Finite Group Databasebeta

← #10877168

Subgroups — #10877168

Order 768.

740 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3092C211yesmaximal, characteristic
2#82916C211yesmaximal, characteristic
3#82914C211yesmaximal, characteristic
4C22×#807C211yesmaximal, characteristic, complemented
5#71562C211yesmaximal, characteristic, complemented
6#70670C212yesmaximal, complemented
7#70669C212yesmaximal, complemented
8#70301C212yesmaximal, complemented
9#70300C212yesmaximal, complemented
10#69982C211yesmaximal, characteristic, complemented
11#69973C211yesmaximal, characteristic, complemented
12#27675—31nomaximal, Sylow
13C3×#525C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×#948C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C23×C3⋊C8C2211yescharacteristic
18C3×#484—21no—
19C3×#483—21no—
20C3×#481—21no—
21C3×#480—21no—
22C3×#428C2212yes—
23C3×#417C2212yes—
24C3×#407C2211yescharacteristic
25#5234C2211yescharacteristic
26#5232C2212yes—
27#5207C2211yescharacteristic
28#5207C2211yescharacteristic
29C2×#822C2212yes—
30#5044—21no—
31#5043—21no—
32#5034—21no—
33#5033—21no—
34#5011—21no—
35#5011—21no—
36D4×C3⋊C8—21no—
37D4×C3⋊C8—21no—
38#4999C2211yescharacteristic
39#4997C2211yescharacteristic
40#4995C2212yes—
41C2×#807—21no—
42C2×#807—21no—
43C2×#807—21no—
44C2×#807—21no—
45C2×#807C2212yes—
46C2×#807C2212yes—
47#4558C2212yes—
48#4554C2212yes—
49#4554C2212yes—
50#4530C2212yes—
51#4530C2212yes—
52#3092—31no—
53#3092D311yescharacteristic
54#2929—31no—
55C22×#107—31no—
56#2089—32no—
57#2078—32no—
58#1952—32no—
59#1821—31no—
60#1821—31no—
61#1816—32no—
62#1810—31no—
63C2×C3×#41C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C2×C3×M16C2312yes—
68C3×#123—21no—
69C3×C22⋊Q8D411yescharacteristic
70C3×C22⋊Q8D411yescharacteristic
71C3×#121D411yescharacteristic
72C3×#121D411yescharacteristic
73C3×#120—21no—
74C3×C4×D4—21no—
75C3×C4×D4—21no—
76C3×#117C2311yescharacteristic
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2312yes—
83#948—24no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90C2×#208—21no—
91C2×#208C2312yes—
92C2×#208C2311yescharacteristic
93C2×#208C2311yescharacteristic
94C22×C3⋊C8—24no—
95C22×C3⋊C8—21no—
96C22×C3⋊C8D411yescharacteristic
97C22×C3⋊C8D411yescharacteristic
98C22×C3⋊C8D411yescharacteristic
99C22×C3⋊C8D411yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2311yescharacteristic
102C3×#107—22no—
103C3×#106—22no—
104C3×#103—22no—
105C3×#102—22no—
106C3×C22⋊C8—21no—
107C3×C22⋊C8—21no—
108#835—22no—
109#832—22no—
110#830—21no—
111#830—21no—
112#830—21no—
113#830—21no—
114#830—21no—
115#830—21no—
116#822—24no—
117#808—22no—
118#807—22no—
119#807—22no—
120#807—22no—
121#807—22no—
122#807—22no—
123C4⋊(C3⋊C8)—21no—
124C4⋊(C3⋊C8)—21no—
125#803—21no—
126C4×C3⋊C8—21no—
127C2×#131—31no—
128C23×C8—31no—
129#525C2×D311yescharacteristic
130C22×C4⋊C4—31no—
131C22×C4⋊C4C2×D311yescharacteristic
132#484—21no—
133#484—61no—
134#483—21no—
135#483—61no—
136#481—21no—
137#481—61no—
138#480—21no—
139#480—61no—
140#435—61no—
141#435—61no—
142C8×D4—61no—
143C8×D4—61no—
144C2×#108—32no—
145#428—32no—
146#428C2×D312yes—
147C2×#107—32no—
148C2×#107—32no—
149C2×#107—61no—
150C2×#107—61no—
