Finite Group Databasebeta

← #10883507

Subgroups — #10883507

Order 768.

790 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82441C211yesmaximal, characteristic
2C2×C4×#216C211yesmaximal, characteristic, complemented
3#79067C211yesmaximal, characteristic
4#79041C211yesmaximal, characteristic
5C3×#1801C211yesmaximal, characteristic, complemented
6#70528C211yesmaximal, characteristic, complemented
7#70516C211yesmaximal, characteristic, complemented
8#70125C212yesmaximal, complemented
9#70123C212yesmaximal, complemented
10#69010C212yesmaximal, complemented
11#69007C212yesmaximal, complemented
12#28176—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C22×#216C2211yescharacteristic
15C2×#942C2212yes—
16C2×#909C2212yes—
17C2×#907C2212yes—
18#5477C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C3×#423C2211yescharacteristic
21C3×#423C2211yescharacteristic
22C2×C3×#101C2212yes—
23C2×#835C2211yescharacteristic
24C2×#824C2212yes—
25#5133C2212yes—
26#5129C2212yes—
27#5031—21no—
28C4×#216—21no—
29C4×#216—21no—
30C4×#216—21no—
31C4×#216—21no—
32#5028—22no—
33#5027—21no—
34C2×#810C2211yescharacteristic
35#4966C2212yes—
36C2×#807C2211yescharacteristic
37#4924C2211yescharacteristic
38C2×C4×C3⋊C8C2211yescharacteristic
39#4849—21no—
40#4848—21no—
41#4845—22no—
42#4838—21no—
43#4838—21no—
44#4834—21no—
45#4834—21no—
46C3×#343C2212yes—
47#4558C2212yes—
48#4558C2212yes—
49#4554C2212yes—
50#3037—31no—
51#3037—31no—
52C2×C4×Q16—31no—
53#2931—31no—
54#2078—32no—
55#2020—32no—
56#1908—32no—
57#1908—32no—
58#1837—31no—
59#1801—31no—
60#1801D311yescharacteristic, complemented
61C22×C3×Q8C2311yescharacteristic
62C2×#235C2312yes—
63C22×Dic6C2311yescharacteristic
64C2×C3×M16C2312yes—
65C2×C3×M16C2312yes—
66C3×C4×Q8—21no—
67C3×C4×Q8—21no—
68C3×C4×Q8—21no—
69C3×C4×Q8—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#216—22no—
75C2×#216—22no—
76C2×#216—22no—
77C2×#216—21no—
78C2×#216—21no—
79#942—24no—
80#942—42no—
81C2×#215—22no—
82#930—21no—
83#930—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×Dic3⋊C4—21no—
86C2×C4×Dic3—21no—
87C2×#208C2312yes—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2311yescharacteristic
90#909—24no—
91#909—42no—
92#907—42no—
93#907—42no—
94C2×Dic12—22no—
95C2×#204—22no—
96C2×ΓL(1,𝔽25)—22no—
97C2×D3×C8—22no—
98#875D412yes—
99#874D411yescharacteristic
100C4×D12—21no—
101C4×D12—21no—
102#869D411yescharacteristic
103C4×Dic6—21no—
104C4×Dic6—21no—
105C4×Dic6—21no—
106C4×Dic6—21no—
107C3×C4⋊C8—21no—
108C3×C4⋊C8—21no—
109C3×C4⋊C8—21no—
110C3×C4⋊C8—21no—
111C3×#101—42no—
112#835—21no—
113#835—21no—
114#824—42no—
115#821—22no—
116(C2×D3)⋊C8—22no—
117(C2×D3)⋊C8—22no—
118#816—22no—
119#810—22no—
120#810—21no—
121#810—21no—
122#809—22no—
123#807—21no—
124#807—21no—
125C4⋊(C3⋊C8)—21no—
126C4⋊(C3⋊C8)—21no—
127C4×C3⋊C8D411yescharacteristic
128C4×C3⋊C8D411yescharacteristic
129C4×C3⋊C8D411yescharacteristic
130C4×C3⋊C8D411yescharacteristic
131C2×#137—32no—
132C2×#137—32no—
133C22×Q16—31no—
134C2×#131—32no—
135#532—31no—
136C2×C4×Q8—31no—
137C2×C4×Q8C2×D311yescharacteristic
138#464—62no—
139#464—62no—
140#462—61no—
141#462—61no—
142#460—61no—
143#460—61no—
144C4×Q16—61no—
145C4×Q16—61no—
146C4×Q16—61no—
147C4×Q16—61no—
148#436—61no—
149#436—61no—
150C8×D4—61no—
151C8×D4—61no—
152C2×#107—31no—
153#423—31no—
154#423C2×D311yescharacteristic
155#423C2×D311yescharacteristic
156#417—32no—
157#417—32no—
158C2×#103—31no—
159C2×#103—31no—
160C2×#101—32no—
161C2×#101C2×D312yes—
162#408—32no—
163C2×C4×C8—31no—
