Finite Group Databasebeta

← #10883534

Subgroups — #10883534

Order 768.

782 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82441C211yesmaximal, characteristic
2C2×#5030C211yesmaximal, characteristic, complemented
3#79067C211yesmaximal, characteristic
4#79042C211yesmaximal, characteristic
5C3×#1801C211yesmaximal, characteristic, complemented
6#70528C211yesmaximal, characteristic, complemented
7#70517C211yesmaximal, characteristic, complemented
8#70125C212yesmaximal, complemented
9#70123C212yesmaximal, complemented
10#69011C212yesmaximal, complemented
11#69007C212yesmaximal, complemented
12#28179—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—
28#5030—22no—
29#5030—22no—
30#5028—22no—
31#5027—21no—
32C2×#810C2211yescharacteristic
33#4966C2212yes—
34C2×#808C2211yescharacteristic
35#4924C2211yescharacteristic
36C2×#803C2211yescharacteristic
37#4849—21no—
38#4848—21no—
39#4845—22no—
40#4836—22no—
41#4836—22no—
42C3×#343C2212yes—
43#4558C2212yes—
44#4558C2212yes—
45#4554C2212yes—
46#3037—31no—
47#3037—31no—
48C2×#441—31no—
49#2932—31no—
50#2079—32no—
51#2020—32no—
52#1908—32no—
53#1908—32no—
54#1838—31no—
55#1801—31no—
56#1801D311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C2×#235C2312yes—
59C22×Dic6C2311yescharacteristic
60C2×C3×M16C2312yes—
61C2×C3×M16C2312yes—
62C3×C4×Q8—21no—
63C3×C4×Q8—21no—
64C3×C4×Q8—21no—
65C3×C4×Q8—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70C2×#216—22no—
71C2×#216—22no—
72C2×#216—22no—
73C2×#216—22no—
74#942—24no—
75#942—42no—
76C2×#215—22no—
77#930—21no—
78#930—21no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×C4×Dic3—21no—
82C2×#208C2312yes—
83C22×C3⋊C8C2311yescharacteristic
84C22×C3⋊C8C2311yescharacteristic
85#909—24no—
86#909—42no—
87#907—42no—
88#907—42no—
89C2×Dic12—22no—
90C2×#204—22no—
91C2×ΓL(1,𝔽25)—22no—
92C2×D3×C8—22no—
93#875D412yes—
94#874D411yescharacteristic
95C4×D12—21no—
96C4×D12—21no—
97#869D411yescharacteristic
98C4×Dic6—21no—
99C4×Dic6—21no—
100C4×Dic6—21no—
101C4×Dic6—21no—
102C3×C4⋊C8—21no—
103C3×C4⋊C8—21no—
104C3×C4⋊C8—21no—
105C3×C4⋊C8—21no—
106C3×#101—42no—
107#835—22no—
108#824—42no—
109#821—22no—
110(C2×D3)⋊C8—24no—
111#816—22no—
112#810—22no—
113#810—22no—
114#809—22no—
115#808—22no—
116C4⋊(C3⋊C8)—21no—
117C4⋊(C3⋊C8)—21no—
118#803D412yes—
119#803D412yes—
120C2×#137—32no—
121C2×#137—32no—
122C22×Q16—31no—
123C2×#131—32no—
124#532—31no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127#464—62no—
128#464—62no—
129#462—61no—
130#462—61no—
131#460—61no—
132#460—61no—
133#441—62no—
134#441—62no—
135#435—62no—
136#435—62no—
137C2×#106—31no—
138#423—31no—
139#423C2×D311yescharacteristic
140#423C2×D311yescharacteristic
141#417—32no—
142#417—32no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#101—32no—
146C2×#101C2×D312yes—
147#408—32no—
148C2×#97—31no—
149#343—32no—
150#343C2×D312yes—
151#341—32no—
152#341—32no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8#4112yes—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4#4111yescharacteristic
162C22×C3×C4#4111yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×#67—22no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167#235—22no—
168#235—22no—
169#235—22no—
170#235—22no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12C2×D412yes—
174C2×C4×D3—22no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—21no—
179C2×Dic6—21no—
180C2×Dic6C2×D412yes—
181C2×Dic6#4112yes—
182C3×M16—22no—
183C3×M16—22no—
184C3×M16—22no—
185C3×M16—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—41no—
191C3×C4⋊C4—41no—
192C3×C42—41no—
193C3×C42—41no—
194C3×C42C2×D411yescharacteristic
195C3×C42C2×D411yescharacteristic
196C3×C42C2×D411yescharacteristic
197C3×C42C2×D411yescharacteristic
198#216—28no—
199#216—44no—
200#215—44no—
201(C2×D3)⋊C4—22no—
202(C2×D3)⋊C4—22no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207Dic3⋊C4—41no—
208Dic3⋊C4—41no—
209C4×Dic3—41no—
210C4×Dic3—41no—
211#208—22no—
212#208—22no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8C2×D412yes—
219C2×C3⋊C8C2×D412yes—
220Dic12—44no—
221#204—44no—
222ΓL(1,𝔽25)—44no—
223D3×C8—44no—
224C2×#41—32no—
225C22×Q8—31no—
226C22×Q8C22×D311yescharacteristic
227#137—64no—
228#137—64no—
229#137—122no—
230#137—122no—
231C2×Q16—62no—
232C2×Q16—62no—
233C2×Q16—62no—
234C2×Q16—62no—
235C2×Q16—62no—
236C2×SD16—62no—
237C2×SD16—62no—
238#131—122no—
239#131—122no—
240C2×M16—32no—
241C2×M16—62no—
242C2×M16C22×D312yes—
243C2×M16C22×D312yes—
244C22×C8—31no—
245C22×C8—31no—
246C22×C8—62no—
247C4⋊Q8—31no—
248#127—31no—
249#124—32no—
250#121—61no—
251#121—61no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—61no—
257C4×Q8—61no—
258C4×Q8—61no—
259C4×Q8—61no—
260C4×D4—61no—