151C2×#107—61no—
152C2×#107—61no—
153#424—31no—
154#417—32no—
155#417C2×D312yes—
156#408—31no—
157#407—31no—
158#407—31no—
159#407C2×D311yescharacteristic
160#405—31no—
161#343—32no—
162#343—32no—
163#343—32no—
164#343—32no—
165#341—32no—
166C24×C3#4111yescharacteristic
167C3×#41—22no—
168C3×#41—22no—
169C2×C3×Q8C2×D411yescharacteristic
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4C2×D411yescharacteristic
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—22no—
178C22×C3×C4—41no—
179C22×C3×C4—41no—
180C22×C3×C4—82no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4#4111yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—22no—
188C22×Dic3—41no—
189C22×Dic3—41no—
190C22×Dic3—82no—
191C3×M16—22no—
192C3×M16—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4C2×D411yescharacteristic
201C3×C4⋊C4C2×D411yescharacteristic
202C3×C22⋊C4—21no—
203C3×C22⋊C4—21no—
204C3×C22⋊C4—21no—
205C3×C22⋊C4—21no—
206C3×C22⋊C4—21no—
207C3×C42—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—41no—
211C4⋊Dic3—41no—
212#208—22no—
213#208—22no—
214#208—22no—
215#208—22no—
216#208—42no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—42no—
226C2×C3⋊C8C2×D411yescharacteristic
227C2×C3⋊C8C2×D411yescharacteristic
228C2×C3⋊C8C2×D411yescharacteristic
229C2×C3⋊C8C2×D411yescharacteristic
230C2×C3⋊C8#4114yes—
231C2×#41C22×D311yescharacteristic
232C23×C4—21no—
233C23×C4—61no—
234C23×C4C22×D311yescharacteristic
235#131—64no—
236C2×M16—31no—
237C2×M16—32no—
238C2×M16—31no—
239C2×M16—61no—
240C2×M16C22×D312yes—
241C22×C8—31no—
242C22×C8—31no—
243C22×C8—31no—
244C22×C8—31no—
245C22×C8—31no—
246C22×C8—31no—
247C22×C8—64no—
248C22×C8—61no—
249#123—21no—
250C22⋊Q8#6711yescharacteristic
251C22⋊Q8#6711yescharacteristic
252#121#6711yescharacteristic
253#121#6711yescharacteristic
254#120—21no—
255C4×D4—21no—
256C4×D4—21no—
257#117C22×D311yescharacteristic
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—32no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—61no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4C22×D312yes—
270#108—64no—
271#107—22no—
272#107—62no—
273#107—62no—
274#107—62no—
275#107—62no—
276#107—62no—
277#106—22no—
278#106—62no—
279C4⋊C8—61no—
280C4⋊C8—61no—
281#103—22no—
282#103—62no—
283#102—22no—
284#102—62no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—61no—
288C22⋊C8—61no—
289C22⋊C8—61no—
290C22⋊C8—61no—
291C22⋊C8—61no—
292C22⋊C8—61no—
293#97—61no—
294C4×C8—61no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—21no—
299C23×C3#12111yescharacteristic
300C23×C3#12111yescharacteristic
301C23×C3C2×#4111yescharacteristic
302C3×Q8—22no—
303C3×D4—22no—
304C3×D4—42no—
305C3×D4—42no—
306C3×D4—42no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—82no—
321C2×C3×C4—82no—
322C2×C3×C4—82no—
323C2×C3×C4#12111yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4112yes—
328C2×C3×C42−1+412yes—
329C2×Dic3—42no—
330C2×Dic3—82no—
331C2×Dic3—82no—
332C2×Dic3—82no—
333C3×C8—42no—
334C3⋊C8—21no—
335C3⋊C8—21no—
336C3⋊C8—41no—
337C3⋊C8—41no—
338C3⋊C8—42no—
339C3⋊C8#12312yes—
340C3⋊C8#12312yes—
341C24#23711yescharacteristic
342#41—22no—
343#41—22no—
344C2×Q8C2×#6711yescharacteristic
