Finite Group Databasebeta

← #10900901

Subgroups — #10900901

Order 768.

1596 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1596 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C2×#5493C211yesmaximal, characteristic
2C3×#2483C211yesmaximal, characteristic, complemented
3#82126C212yesmaximal, complemented
4#71542C211yesmaximal, characteristic, complemented
5#69596C214yesmaximal, complemented
6#69539C214yesmaximal, complemented
7#69498C212yesmaximal, complemented
8#29635—31nomaximal, Sylow
9C23×#67C2212yes—
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C2×#930—21no—
13C2×#930—21no—
14C2×#929C2214yescomplemented
15#5493—24no—
16#5493—22no—
17#5493C2214yes—
18C2×#886—22no—
19C2×#882—22no—
20C3×#397—22no—
21C3×#385—21no—
22C3×#385C2212yescomplemented
23C3×#381—21no—
24C3×#381C2212yescomplemented
25C3×#379C2211yescharacteristic
26#5247C2212yes—
27#5213C2214yescomplemented
28#4959—24no—
29#4955—22no—
30C2×#806C2218yescomplemented
31#4946—22no—
32#4526C2214yescomplemented
33C2×#534—31no—
34#2483D311yescharacteristic, complemented
35#2405—32no—
36#2044—34no—
37#2022—34no—
38#1819—31no—
39#1781—32no—
40C25×C3C2311yescharacteristic
41C24×D3D412yes—
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C22×#67—22no—
47C22×#67—22no—
48C22×#67—22no—
49C22×#67—22no—
50C22×#67—48no—
51C22×#67D414yes—
52C22×#67C2314yes—
53C23×Dic3D412yes—
54C22×D12—21no—
55C22×D12—21no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4D412yes—
66C2×C3×C22⋊C4C2312yes—
67C2×C3×C22⋊C4C2312yes—
68C2×#217—22no—
69C2×#217—22no—
70C2×#217C2314yes—
71#930—24no—
72#930—24no—
73#930—24no—
74#930—24no—
75#929—48no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—24no—
79C2×(C2×D3)⋊C4—24no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×Dic3⋊C4—24no—
83#886—28no—
84#886—28no—
85#882—28no—
86#882—28no—
87C3×#95—22no—
88C3×#95—22no—
89C3×#95—22no—
90C3⋊#95—24no—
91#806—48no—
92C23×D4—32no—
93#534—34no—
94#534—64no—
95#534—62no—
96C2×#123—34no—
97C2×#123—62no—
98C2×#121—61no—
99C2×#121—61no—
100C2×#120—62no—
101C22×C22⋊C4C2×D311yescharacteristic
102C22×C22⋊C4C2×D311yescharacteristic
103C2×#99—38no—
104#397—22no—
105#394—64no—
106#386—34no—
107#385—21no—
108#385—62no—
109#385C2×D312yescomplemented
110#381—21no—
111#381C2×D312yescomplemented
112#379—32no—
113#379—62no—
114#379C2×D311yescharacteristic
115#342—34no—
116C24×C3—24no—
117C24×C3—42no—
118C24×C3—81no—
119C24×C3C2×D412yes—
120C24×C3C2×D412yes—
121C24×C3C2×D411yescharacteristic
122C24×C3C2×D411yescharacteristic
123C24×C3C2411yescharacteristic
124C23×D3—22no—
125C23×D3—22no—
126C23×D3—44no—
127C23×D3—44no—
128C23×D3—82no—
129C23×D3C2×D414yes—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—24no—
140C22×C3×C4—42no—
141C22×C3×C4—81no—
142C22×C3×C4—81no—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C2×#67—24no—
146C2×#67—24no—
147C2×#67—24no—
148C2×#67—24no—
149C2×#67—28no—
150C2×#67—28no—
151C2×#67—44no—
152C2×#67—44no—
153C2×#67—48no—
154C2×#67—48no—
155C2×#67—48no—
156C2×#67—48no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—24no—
160C22×Dic3—44no—
161C22×Dic3C2×D414yes—
162C2×D12—24no—
163C2×D12—24no—
164C2×D12—82no—
165C2×D12—82no—
166C3×C4⋊C4—82no—
167C3×C22⋊C4—24no—
168C3×C22⋊C4—24no—
169C3×C22⋊C4—24no—
170C3×C22⋊C4—42no—
171C3×C22⋊C4—42no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—81no—
175#217—28no—
176#217—44no—
177#217—82no—
178(C2×D3)⋊C4—28no—
179(C2×D3)⋊C4—28no—
180(C2×D3)⋊C4—48no—
181(C2×D3)⋊C4—84no—
182C4⋊Dic3—24no—
183C4⋊Dic3—24no—
184Dic3⋊C4—48no—
185C25—32no—
186C25C22×D311yescharacteristic
187C22×D4—34no—
188C22×D4—34no—
189C22×D4—61no—
190C22×D4—61no—
191C22×D4—62no—
192C22×D4—62no—
193C22×D4—62no—