164#343—32no—
165#343C2×D312yes—
166#341—32no—
167#341—32no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—21no—
171C2×C3×Q8—21no—
172C2×C3×Q8—21no—
173C2×C3×Q8#4112yes—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4#4111yescharacteristic
177C22×C3×C4#4111yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—22no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182#235—22no—
183#235—22no—
184#235—22no—
185#235—22no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12C2×D412yes—
189C2×C4×D3—22no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6C2×D412yes—
196C2×Dic6#4112yes—
197C3×M16—22no—
198C3×M16—22no—
199C3×M16—22no—
200C3×M16—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—41no—
207C3×C42—41no—
208C3×C42—41no—
209C3×C42C2×D411yescharacteristic
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D411yescharacteristic
212C3×C42C2×D411yescharacteristic
213#216—24no—
214#216—24no—
215#216—44no—
216#215—44no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—22no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223Dic3⋊C4—41no—
224Dic3⋊C4—41no—
225C4×Dic3—41no—
226C4×Dic3—41no—
227#208—22no—
228#208—22no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8C2×D411yescharacteristic
235C2×C3⋊C8C2×D411yescharacteristic
236C2×C3⋊C8C2×D411yescharacteristic
237C2×C3⋊C8C2×D411yescharacteristic
238Dic12—44no—
239#204—44no—
240ΓL(1,𝔽25)—44no—
241D3×C8—44no—
242C2×#41—32no—
243C22×Q8—31no—
244C22×Q8C22×D311yescharacteristic
245#137—64no—
246#137—64no—
247#137—122no—
248#137—122no—
249C2×Q16—62no—
250C2×Q16—61no—
251C2×Q16—61no—
252C2×Q16—62no—
253C2×Q16—62no—
254C2×Q16—62no—
255C2×SD16—62no—
256C2×SD16—62no—
257#131—122no—
258#131—122no—
259C2×M16—32no—
260C2×M16—62no—
261C2×M16C22×D312yes—
262C2×M16C22×D312yes—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8—62no—
266C4⋊Q8—31no—
267#127—31no—
268#124—32no—
269#121—61no—
270#121—61no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—61no—
276C4×Q8—61no—
277C4×Q8—61no—
278C4×Q8—61no—
279C4×D4—61no—
280C4×D4—61no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C42—21no—
286C2×C42—61no—
287C2×C42C22×D311yescharacteristic
288#107—61no—
289#107—61no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—61no—
295C4⋊C8—61no—
296#103—61no—
297#103—61no—
298#103—62no—
299#103—62no—
300#103—61no—
301#103—61no—
302#102—62no—
303#102—62no—
304#101—42no—
305#101—122no—
306C22⋊C8—62no—
307C22⋊C8—62no—
308C4×C8—31no—
309C4×C8—31no—
310C4×C8—31no—
311C4×C8—31no—
312C23×C3C2×#4111yescharacteristic
313C22×D3—22no—
314C3×Q8—22no—
315C3×Q8—22no—
316C3×Q8—41no—
317C3×Q8—41no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—81no—
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4#12111yescharacteristic
327C2×C3×C4#12111yescharacteristic
328C2×C3×C4#12111yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334C2×C3×C4H(5,𝔽2)11yescharacteristic
335#67—42no—
336#67—42no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342C2×Dic3—81no—
343D12—22no—
344D12—22no—
345D12—42no—
346D12—42no—
347C4×D3—42no—
348C4×D3—42no—
349Dic6—22no—
350Dic6—22no—
351Dic6—22no—
352Dic6—22no—
353Dic6—41no—
354Dic6—41no—
355C3×C8—42no—
356C3×C8—42no—
357C3⋊C8—21no—
358C3⋊C8—21no—
359C3⋊C8—21no—
360C3⋊C8—21no—
361C3⋊C8—42no—
362#41—62no—
363#41—62no—
364#41—62no—
365#41—62no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—32no—
372C2×Q8—32no—
373C2×Q8—61no—
374C2×Q8—61no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8#23912yes—
379C2×D4—32no—
