Finite Group Databasebeta

← #10844950

Subgroups — #10844950

Order 768.

764 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2962C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic, complemented
3#83336C211yesmaximal, characteristic, complemented
4C2×#4843C211yesmaximal, characteristic
5#70542C212yesmaximal, complemented
6#70540C212yesmaximal, complemented
7#70474C211yesmaximal, characteristic, complemented
8#69903C212yesmaximal, complemented
9#69892C212yesmaximal, complemented
10#69876C211yesmaximal, characteristic, complemented
11C4×#823C211yesmaximal, characteristic, complemented
12#25312—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14C22×D24C2212yes—
15C2×#874C2211yescharacteristic
16C2×C4×D12C2211yescharacteristic
17C3×#446—22no—
18C3×#446—21no—
19C3×C8×Q8—21no—
20C3×#424C2212yes—
21C2×C3×C4⋊C8C2211yescharacteristic
22C3×#405C2212yes—
23C3×C4×M16C2211yescharacteristic
24C2×#823C2212yes—
25C2×#823C2211yescharacteristic
26C2×#823C2211yescharacteristic
27C2×#823C2212yes—
28C2×#821C2212yes—
29C2×#819C2212yes—
30#5088—21no—
31#5088—21no—
32#5087—21no—
33#5084—21no—
34#5067—22no—
35#5066—22no—
36#5001C2212yes—
37#5001C2212yes—
38#5001C2212yes—
39#4995C2212yes—
40#4924C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4843—21no—
43#4843—22no—
44#4843—21no—
45#4554C2211yescharacteristic
46#4554C2211yescharacteristic
47#4552C2212yes—
48#3138—31no—
49#3077—31no—
50C2×#459—31no—
51#2962D311yescharacteristic, complemented
52#2063—32no—
53#2045—32no—
54#1978—32no—
55#1965—31no—
56#1950—31no—
57#1931—32no—
58C4×#100—31no—
59C22×D12—21no—
60C22×D12—21no—
61C22×D12C2312yes—
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C2×C3×M16C2311yescharacteristic
65C2×C3×M16C2311yescharacteristic
66C22×C3×C8C2312yes—
67C3×C4⋊Q8D411yescharacteristic
68C3×C4⋊Q8D411yescharacteristic
69C3×#125D412yes—
70C3×C22⋊Q8—21no—
71C3×C22⋊Q8—21no—
72C3×C4×Q8—21no—
73C3×C4×Q8—21no—
74C3×#117C2312yes—
75C2×C3×C42C2311yescharacteristic
76C2×(C2×D3)⋊C4—21no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×#208C2312yes—
79C2×#208C2311yescharacteristic
80C2×#208C2311yescharacteristic
81C2×D24—24no—
82C2×D24—22no—
83C2×D24—22no—
84C2×D24—42no—
85#874—21no—
86#874—21no—
87#874—21no—
88#874—21no—
89C4×D12—21no—
90C4×D12—21no—
91C4×D12—21no—
92C4×D12—21no—
93C3×C4⋊C8—22no—
94C3×C4⋊C8—22no—
95C3×C4⋊C8D411yescharacteristic
96C3×C4⋊C8D411yescharacteristic
97C3×C4⋊C8D412yes—
98C3×#97—22no—
99C3×#97—21no—
100C3×C4×C8—22no—
101C3×C4×C8—21no—
102#823—22no—
103#823—22no—
104#823—24no—
105#823—42no—
106#821—22no—
107#821—22no—
108#819—42no—
109#809—22no—
110#809—22no—
111#809—22no—
112#809—22no—
113#809—22no—
114#809—22no—
115#808—22no—
116#807—22no—
117C4⋊(C3⋊C8)—21no—
118C4⋊(C3⋊C8)—21no—
119#803—21no—
120C4×C3⋊C8—21no—
121C22×D8—32no—
122#533C2×D311yescharacteristic
123C2×#127—31no—
124C2×C4×D4—31no—
125#490—62no—
126#486—61no—
127#486—61no—
128#478—62no—
129#476—61no—
130#474—61no—
131#459—61no—
132#459—62no—
133#459—61no—
134#446—22no—
135#446—21no—
136C8×Q8—21no—
137C2×#108—32no—
138#428—32no—
139#424C2×D312yes—
140#423—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142#418—32no—
143#418—32no—
144#418—32no—
145C2×#102—32no—
146C2×#100—32no—
147C2×#100—31no—
148C2×#100—31no—
149C2×#100—32no—
150#405C2×D312yes—
151C4×M16—31no—
152C4×M16C2×D311yescharacteristic
153#343—31no—
154#343—31no—
155#341—32no—
156C23×D3—21no—