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4C2×#6711yescharacteristic
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—62no—
356C22×C4—82no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4—242no—
360C22×C4D3×D411yescharacteristic
361C22×C4D3×D411yescharacteristic
362C22×C4#23711yescharacteristic
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365C22×C4C23×D311yescharacteristic
366M16—22no—
367M16—22no—
368M16—62no—
369M16—62no—
370M16—62no—
371M16—62no—
372M16—122no—
373C2×C8—22no—
374C2×C8—31no—
375C2×C8—31no—
376C2×C8—34no—
377C2×C8—31no—
378C2×C8—31no—
379C2×C8—62no—
380C2×C8—62no—
381C2×C8—62no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—62no—
387C2×C8—122no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4—121no—
397C4⋊C4—121no—
398C4⋊C4C2×#6711yescharacteristic
399C4⋊C4C2×#6711yescharacteristic
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C42—21no—
406C22×C3—28no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3#47212yes—
417C22×C3#47312yes—
418C22×C3C2×#12111yescharacteristic
419C22×C3#55212yes—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4—82no—
426C3×C4C2×#12311yescharacteristic
427C3×C4#53611yescharacteristic
428C3×C4#54712yes—
429Dic3—82no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—21no—
434C23#93911yescharacteristic
435C23#93911yescharacteristic
436C23C2×#23711yescharacteristic
437Q8—22no—
438D4—22no—
439D4—42no—
440D4—42no—
441D4—42no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—122no—
459C2×C4—242no—
460C2×C4—242no—
461C2×C4—242no—
462C2×C4#93911yescharacteristic
463C2×C4#93911yescharacteristic
464C2×C4C2×D3×D411yescharacteristic
465C2×C4D3×#4112yes—
466C2×C4#101012yes—
467C2×C4C22×#6711yescharacteristic
468C8—32no—
469C8—32no—
470C8—42no—
471C8—61no—
472C8—61no—
473C8—121no—
474C8—121no—
475C8—122no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—22no—
479C2×C3—81no—
480C2×C3#248611yescharacteristic
481C2×C3C2×#47211yescharacteristic
482C2×C3C2×#47311yescharacteristic
483C22—28no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22#519912yes—
494C22#520412yes—
495C22#565012yes—
496C22C2×#93911yescharacteristic
497C4—21no—
498C4—21no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—82no—
503C4—242no—
504C4#564811yescharacteristic
505C4D3×#12311yescharacteristic
506C4#582412yes—
507C3#2767511yescharacteristic, complemented, Sylow
508C2—22no—
509C2—22no—
510C2—22no—
511C2—81no—
512C2#8285611yescharacteristic
513C2C2×#519911yescharacteristic
514C2C2×#520411yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3092C2
socleC22×C3C2×#121
prime radicalC3×#525C22
hypercenter#3092D3
hypocenterC3#27675
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3092

Complements

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

NH ≅ G/Ntype
4. C22×#807C2semidirect
5. #71562C2semidirect
6. #70670C2semidirect
7. #70669C2semidirect
8. #70301C2semidirect
9. #70300C2semidirect
10. #69982C2semidirect
11. #69973C2semidirect
507. C3#27675semidirect

Maximal subgroups

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

subgroupindex
C3×#30922
#829162
#829142
C22×#8072
#715622
#706702
#706692
#703012
#703002
#699822
#699732
#276753