Finite Group Databasebeta

← #10836645

Subgroups — #10836645

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3079C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic, complemented
3C4×#949C211yesmaximal, characteristic
4#82515C211yesmaximal, characteristic, complemented
5#71790C211yesmaximal, characteristic, complemented
6#70673C212yesmaximal, complemented
7#70669C212yesmaximal, complemented
8#70527C211yesmaximal, characteristic, complemented
9#70520C211yesmaximal, characteristic, complemented
10#70307C212yesmaximal
11#70301C212yesmaximal, complemented
12#24720—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C3×C4×#41C2211yescharacteristic
15C2×#949C2211yescharacteristic
16C2×C4×D12C2211yescharacteristic
17C3×#477—21no—
18C3×#476—21no—
19C3×#475—21no—
20C3×#474—21no—
21C3×#428C2212yes—
22C3×#423C2211yescharacteristic
23C3×#417C2212yes—
24#5234C2212yes—
25#5232C2211yescharacteristic
26C2×#823C2212yes—
27C2×#822C2212yes—
28#5087—21no—
29#5087—21no—
30#5087—22no—
31#5026—21no—
32C4×#215—21no—
33#5014—21no—
34C4×#213—21no—
35#5001C2211yescharacteristic
36#5000—22no—
37#4997C2212yes—
38#4995C2211yescharacteristic
39C2×#809C2212yes—
40C2×#807C2212yes—
41#4926—21no—
42#4925—21no—
43#4924C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C4×#208C2211yescharacteristic
46#4558C2212yes—
47#4554C2212yes—
48#4530C2212yes—
49#4530C2212yes—
50#3079D311yescharacteristic, complemented
51C2×#474—31no—
52C4×AGL(1,ℤ/8)—31no—
53#2925—31no—
54#2078—32no—
55#2051—32no—
56#1997—31no—
57#1952—32no—
58#1950—31no—
59#1931—32no—
60#1800—31no—
61C2×C3×#41C2311yescharacteristic
62C22×C3×Q8—21no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C2×C3×M16C2312yes—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4×Q8D411yescharacteristic
71C3×C4×Q8D411yescharacteristic
72C3×C4×D4—21no—
73C3×C4×D4—21no—
74C3×C4×D4D411yescharacteristic
75C3×C4×D4D411yescharacteristic
76C3×#117—21no—
77C3×#117C2311yescharacteristic
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82#949—22no—
83#949—22no—
84C2×#215—21no—
85C2×#213—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×C4⋊Dic3C2311yescharacteristic
88C2×#208C2312yes—
89C2×#208C2311yescharacteristic
90C2×#208C2311yescharacteristic
91C22×C3⋊C8C2312yes—
92C4×D12—21no—
93C4×D12—21no—
94C4×D12—21no—
95C4×D12—21no—
96C3×#107—22no—
97C3×#106—22no—
98C3×C4⋊C8—21no—
99C3×C4⋊C8—21no—
100C3×#103—22no—
101C3×#102—22no—
102#835—21no—
103#832—21no—
104#830—22no—
105#830—22no—
106#823—24no—
107#822—24no—
108#809—21no—
109#809—21no—
110#809—22no—
111#809—22no—
112#808—21no—
113#807—21no—
114#807—22no—
115#807—22no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118C4⋊(C3⋊C8)D411yescharacteristic
119C4⋊(C3⋊C8)D411yescharacteristic
120C4⋊(C3⋊C8)D412yes—
121#803—21no—
122#803—22no—
123C4×C3⋊C8—21no—
124C4×C3⋊C8—22no—
125C2×AGL(1,ℤ/8)—31no—
126C2×C4⋊Q8C2×D311yescharacteristic
127C4×#41C2×D311yescharacteristic
128C2×C4×D4—31no—
129#477—21no—
130#476—21no—
131#475—21no—
132#474—21no—
133#474—61no—
134#474—61no—
135#474—62no—
136#442—61no—
137#440—61no—
138C4×SD16—61no—
139C4×D8—61no—
140#433—62no—
141#432—61no—
142#431—61no—
143C2×#108—32no—
144#428—31no—
145#428C2×D312yes—
146C2×#107—32no—
147#423—31no—
148#423C2×D311yescharacteristic
149C2×C4⋊C8—31no—
150#418—31no—
151#417—31no—
152#417C2×D312yes—
153C2×#102—32no—
154C2×#100—32no—
155#408—32no—
156#405—32no—
157C4×M16—31no—
158#343—32no—
159#343—32no—
160#343—32no—
161#341—32no—
162C23×D3—21no—
163C3×#41—21no—
164C3×#41—21no—
