescharacteristic
497C22C2×#93511yescharacteristic
498C22#580511yescharacteristic
499C22C2×#94911yescharacteristic
500C22C2×#95011yescharacteristic
501C22#582211yescharacteristic
502C22C2×#95111yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—42no—
508C4—41no—
509C4—81no—
510C4—81no—
511C4—242no—
512C4—242no—
513C4#549512yes—
514C4#549512yes—
515C3#2758411yescharacteristic, complemented, Sylow
516C2—42no—
517C2—82no—
518C2#8211311yescharacteristic
519C2#8252911yescharacteristic
520C2#8253511yescharacteristic
521C2#8291311yescharacteristic
522C2#8291411yescharacteristic
523C2#8330711yescharacteristic
524C2#8331811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2297C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenter#2297D3
hypocenterC3#27584
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #82062C2semidirect
4. #71576C2semidirect
5. #71531C2semidirect
6. #69684C2semidirect
7. #69670C2semidirect
8. #69666C2semidirect
9. #69656C2semidirect
10. #69640C2semidirect
11. #69636C2semidirect
515. C3#27584semidirect

Maximal subgroups

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

subgroupindex
C3×#22972
#820622
#721352
#715762
#715312
#696842
#696702
#696662
#696562
#696402
#696362
#275843