Finite Group Databasebeta

← #10943907

Subgroups — #10943907

Order 768.

3940 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 3940 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#1004C211yesmaximal, characteristic
2C3×#2435C211yesmaximal, characteristic, complemented
3C2×#5061C214yesmaximal, complemented
4#82735C212yesmaximal, complemented
5#72129C211yesmaximal, characteristic, complemented
6#69662C214yesmaximal, complemented
7#69454C212yesmaximal, complemented
8#32891—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C23×#67—22no—
11C2×#1004—22no—
12C2×#1004—216no—
13C2×#1004—24no—
14C2×#1004C2214yes—
15C22×#237—22no—
16C22×D3×D4—22no—
17C22×#235C2212yes—
18C2×C3×#124C2212yescomplemented
19C2×C3×C22⋊Q8C2211yescharacteristic, complemented
20C2×#942C2214yescomplemented
21C3×#394—22no—
22C3×#390C2212yescomplemented
23C3×#379—22no—
24C3×C4×C22⋊C4C2211yescharacteristic, complemented
25C2×#833C2214yescomplemented
26#5207C2212yescomplemented
27#5061—216no—
28#5050—22no—
29#5047—24no—
30#5046—22no—
31#4966C2214yescomplemented
32C2×#805C2218yescomplemented
33C22×H(5,𝔽2)—31no—
34C2×#455—34no—
35#3007—32no—
36#2435D311yescharacteristic, complemented
37#1792—34no—
38#1789—32no—
39#1783—31no—
40C25×C3D412yes—
41C24×D3—22no—
42C22×C3×Q8C2311yescharacteristic, complemented
43C22×C3×D4—28no—
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4D412yes—
47C22×C3×D4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C22×#67—24no—
50C22×#67—232no—
51C22×#67—28no—
52C22×#67—22no—
53C22×#67—28no—
54C23×Dic3—22no—
55#1004—232no—
56#1004—232no—
57#1004—232no—
58#1004—28no—
59#1004—216no—
60#1004D4116yes—
61C2×#237—28no—
62C2×#237—232no—
63C2×#237—28no—
64C2×D3×D4—232no—
65C2×D3×D4—28no—
66C2×D3×D4—28no—
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—216no—
70C2×#235D414yes—
71C2×#235C2314yes—
72C22×D12D412yes—
73C22×C4×D3—22no—
74C22×Dic6D412yes—
75C3×#124—28no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4C2311yescharacteristic, complemented
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C42C2312yescomplemented
85C2×#216—24no—
86#942—216no—
87C2×#215—24no—
88C2×#208C2314yescomplemented
89C3×#95—21no—
90#833—28no—
91#830—22no—
92#830—22no—
93#810—24no—
94#809—24no—
95#805—216no—
96C2×H(5,𝔽2)—34no—
97C2×H(5,𝔽2)—64no—
98C2×H(5,𝔽2)—616no—
99C2×H(5,𝔽2)—62no—
100C22×#41—32no—
101C22×#41—62no—
102C23×D4—62no—
103C23×D4—62no—
104C23×D4C2×D311yescharacteristic
105C2×#137—34no—
106C2×#124C2×D312yescomplemented
107C2×C22⋊Q8C2×D311yescharacteristic, complemented
108#455—616no—
109#452—64no—
110#451—62no—
111#450—62no—
112C2×#104—38no—
113#417—34no—
114C2×#100—34no—
115#407—32no—
116#394—22no—
117#390C2×D312yescomplemented
118#379—22no—
119C4×C22⋊C4C2×D311yescharacteristic, complemented
120C24×C3—22no—
121C24×C3—28no—
122C24×C3—22no—
123C24×C3—44no—
124C24×C3—82no—
125C24×C3C2×D411yescharacteristic
126C24×C3C2×D414yes—
127C23×D3—22no—
128C23×D3—216no—
129C23×D3—28no—
130C23×D3—84no—
131C23×D3—84no—
132C2×C3×Q8—22no—
133C2×C3×Q8—24no—
134C2×C3×D4—216no—
135C2×C3×D4—216no—
136C2×C3×D4—28no—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4C2×D418yes—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C2×#67—216no—
150C2×#67—232no—
151C2×#67—216no—
152C2×#67—264no—
153C2×#67—232no—
154C2×#67—24no—
155C2×#67—232no—
156C2×#67—24no—
157C22×Dic3—216no—
158C22×Dic3—28no—