261C4×D4—61no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4C22×D311yescharacteristic
266C2×C42—21no—
267C2×C42—61no—
268C2×C42C22×D311yescharacteristic
269#106—62no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—61no—
275C4⋊C8—61no—
276#103—62no—
277#103—62no—
278#103—62no—
279#103—62no—
280#102—62no—
281#102—62no—
282#101—42no—
283#101—122no—
284C22⋊C8—64no—
285#97—32no—
286#97—32no—
287C23×C3C2×#4111yescharacteristic
288C22×D3—22no—
289C3×Q8—22no—
290C3×Q8—22no—
291C3×Q8—41no—
292C3×Q8—41no—
293C2×C3×C4—22no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—81no—
300C2×C3×C4#12111yescharacteristic
301C2×C3×C4#12111yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×C3×C4#12111yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310#67—42no—
311#67—42no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—81no—
318D12—22no—
319D12—22no—
320D12—81no—
321D12—81no—
322C4×D3—42no—
323C4×D3—42no—
324Dic6—22no—
325Dic6—22no—
326Dic6—22no—
327Dic6—22no—
328Dic6—41no—
329Dic6—41no—
330C3×C8—42no—
331C3×C8—42no—
332C3⋊C8—22no—
333C3⋊C8—22no—
334C3⋊C8—42no—
335#41—62no—
336#41—62no—
337#41—62no—
338#41—62no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—32no—
345C2×Q8—32no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8#23912yes—
352C2×D4—32no—
353C2×D4—62no—
354C2×D4—61no—
355C2×D4—61no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—61no—
359C22×C4—62no—
360C22×C4—61no—
361C22×C4#23511yescharacteristic
362C22×C4#23511yescharacteristic
363C22×C4C23×D311yescharacteristic
364Q16—68no—
365Q16—124no—
366Q16—124no—
367SD16—124no—
368SD16—124no—
369M16—22no—
370M16—22no—
371M16—22no—
372M16—22no—
373M16—62no—
374M16—62no—
375M16—124no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—32no—
379C2×C8—32no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—61no—
383C2×C8—61no—
384C2×C8—62no—
385C2×C8—124no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—121no—
395C4⋊C4—121no—
396C22⋊C4—62no—
397C22⋊C4—62no—
398C42—41no—
399C42—41no—
400C42—121no—
401C42—121no—
402C42D3×D411yescharacteristic
403C42D3×D411yescharacteristic
404C42D3×D411yescharacteristic
405C42D3×D411yescharacteristic
406C22×C3—21no—
407C22×C3—21no—
408C22×C3C2×#12111yescharacteristic
409C22×C3#55311yescharacteristic
410C22×C3#55311yescharacteristic
411C2×D3—42no—
412C2×D3—42no—
413C2×D3—42no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—42no—
419C3×C4—81no—
420C3×C4#52511yescharacteristic
421C3×C4#53511yescharacteristic
422C3×C4#54611yescharacteristic
423C3×C4#54611yescharacteristic
424Dic3—42no—
425Dic3—42no—
426Dic3—81no—
427C23—62no—
428C23C2×#23511yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—41no—
432Q8—41no—
433Q8—62no—
434Q8—62no—
435Q8—62no—
436Q8—62no—
437Q8—121no—
438Q8—121no—
439D4—62no—
440D4—62no—
441D4—122no—
442D4—122no—
443D4—241no—
444D4—241no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—81no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—121no—
459C2×C4—241no—
460C2×C4#88411yescharacteristic
461C2×C4#88411yescharacteristic
462C2×C4#88411yescharacteristic
463C2×C4#88411yescharacteristic
464C2×C4C2×D3×D411yescharacteristic
465C2×C4C2×D3×D411yescharacteristic
466C2×C4#100411yescharacteristic
467C2×C4C2×#23911yescharacteristic
468C2×C4D3×#4111yescharacteristic
469C2×C4#100911yescharacteristic
470C8—42no—
471C8—42no—
472C8—62no—
473C8—62no—
474C8—122no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3#253511yescharacteristic
478C2×C3#306911yescharacteristic
479C2×C3#306911yescharacteristic
480D3—82no—
481C22—21no—
482C22—21no—
483C22—122no—
484C22—122no—
485C22—122no—
486C22C2×#88411yescharacteristic
487C22#551411yescharacteristic
488C22#553011yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—42no—
494C4—81no—
495C4—122no—
496C4—122no—
497C4—241no—
498C4#560711yescharacteristic
499C4#561211yescharacteristic
500C4#562711yescharacteristic
501C4#563011yescharacteristic
502C3#2817911yescharacteristic, complemented, Sylow
503C2—21no—
504C2—21no—
505C2—242no—
506C2#7393311yescharacteristic
507C2#7871211yescharacteristic
508C2#7872111yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1801C2
socleC22×C3C2×#121
prime radical#5477C22
hypercenter#1801D3
hypocenterC3#28179
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×#5030C2semidirect
5. C3×#1801C2semidirect
6. #70528C2semidirect
7. #70517C2semidirect
8. #70125C2semidirect
9. #70123C2semidirect
10. #69011C2semidirect
11. #69007C2semidirect
56. #1801D3semidirect
502. C3#28179semidirect

Maximal subgroups

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

subgroupindex
#824412
C2×#50302
#790672
#790422
C3×#18012
#705282
#705172
#701252
#701232
#690112
#690072
#281793