Finite Group Databasebeta

← #10828044

Subgroups — #10828044

Order 768.

1286 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87273C211yesmaximal, characteristic
2C3×#2311C211yesmaximal, characteristic, complemented
3#82139C212yesmaximal, complemented
4#72137C211yesmaximal, characteristic, complemented
5#71591C212yesmaximal, complemented
6#69695C214yesmaximal, complemented
7#69662C212yesmaximal, complemented
8#69642C212yesmaximal, complemented
9#24025—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C2×#1004C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic, complemented
13C22×C3×C4⋊C4C2211yescharacteristic, complemented
14#5829—22no—
15#5829—22no—
16#5824—22no—
17#5824—22no—
18#5816C2212yes—
19#5804—22no—
20#5798C2211yescharacteristic
21#5791C2212yes—
22C2×#929—22no—
23C3×#391C2212yescomplemented
24C3×#390C2212yes—
25C3×#386—22no—
26C3×#380—22no—
27C2×#836C2212yescomplemented
28C2×#833C2212yes—
29#5207C2212yescomplemented
30#4971—22no—
31#4970—22no—
32#4968—22no—
33#4967—22no—
34#4966C2214yes—
35#4964C2214yescomplemented
36C2×#805C2214yes—
37#4523C2214yescomplemented
38#3485—31no—
39#3092—32no—
40#2311D311yescharacteristic, complemented
41#2038—34no—
42#1897—32no—
43#1856—32no—
44#1792—32no—
45#1785—31no—
46C2×C3×#41—24no—
47C2×C3×#41—22no—
48C2×C3×#41—22no—
49C2×C3×#41—42no—
50C22×C3×Q8C2311yescharacteristic
51C22×C3×D4D412yes—
52C22×C3×D4C2311yescharacteristic
53C23×C3×C4—21no—
54C23×C3×C4D412yes—
55C23×C3×C4C2311yescharacteristic
56C22×#67—22no—
57#1004—24no—
58#1004—42no—
59C2×#237—22no—
60C2×D3×D4—22no—
61C2×#235C2312yes—
62C3×#117—42no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C42C2312yes—
71#953—22no—
72C2×#217—22no—
73#947—22no—
74#946—22no—
75#946—22no—
76#944—22no—
77#939—24no—
78#939—24no—
79#937—24no—
80#935—24no—
81D4×Dic3—22no—
82D4×Dic3—22no—
83#930D412yes—
84#930D412yes—
85#929—24no—
86#929—24no—
87#929—22no—
88#929—24no—
89C4×#67—22no—
90C4×#67—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—22no—
93#926C2312yes—
94#924D412yes—
95#924D412yes—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×#208C2314yes—
99C3×#95—24no—
100#836—42no—
101#833—24no—
102#830—22no—
103#830—22no—
104#810—24no—
105#809—24no—
106#808—24no—
107#807—24no—
108#805—44no—
109C2×H(5,𝔽2)—31no—
110C22×#41C2×D311yescharacteristic
111#547—62no—
112#547—62no—
113#546—62no—
114#546—62no—
115#536—32no—
116#534—62no—
117#525—32no—
118C2×#123—62no—
119#518—31no—
120C2×#117C2×D311yescharacteristic, complemented
121C22×C4⋊C4C2×D311yescharacteristic, complemented
122#484—62no—
123#483—62no—
124#481—62no—
125#480—62no—
126#428—34no—
127C2×#104—34no—
128#417—34no—
129C2×#101—32no—
130C2×#100—32no—
131#407—32no—
132#391C2×D312yescomplemented
133#390C2×D312yes—
134#386—22no—
135#380—22no—
136#339—34no—
137C24×C3C2×D411yescharacteristic
138C24×C3C2×D412yes—
139C23×D3—22no—
140C3×#41—48no—
141C3×#41—48no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×D4—24no—
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—42no—
150C2×C3×D4#4114yes—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—82no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×#67—24no—