157C23×D3—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C22×C3×C4#4111yescharacteristic
161C22×C3×C4#4111yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—21no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—41no—
176C2×D12—41no—
177C2×D12—41no—
178C2×D12—41no—
179C2×D12—82no—
180C2×D12—82no—
181C2×D12#4112yes—
182C2×C4×D3—41no—
183C2×C4×D3—41no—
184C2×C4×D3—82no—
185C3×M16—21no—
186C3×M16—21no—
187C3×M16—21no—
188C3×M16#4112yes—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8C2×D412yes—
194C2×C3×C8C2×D412yes—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4C2×D412yes—
200C3×C4⋊C4C2×D412yes—
201C3×C22⋊C4—22no—
202C3×C42—22no—
203C3×C42C2×D411yescharacteristic
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D411yescharacteristic
207(C2×D3)⋊C4—41no—
208(C2×D3)⋊C4—41no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211#208—22no—
212#208—21no—
213#208—21no—
214#208—21no—
215#208—22no—
216#208—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—22no—
220D24—44no—
221D24—44no—
222C22×D4—31no—
223C22×D4—32no—
224C22×D4—61no—
225C22×D4—61no—
226C23×C4—61no—
227C2×D8—64no—
228C2×D8—62no—
229C2×D8—62no—
230C2×D8—122no—
231C2×M16—31no—
232C2×M16—32no—
233C2×M16—31no—
234C2×M16C22×D311yescharacteristic
235C2×M16C22×D311yescharacteristic
236C22×C8C22×D312yes—
237C4⋊Q8D1211yescharacteristic
238C4⋊Q8D1211yescharacteristic
239#127—61no—
240#127—61no—
241#127—61no—
242#127—61no—
243#125D1212yes—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×D4—61no—
249C4×D4—61no—
250C4×D4—61no—
251C4×D4—61no—
252#117C22×D312yes—
253C2×C4⋊C4—31no—
254C2×C22⋊C4—61no—
255C2×C42C22×D311yescharacteristic
256#108—122no—
257#107—62no—
258#106—62no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—61no—
262C4⋊C8—61no—
263C4⋊C8#6711yescharacteristic
264C4⋊C8#6711yescharacteristic
265C4⋊C8#6712yes—
266#102—62no—
267#102—62no—
268#102—62no—
269#102—62no—
270#102—62no—
271#102—62no—
272#102—62no—
273#102—62no—
274#100—62no—
275#100—62no—
276#100—64no—
277#100—122no—
278#97—22no—
279#97—21no—
280#97—61no—
281C4×C8—22no—
282C4×C8—21no—
283C4×C8—61no—
284C23×C3C2×#4111yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—81no—
292C22×D3—81no—
293C22×D3—82no—
294C22×D3—82no—
295C3×Q8—42no—
296C3×Q8—81no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—41no—
301C2×C3×C4#12111yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×C3×C4#12111yescharacteristic
304C2×C3×C4#12111yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311C2×Dic3—41no—
312C2×Dic3—81no—
313D12—22no—
314D12—22no—
315D12—42no—
316D12—41no—
317D12—42no—
318D12—42no—
319D12—41no—
320D12—42no—
321D12—82no—
322D12—82no—
323D12—82no—
324D12—82no—
325C4×D3—82no—
326C4×D3—82no—
327C3×C8—22no—
328C3×C8—22no—
329C3×C8—22no—
330C3×C8—41no—
331C3⋊C8—41no—
332C3⋊C8—41no—
333C3⋊C8—42no—
334C24—61no—
335C24—62no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×D4—32no—
339C2×D4—61no—
340C2×D4—62no—
341C2×D4—61no—
342C2×D4—62no—
343C2×D4—61no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—61no—
347C2×D4—62no—
348C2×D4—61no—
349C2×D4—62no—
350C2×D4—121no—
351C2×D4—121no—
352C2×D4—121no—
353C2×D4—121no—
354C2×D4—242no—
355C2×D4—242no—
356C22×C4—61no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4—242no—
360C22×C4#23511yescharacteristic
361C22×C4#23511yescharacteristic
362C22×C4C23×D311yescharacteristic
363D8—124no—
364D8—124no—
365M16—21no—
366M16—21no—