165C3×#41—21no—
166C3×#41—21no—
167C2×C3×Q8—41no—
168C2×C3×Q8—41no—
169C2×C3×Q8—82no—
170C2×C3×Q8C2×D411yescharacteristic
171C2×C3×D4—21no—
172C2×C3×D4C2×D411yescharacteristic
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4#4111yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C22×Dic3—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—21no—
186C2×D12#4112yes—
187C2×C4×D3—41no—
188C2×C4×D3—41no—
189C2×C4×D3—82no—
190C3×M16—22no—
191C3×M16—22no—
192C2×C3×C8—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—41no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4C2×D411yescharacteristic
200C3×C4⋊C4C2×D411yescharacteristic
201C3×C22⋊C4—21no—
202C3×C22⋊C4—21no—
203C3×C22⋊C4—21no—
204C3×C42—21no—
205C3×C42—21no—
206C3×C42—21no—
207C3×C42C2×D411yescharacteristic
208C3×C42C2×D411yescharacteristic
209C3×C42C2×D411yescharacteristic
210C3×C42C2×D411yescharacteristic
211#215—24no—
212#213—24no—
213(C2×D3)⋊C4—41no—
214(C2×D3)⋊C4—41no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217#208—21no—
218#208—22no—
219#208—21no—
220#208—21no—
221#208—22no—
222#208#4112yes—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8C2×D412yes—
229C2×C3⋊C8C2×D412yes—
230C2×#41C22×D311yescharacteristic
231C22×Q8—21no—
232C22×D4—31no—
233C23×C4—61no—
234AGL(1,ℤ/8)—62no—
235AGL(1,ℤ/8)—62no—
236C2×SD16—61no—
237C2×D8—61no—
238C2×M16—31no—
239C2×M16—31no—
240C2×M16—32no—
241C2×M16C22×D312yes—
242C22×C8—32no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4×Q8#6711yescharacteristic
248C4×Q8#6711yescharacteristic
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—61no—
252C4×D4—61no—
253C4×D4—61no—
254C4×D4—61no—
255C4×D4#6711yescharacteristic
256C4×D4#6711yescharacteristic
257#117—21no—
258#117C22×D311yescharacteristic
259C2×C4⋊C4—21no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4C22×D312yes—
262C2×C22⋊C4—61no—
263C2×C42—21no—
264C2×C42C22×D311yescharacteristic
265#108—64no—
266#107—22no—
267#107—62no—
268#107—62no—
269#107—61no—
270#106—22no—
271#106—61no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—31no—
275C4⋊C8—31no—
276C4⋊C8—32no—
277C4⋊C8—61no—
278C4⋊C8—61no—
279#103—22no—
280#103—61no—
281#102—22no—
282#102—62no—
283#102—62no—
284#102—61no—
285#102—61no—
286#102—61no—
287#100—64no—
288C22⋊C8—62no—
289C22⋊C8—62no—
290#97—62no—
291#97—61no—
292C4×C8—62no—
293C4×C8—61no—
294C23×C3—21no—
295C23×C3C2×#4111yescharacteristic
296C22×D3—22no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—81no—
300C22×D3—81no—
301C3×Q8—21no—
302C3×Q8—21no—
303C3×Q8—82no—
304C3×Q8—82no—
305C3×D4—21no—
306C3×D4—21no—
307C3×D4—41no—
308C3×D4—41no—
309C2×C3×C4—22no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—81no—
319C2×C3×C4—82no—
320C2×C3×C4#12111yescharacteristic
321C2×C3×C4#12111yescharacteristic
322C2×C3×C4#12111yescharacteristic
323C2×C3×C4#12111yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C42−1+411yescharacteristic
329C2×C3×C42−1+411yescharacteristic
330C2×Dic3—41no—
331C2×Dic3—81no—
332D12—22no—
333D12—22no—
334D12—41no—
335D12—41no—
336C4×D3—82no—
337C4×D3—82no—
338C3×C8—42no—
339C3⋊C8—22no—
340C3⋊C8—22no—
341C3⋊C8—22no—
342C3⋊C8—41no—
343C3⋊C8—42no—
344C24—61no—
345#41—21no—
346#41—21no—
347#41—21no—
348#41—21no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—82no—
352C2×Q8C2×#6711yescharacteristic
353C2×D4—21no—
354C2×D4—32no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4C2×#6711yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—61no—
366C22×C4—121no—