167C2×#67—22no—
168C2×#67—22no—
169C2×#67—22no—
170C2×#67—44no—
171C2×#67—44no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175#237—48no—
176D3×D4—48no—
177#235—24no—
178#235—24no—
179C2×D12C2×D412yes—
180C2×C4×D3—22no—
181C2×Dic6C2×D412yes—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—41no—
185C3×C4⋊C4—41no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—84no—
188C3×C42—24no—
189#217—24no—
190#217—22no—
191#217—22no—
192#217—22no—
193(C2×D3)⋊C4—24no—
194(C2×D3)⋊C4—24no—
195(C2×D3)⋊C4—22no—
196(C2×D3)⋊C4—22no—
197C4⋊Dic3C2×D412yes—
198C4⋊Dic3C2×D412yes—
199Dic3⋊C4—24no—
200Dic3⋊C4—24no—
201Dic3⋊C4—22no—
202Dic3⋊C4—22no—
203C4×Dic3—22no—
204#208—24no—
205#208—24no—
206C2×C3⋊C8—24no—
207H(5,𝔽2)—64no—
208H(5,𝔽2)—122no—
209C2×#41—24no—
210C2×#41—22no—
211C2×#41—22no—
212C2×#41—32no—
213C2×#41—42no—
214C2×#41—62no—
215C22×Q8C22×D311yescharacteristic
216C22×D4—62no—
217C22×D4—62no—
218C22×D4#6712yes—
219C22×D4C22×D311yescharacteristic
220C23×C4—21no—
221C23×C4#6712yes—
222C23×C4C22×D311yescharacteristic
223C2×M16—34no—
224C4⋊Q8—62no—
225#124—62no—
226#123—64no—
227#123—64no—
228#123—64no—
229#123—62no—
230#123—64no—
231C22⋊Q8—32no—
232C22⋊Q8—32no—
233C22⋊Q8—62no—
234C22⋊Q8—62no—
235#121—32no—
236#121—32no—
237#121—64no—
238#121—64no—
239#120—64no—
240#120—62no—
241C4×D4—62no—
242C4×D4—62no—
243C4×D4—62no—
244C4×D4—62no—
245#117—32no—
246#117—42no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4C22×D312yes—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—62no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C42C22×D312yes—
260#107—64no—
261#106—64no—
262#104—124no—
263#103—64no—
264#102—64no—
265#101—122no—
266#100—64no—
267C22⋊C8—62no—
268C22⋊C8—62no—
269#95—24no—
270C23×C3—22no—
271C23×C3—21no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3#12012yes—
279C23×C3#12314yes—
280C23×C3C22×D411yescharacteristic
281C22×D3—22no—
282C22×D3—44no—
283C22×D3—82no—
284C22×D3—82no—
285C3×Q8—44no—
286C3×D4—28no—
287C3×D4—48no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—44no—
296C2×C3×C4—44no—
297C2×C3×C4—42no—
298C2×C3×C4—44no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—84no—
303C2×C3×C4#12011yescharacteristic
304C2×C3×C4#12011yescharacteristic
305C2×C3×C4C22×D412yes—
306C2×C3×C4C2×#4112yes—
307C2×C3×C4H(5,𝔽2)12yes—
308#67—48no—
309#67—42no—
310#67—42no—
311#67—82no—
312#67—82no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—44no—
317C2×Dic3—82no—
318C2×Dic3—82no—
319D12—24no—
320C4×D3—44no—
321Dic6—24no—
322C3⋊C8—44no—
323C24—62no—
324C24C2×#6712yes—
325C24C2×#6711yescharacteristic
326#41—48no—
327#41—48no—
328#41—64no—
329#41—64no—
330#41—128no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—32no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—32no—
340C2×D4—42no—
341C2×D4—64no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—62no—
345C2×D4—128no—
346C2×D4—124no—
347C2×D4—124no—
348C2×D4#23514yes—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—82no—
363C22×C4D3×D412yes—
364C22×C4D3×D412yes—
365C22×C4C2×#6712yes—
366C22×C4C2×#6711yescharacteristic
367C22×C4C23×D311yescharacteristic