380C2×D4—62no—
381C2×D4—61no—
382C2×D4—61no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—61no—
386C22×C4—62no—
387C22×C4—61no—
388C22×C4#23511yescharacteristic
389C22×C4#23511yescharacteristic
390C22×C4C23×D311yescharacteristic
391Q16—64no—
392Q16—64no—
393Q16—124no—
394Q16—124no—
395SD16—124no—
396SD16—124no—
397M16—22no—
398M16—22no—
399M16—22no—
400M16—22no—
401M16—62no—
402M16—62no—
403M16—124no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—31no—
407C2×C8—31no—
408C2×C8—31no—
409C2×C8—31no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—62no—
415C2×C8—124no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—121no—
425C4⋊C4—121no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C42—41no—
429C42—41no—
430C42—121no—
431C42—121no—
432C42D3×D411yescharacteristic
433C42D3×D411yescharacteristic
434C42D3×D411yescharacteristic
435C42D3×D411yescharacteristic
436C22×C3—21no—
437C22×C3—21no—
438C22×C3C2×#12111yescharacteristic
439C22×C3#55311yescharacteristic
440C22×C3#55311yescharacteristic
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—42no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—42no—
448C3×C4—81no—
449C3×C4#47312yes—
450C3×C4#52511yescharacteristic
451C3×C4#53511yescharacteristic
452C3×C4#54611yescharacteristic
453C3×C4#54611yescharacteristic
454Dic3—42no—
455Dic3—42no—
456Dic3—81no—
457C23—62no—
458C23C2×#23511yescharacteristic
459Q8—22no—
460Q8—22no—
461Q8—41no—
462Q8—41no—
463Q8—62no—
464Q8—62no—
465Q8—62no—
466Q8—62no—
467Q8—121no—
468Q8—121no—
469D4—62no—
470D4—62no—
471D4—122no—
472D4—122no—
473D4—122no—
474D4—122no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—81no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—121no—
489C2×C4—241no—
490C2×C4#88411yescharacteristic
491C2×C4#88411yescharacteristic
492C2×C4#88411yescharacteristic
493C2×C4#88411yescharacteristic
494C2×C4C2×D3×D411yescharacteristic
495C2×C4C2×D3×D411yescharacteristic
496C2×C4#100411yescharacteristic
497C2×C4C2×#23911yescharacteristic
498C2×C4D3×#4111yescharacteristic
499C2×C4#100911yescharacteristic
500C8—42no—
501C8—42no—
502C8—61no—
503C8—61no—
504C8—61no—
505C8—61no—
506C8—122no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3#253511yescharacteristic
510C2×C3C2×#47311yescharacteristic
511C2×C3#306711yescharacteristic
512D3—82no—
513C22—21no—
514C22—21no—
515C22—122no—
516C22—122no—
517C22—122no—
518C22C2×#88411yescharacteristic
519C22#551411yescharacteristic
520C22#553011yescharacteristic
521C4—21no—
522C4—21no—
523C4—21no—
524C4—42no—
525C4—81no—
526C4—122no—
527C4—122no—
528C4—241no—
529C4#475512yes—
530C4#560711yescharacteristic
531C4#561211yescharacteristic
532C4#562711yescharacteristic
533C4#563011yescharacteristic
534C3#2817611yescharacteristic, complemented, Sylow
535C2—21no—
536C2—21no—
537C2—242no—
538C2#7393311yescharacteristic
539C2C2×#475511yescharacteristic
540C2#7871911yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1801C2
socleC22×C3C2×#121
prime radical#5477C22
hypercenter#1801D3
hypocenterC3#28176
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. C2×C4×#216C2semidirect
5. C3×#1801C2semidirect
6. #70528C2semidirect
7. #70516C2semidirect
8. #70125C2semidirect
9. #70123C2semidirect
10. #69010C2semidirect
11. #69007C2semidirect
60. #1801D3semidirect
534. C3#28176semidirect

Maximal subgroups

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

subgroupindex
#824412
C2×C4×#2162
#790672
#790412
C3×#18012
#705282
#705162
#701252
#701232
#690102
#690072
#281763