367C22×C4—121no—
368C22×C4—242no—
369C22×C4#23711yescharacteristic
370C22×C4#23711yescharacteristic
371C22×C4C23×D311yescharacteristic
372SD16—64no—
373D8—64no—
374M16—22no—
375M16—22no—
376M16—32no—
377M16—61no—
378M16—62no—
379M16—61no—
380M16—61no—
381M16—62no—
382C2×C8—22no—
383C2×C8—32no—
384C2×C8—32no—
385C2×C8—62no—
386C2×C8—62no—
387C2×C8—61no—
388C2×C8—62no—
389C2×C8—61no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4C2×#6711yescharacteristic
399C4⋊C4C2×#6711yescharacteristic
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—121no—
404C22⋊C4—121no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42D3×D411yescharacteristic
409C42D3×D411yescharacteristic
410C42C2×#6711yescharacteristic
411C42C2×#6711yescharacteristic
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3C2×#12111yescharacteristic
418C22×C3#55211yescharacteristic
419C22×C3#55611yescharacteristic
420C2×D3—42no—
421C2×D3—82no—
422C2×D3—81no—
423C2×D3—81no—
424C2×D3—82no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—82no—
432C3×C4#47212yes—
433C3×C4#52511yescharacteristic
434C3×C4#53611yescharacteristic
435C3×C4#54711yescharacteristic
436C3×C4#54811yescharacteristic
437Dic3—81no—
438C23—21no—
439C23—62no—
440C23—121no—
441C23—121no—
442C23—241no—
443C23—241no—
444C23C2×#23711yescharacteristic
445Q8—21no—
446Q8—21no—
447Q8—82no—
448Q8—82no—
449D4—21no—
450D4—21no—
451D4—41no—
452D4—41no—
453D4—62no—
454D4—62no—
455D4—121no—
456D4—121no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—121no—
469C2×C4—242no—
470C2×C4—242no—
471C2×C4—241no—
472C2×C4#93911yescharacteristic
473C2×C4#93911yescharacteristic
474C2×C4#93911yescharacteristic
475C2×C4#93911yescharacteristic
476C2×C4C2×D3×D411yescharacteristic
477C2×C4#100711yescharacteristic
478C2×C4#100711yescharacteristic
479C2×C4D3×#4111yescharacteristic
480C2×C4D3×#4111yescharacteristic
481C2×C4C22×#6711yescharacteristic
482C8—42no—
483C8—62no—
484C8—62no—
485C8—62no—
486C8—122no—
487C8—121no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—81no—
491C2×C3#241511yescharacteristic
492C2×C3C2×#47211yescharacteristic
493C2×C3#306911yescharacteristic
494D3—82no—
495C22—21no—
496C22—21no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—122no—
501C22—242no—
502C22—242no—
503C22—241no—
504C22—241no—
505C22#563911yescharacteristic
506C22#564111yescharacteristic
507C22C2×#93911yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—82no—
515C4—241no—
516C4#519912yes—
517C4#563111yescharacteristic
518C4#563311yescharacteristic
519C4#581611yescharacteristic
520C4#582811yescharacteristic
521C3#2472011yescharacteristic, complemented, Sylow
522C2—21no—
523C2—21no—
524C2—81no—
525C2—242no—
526C2#8281411yescharacteristic
527C2C2×#519911yescharacteristic
528C2#8397811yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3079C2
socleC22×C3C2×#121
prime radicalC2×#949C22
hypercenter#3079D3
hypocenterC3#24720
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3079

Complements

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

NH ≅ G/Ntype
1. C3×#3079C2semidirect
2. C2×#5087C2semidirect
4. #82515C2semidirect
5. #71790C2semidirect
6. #70673C2semidirect
7. #70669C2semidirect
8. #70527C2semidirect
9. #70520C2semidirect
11. #70301C2semidirect
50. #3079D3semidirect
521. C3#24720semidirect

Maximal subgroups

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

subgroupindex
C3×#30792
C2×#50872
C4×#9492
#825152
#717902
#706732
#706692
#705272
#705202
#703072
#703012
#247203