159C22×Dic3—22no—
160#237—264no—
161#237—232no—
162#237—232no—
163D3×D4—264no—
164D3×D4—232no—
165D3×D4—232no—
166#235—216no—
167#235—216no—
168#235—28no—
169#235—28no—
170#235—28no—
171#235C2×D4116yes—
172C2×D12—22no—
173C2×D12—28no—
174C2×D12C2×D414yes—
175C2×C4×D3—216no—
176C2×C4×D3—24no—
177C2×C4×D3—24no—
178C2×Dic6—22no—
179C2×Dic6—28no—
180C2×Dic6C2×D414yes—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—48no—
187C3×C22⋊C4—84no—
188C3×C42—24no—
189#216—48no—
190#215—48no—
191#208—28no—
192C2×C3⋊C8—24no—
193C25—62no—
194C25#6712yes—
195H(5,𝔽2)—316no—
196H(5,𝔽2)—616no—
197H(5,𝔽2)—632no—
198H(5,𝔽2)—632no—
199H(5,𝔽2)—632no—
200H(5,𝔽2)—68no—
201C2×#41—34no—
202C2×#41—34no—
203C2×#41—616no—
204C2×#41—68no—
205C2×#41—632no—
206C2×#41—68no—
207C2×#41—62no—
208C2×#41—62no—
209C22×Q8—32no—
210C22×Q8C22×D311yescharacteristic, complemented
211C22×D4—28no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—32no—
215C22×D4—68no—
216C22×D4—632no—
217C22×D4—68no—
218C22×D4—64no—
219C22×D4—632no—
220C22×D4—68no—
221C22×D4—62no—
222C22×D4—68no—
223C22×D4#6712yes—
224C22×D4C22×D312yes—
225C23×C4—62no—
226C23×C4—62no—
227C23×C4C22×D311yescharacteristic
228#137—616no—
229C2×Q16—64no—
230C2×SD16—64no—
231C2×M16—34no—
232#124—28no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C22×D311yescharacteristic, complemented
237C2×C22⋊C4—21no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—24no—
241C2×C42C22×D312yescomplemented
242#104—616no—
243#103—64no—
244#102—64no—
245#100—68no—
246C22⋊C8—62no—
247C22⋊C8—62no—
248#95—21no—
249C23×C3—28no—
250C23×C3—28no—
251C23×C3—24no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—24no—
255C23×C3—41no—
256C23×C3—44no—
257C23×C3—44no—
258C23×C3—44no—
259C23×C3—44no—
260C23×C3—82no—
261C23×C3—82no—
262C23×C3—816no—
263C23×C3#12011yescharacteristic
264C23×C3#12011yescharacteristic
265C23×C3#12018yes—
266C23×C3C22×D411yescharacteristic
267C22×D3—28no—
268C22×D3—24no—
269C22×D3—232no—
270C22×D3—44no—
271C22×D3—44no—
272C22×D3—832no—
273C22×D3—88no—
274C22×D3—88no—
275C3×Q8—44no—
276C3×D4—216no—
277C3×D4—216no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4#12012yes—
289C2×C3×C4C22×D412yes—
290C2×C3×C4C22×D414yes—
291#67—216no—
292#67—264no—
293#67—216no—
294C2×Dic3—28no—
295C2×Dic3—232no—
296C2×Dic3—24no—
297D12—28no—
298D12—24no—
299D12#12018yes—
300C4×D3—216no—
301C4×D3—216no—
302Dic6—28no—
303Dic6—24no—
304Dic6#12018yes—
305C3⋊C8—44no—
306C24—22no—
307C24—28no—
308C24—22no—
309C24—44no—
310C24—616no—
311C24—68no—
312C24—62no—
313C24—82no—
314C24—244no—
315C24—244no—
316C24C2×#6711yescharacteristic
317C24C2×#6714yes—
318#41—316no—
319#41—68no—
320#41—68no—
321#41—616no—
322#41—616no—
323#41—632no—
324#41—664no—
325#41—632no—
326#41—68no—
327C2×Q8—22no—
328C2×Q8—24no—
329C2×Q8—34no—
330C2×Q8—68no—
331C2×Q8—62no—
332C2×D4—216no—
333C2×D4—216no—
334C2×D4—28no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—34no—
339C2×D4—68no—
340C2×D4—632no—
341C2×D4—664no—
342C2×D4—632no—
343C2×D4—632no—
344C2×D4—616no—
345C2×D4—616no—
346C2×D4—664no—
347C2×D4—632no—
348C2×D4—64no—
349C2×D4—632no—
350C2×D4—64no—
351C2×D4—62no—
352C2×D4D3×D418yes—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—616no—
358C22×C4—616no—
359C22×C4—68no—
360C22×C4—62no—
361C22×C4—64no—
362C22×C4—64no—
363C22×C4D3×D412yes—
364C22×C4D3×D412yes—