367M16—21no—
368M16—62no—
369M16—61no—
370M16—61no—
371M16—61no—
372M16—62no—
373M16—61no—
374M16#23912yes—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—61no—
380C2×C8—61no—
381C2×C8—62no—
382C2×C8C2×#6712yes—
383C2×C8C2×#6712yes—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4C2×D1212yes—
391C4⋊C4C2×D1212yes—
392C22⋊C4—22no—
393C22⋊C4—121no—
394C22⋊C4—121no—
395C42—22no—
396C42C2×D1211yescharacteristic
397C42C2×D1211yescharacteristic
398C42C2×#6711yescharacteristic
399C42C2×#6711yescharacteristic
400C22×C3—21no—
401C22×C3—21no—
402C22×C3C2×#12111yescharacteristic
403C22×C3#52911yescharacteristic
404C22×C3#54211yescharacteristic
405C2×D3—42no—
406C2×D3—82no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—82no—
410C2×D3—82no—
411C2×D3—82no—
412C2×D3—82no—
413C2×D3—162no—
414C2×D3—162no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4#47212yes—
421C3×C4#54611yescharacteristic
422C3×C4#54811yescharacteristic
423C3×C4#54811yescharacteristic
424C3×C4#55011yescharacteristic
425Dic3—81no—
426C23—62no—
427C23—121no—
428C23—121no—
429C23—122no—
430C23—122no—
431C23—122no—
432C23—241no—
433C23—241no—
434C23—242no—
435C23—242no—
436C23C2×#23511yescharacteristic
437Q8—42no—
438Q8—81no—
439D4—62no—
440D4—62no—
441D4—121no—
442D4—122no—
443D4—122no—
444D4—122no—
445D4—121no—
446D4—122no—
447D4—242no—
448D4—242no—
449D4—242no—
450D4—242no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—121no—
456C2×C4—242no—
457C2×C4—242no—
458C2×C4—241no—
459C2×C4#93011yescharacteristic
460C2×C4#93011yescharacteristic
461C2×C4#93011yescharacteristic
462C2×C4#93011yescharacteristic
463C2×C4C22×D1211yescharacteristic
464C2×C4C2×#23911yescharacteristic
465C2×C4#100711yescharacteristic
466C2×C4D3×#4111yescharacteristic
467C2×C4#100911yescharacteristic
468C2×C4C22×#6711yescharacteristic
469C8—22no—
470C8—22no—
471C8—22no—
472C8—41no—
473C8—121no—
474C8—121no—
475C8—122no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3#242611yescharacteristic
479C2×C3C2×#47211yescharacteristic
480C2×C3#306811yescharacteristic
481D3—82no—
482D3—162no—
483C22—21no—
484C22—21no—
485C22—122no—
486C22—242no—
487C22—242no—
488C22—241no—
489C22—241no—
490C22—242no—
491C22—242no—
492C22—242no—
493C22—482no—
494C22—482no—
495C22#558011yescharacteristic
496C22#558211yescharacteristic
497C22C2×#93011yescharacteristic
498C4—21no—
499C4—21no—
500C4—21no—
501C4—42no—
502C4—42no—
503C4—241no—
504C4#514212yes—
505C4#557611yescharacteristic
506C4#557711yescharacteristic
507C4#581711yescharacteristic
508C4#582911yescharacteristic
509C3#2531211yescharacteristic, complemented, Sylow
510C2—21no—
511C2—21no—
512C2—242no—
513C2—482no—
514C2#8249511yescharacteristic
515C2C2×#514211yescharacteristic
516C2#8381411yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2962C2
socleC22×C3C2×#121
prime radicalC2×#4843C2
hypercenter#2962D3
hypocenterC3#25312
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. C3×#2962C2semidirect
2. #83350C2semidirect
3. #83336C2semidirect
5. #70542C2semidirect
6. #70540C2semidirect
7. #70474C2semidirect
8. #69903C2semidirect
9. #69892C2semidirect
10. #69876C2semidirect
11. C4×#823C2semidirect
51. #2962D3semidirect
509. C3#25312semidirect

Maximal subgroups

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

subgroupindex
C3×#29622
#833502
#833362
C2×#48432
#705422
#705402
#704742
#699032
#698922
#698762
C4×#8232
#253123