368M16—64no—
369M16—64no—
370C2×C8—64no—
371C4⋊C4—24no—
372C4⋊C4—32no—
373C4⋊C4—32no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—64no—
378C4⋊C4—64no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C22⋊C4—42no—
382C22⋊C4—64no—
383C22⋊C4—64no—
384C22⋊C4—64no—
385C22⋊C4—62no—
386C22⋊C4—62no—
387C22⋊C4—62no—
388C22⋊C4—62no—
389C22⋊C4—62no—
390C22⋊C4—84no—
391C42—24no—
392C42—62no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—24no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—44no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—42no—
404C22×C3—82no—
405C22×C3C2×#12011yescharacteristic
406C22×C3C2×#12311yescharacteristic
407C22×C3#53411yescharacteristic
408C2×D3—42no—
409C2×D3—42no—
410C2×D3—42no—
411C2×D3—164no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—42no—
415C3×C4—82no—
416C3×C4—82no—
417C3×C4#54614yes—
418Dic3—42no—
419Dic3—42no—
420Dic3—42no—
421C23—21no—
422C23—22no—
423C23—21no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—62no—
430C23—124no—
431C23—242no—
432C23—242no—
433C23#92914yes—
434C23#93712yes—
435C23C22×#6711yescharacteristic
436Q8—44no—
437Q8—64no—
438D4—28no—
439D4—48no—
440D4—64no—
441D4—128no—
442D4—122no—
443D4—122no—
444D4—242no—
445D4—242no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—84no—
464C2×C4—124no—
465C2×C4—124no—
466C2×C4—242no—
467C2×C4—242no—
468C2×C4#93711yescharacteristic
469C2×C4#93711yescharacteristic
470C2×C4C2×#23512yes—
471C2×C4C2×D3×D412yes—
472C2×C4#100412yes—
473C8—124no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—44no—
477C2×C3—42no—
478C2×C3#240511yescharacteristic
479C2×C3#302111yescharacteristic
480C2×C3#302211yescharacteristic
481D3—82no—
482C22—21no—
483C22—21no—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—22no—
488C22—44no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—82no—
494C22—122no—
495C22—122no—
496C22—122no—
497C22—484no—
498C22C2×#92911yescharacteristic
499C22C2×#93711yescharacteristic
500C22#580411yescharacteristic
501C4—21no—
502C4—21no—
503C4—42no—
504C4—82no—
505C4—82no—
506C4—122no—
507C4—122no—
508C4—122no—
509C4#549014yes—
510C3#2402511yescharacteristic, complemented, Sylow
511C2—21no—
512C2—21no—
513C2—42no—
514C2—44no—
515C2—242no—
516C2#8212611yescharacteristic
517C2#8339511yescharacteristic
518C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2311C2
socleC22×C3C2×#120
prime radical#87273C2
hypercenter#2311D3
hypocenterC3#24025
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C3×#2311C2semidirect
3. #82139C2semidirect
4. #72137C2semidirect
5. #71591C2semidirect
6. #69695C2semidirect
7. #69662C2semidirect
8. #69642C2semidirect
12. C2×C3×#117C22semidirect
13. C22×C3×C4⋊C4C22semidirect
23. C3×#391C22semidirect
27. C2×#836C22semidirect
29. #5207C22semidirect
35. #4964C22semidirect
37. #4523C22semidirect
40. #2311D3semidirect
120. C2×#117C2×D3semidirect
121. C22×C4⋊C4C2×D3semidirect
132. #391C2×D3semidirect
510. C3#24025semidirect

Maximal subgroups

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

subgroupindex
#872732
C3×#23112
#821392
#721372
#715912
#696952
#696622
#696422
#240253