365C22×C4C2×#6711yescharacteristic
366C22×C4C23×D311yescharacteristic
367Q16—128no—
368SD16—128no—
369M16—68no—
370C2×C8—64no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—48no—
377C22⋊C4—84no—
378C42—24no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—28no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—44no—
388C22×C3—48no—
389C22×C3—416no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—84no—
393C22×C3—88no—
394C22×C3#45512yescomplemented
395C22×C3C2×#12012yes—
396C22×C3C2×#12011yescharacteristic
397C2×D3—28no—
398C2×D3—416no—
399C2×D3—44no—
400C2×D3—44no—
401C2×D3—832no—
402C3×C4—22no—
403C3×C4—82no—
404C3×C4—81no—
405C3×C4—81no—
406C3×C4C2×#12014yes—
407Dic3—28no—
408C23—28no—
409C23—28no—
410C23—22no—
411C23—24no—
412C23—22no—
413C23—24no—
414C23—41no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—68no—
420C23—632no—
421C23—64no—
422C23—82no—
423C23—82no—
424C23—816no—
425C23—124no—
426C23—124no—
427C23—2432no—
428C23—248no—
429C23—248no—
430C23#93711yescharacteristic
431C23#93711yescharacteristic
432C23#93718yes—
433C23C22×#6711yescharacteristic
434Q8—38no—
435Q8—44no—
436Q8—68no—
437Q8—64no—
438D4—216no—
439D4—216no—
440D4—38no—
441D4—68no—
442D4—616no—
443D4—664no—
444D4—616no—
445D4—64no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—616no—
457C2×C4—68no—
458C2×C4—632no—
459C2×C4—64no—
460C2×C4—616no—
461C2×C4#93712yes—
462C2×C4C2×D3×D414yes—
463C2×C4C2×D3×D412yes—
464C8—124no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—48no—
468C2×C3—41no—
469C2×C3#284611yescharacteristic
470C2×C3C2×#45511yescharacteristic
471C2×C3#301911yescharacteristic
472D3—48no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—28no—
480C22—21no—
481C22—44no—
482C22—48no—
483C22—416no—
484C22—44no—
485C22—44no—
486C22—68no—
487C22—84no—
488C22—88no—
489C22—1216no—
490C22—124no—
491C22—124no—
492C22—2432no—
493C22#506112yescomplemented
494C22C2×#93712yes—
495C22C2×#93711yescharacteristic
496C4—22no—
497C4—68no—
498C4—82no—
499C4—81no—
500C4—81no—
501C4D3×#12014yes—
502C3#3289111yescharacteristic, complemented, Sylow
503C2—22no—
504C2—22no—
505C2—41no—
506C2—48no—
507C2—128no—
508C2#8261211yescharacteristic
509C2C2×#506111yescharacteristic
510C2#8339411yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2435C2
socleC22×C3C2×#120
prime radicalC22×#1004C2
hypercenter#2435D3
hypocenterC3#32891
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
2. C3×#2435C2semidirect
3. C2×#5061C2semidirect
4. #82735C2semidirect
5. #72129C2semidirect
6. #69662C2semidirect
7. #69454C2semidirect
18. C2×C3×#124C22semidirect
19. C2×C3×C22⋊Q8C22semidirect
20. C2×#942C22semidirect
22. C3×#390C22semidirect
24. C3×C4×C22⋊C4C22semidirect
25. C2×#833C22semidirect
26. #5207C22semidirect
31. #4966C22semidirect
32. C2×#805C22semidirect
36. #2435D3semidirect
42. C22×C3×Q8C23semidirect
79. C2×C3×C4⋊C4C23semidirect
84. C2×C3×C42C23semidirect
88. C2×#208C23semidirect
106. C2×#124C2×D3semidirect
107. C2×C22⋊Q8C2×D3semidirect
117. #390C2×D3semidirect
119. C4×C22⋊C4C2×D3semidirect
210. C22×Q8C22×D3semidirect
236. C2×C4⋊C4C22×D3semidirect
241. C2×C42C22×D3semidirect
394. C22×C3#455semidirect
493. C22#5061semidirect
502. C3#32891semidirect

Maximal subgroups

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

subgroupindex
C22×#10042
C3×#24352
C2×#50612
#827352
#721292
#696622
#694542
#328913