194C22×D4—62no—
195C22×D4—128no—
196C23×C4—32no—
197C23×C4#6711yescharacteristic
198C23×C4#6711yescharacteristic
199C23×C4C22×D311yescharacteristic
200C23×C4C22×D311yescharacteristic
201#123—68no—
202#123—68no—
203#123—128no—
204#121—64no—
205#121—64no—
206#121—64no—
207#121—64no—
208#120—68no—
209#120—68no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—64no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—34no—
223C2×C22⋊C4—62no—
224C2×C22⋊C4—62no—
225C2×C22⋊C4—64no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—64no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4#6712yes—
230C2×C22⋊C4C22×D312yes—
231C2×C22⋊C4C22×D312yes—
232#99—128no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—64no—
237C23×C3—24no—
238C23×C3—22no—
239C23×C3—24no—
240C23×C3—22no—
241C23×C3—24no—
242C23×C3—22no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3—41no—
247C23×C3—42no—
248C23×C3—41no—
249C23×C3—42no—
250C23×C3—41no—
251C23×C3—88no—
252C23×C3#12012yes—
253C23×C3#12011yescharacteristic
254C23×C3#12011yescharacteristic
255C23×C3C22×D412yes—
256C23×C3C22×D411yescharacteristic
257C23×C3H(5,𝔽2)14yes—
258C22×D3—24no—
259C22×D3—24no—
260C22×D3—24no—
261C22×D3—44no—
262C22×D3—44no—
263C22×D3—44no—
264C22×D3—44no—
265C22×D3—44no—
266C22×D3—44no—
267C22×D3—82no—
268C22×D3—82no—
269C22×D3—816no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—44no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—44no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—81no—
286C2×C3×C4—81no—
287#67—48no—
288#67—48no—
289#67—48no—
290#67—48no—
291C2×Dic3—24no—
292C2×Dic3—24no—
293C2×Dic3—24no—
294C2×Dic3—44no—
295C2×Dic3—44no—
296C2×Dic3—44no—
297C2×Dic3—44no—
298C2×Dic3—44no—
299D12—84no—
300D12—84no—
301C24—24no—
302C24—34no—
303C24—42no—
304C24—62no—
305C24—62no—
306C24—81no—
307C24—124no—
308C24—124no—
309C24—242no—
310C24D3×D412yes—
311C24D3×D412yes—
312C24C2×#6711yescharacteristic
313C24C2×#6711yescharacteristic
314C24C23×D311yescharacteristic
315C2×D4—64no—
316C2×D4—64no—
317C2×D4—64no—
318C2×D4—64no—
319C2×D4—64no—
320C2×D4—68no—
321C2×D4—68no—
322C2×D4—64no—
323C2×D4—124no—
324C2×D4—124no—
325C2×D4—128no—
326C2×D4—128no—
327C2×D4—128no—
328C2×D4—128no—
329C2×D4—242no—
330C2×D4—242no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—34no—
342C22×C4—42no—
343C22×C4—62no—
344C22×C4—62no—
345C22×C4—64no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—124no—
349C22×C4C2×#6712yes—
350C22×C4C2×#6712yes—
351C4⋊C4—64no—
352C4⋊C4—64no—
353C4⋊C4—82no—
354C4⋊C4—128no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—68no—
363C22⋊C4—68no—
364C22⋊C4—68no—
365C22⋊C4—81no—
366C22⋊C4—128no—
367C22⋊C4—124no—
368C22⋊C4—244no—
369C22⋊C4—242no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—21no—
373C22×C3—22no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—22no—
382C22×C3—44no—
383C22×C3—48no—
384C22×C3—44no—
385C22×C3—84no—
386C22×C3—82no—
387C22×C3#45812yes—
388C22×C3C2×#12011yescharacteristic
389C22×C3#53414yes—
390C22×C3#53512yes—
391C2×D3—48no—
392C2×D3—44no—
393C2×D3—44no—
394C2×D3—44no—
395C2×D3—88no—
396C2×D3—88no—
397C2×D3—84no—
398C3×C4—42no—
399C3×C4—42no—
400C3×C4—82no—
401C3×C4—81no—
402C3×C4—81no—
403Dic3—44no—
404Dic3—84no—
405C23—24no—
406C23—22no—
407C23—24no—
408C23—22no—
409C23—24no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23—41no—
419C23—64no—
420C23—64no—
421C23—64no—
422C23—88no—
423C23—124no—
424C23—124no—
425C23—124no—
426C23—124no—
427C23—124no—
428C23—124no—
429C23—242no—
430C23—242no—
431C23—2416no—
432C23#93712yes—
433C23#93711yescharacteristic
434C23#93711yescharacteristic
435C23C2×D3×D412yes—
436C23#100914yes—
437C23C22×#6711yescharacteristic
438D4—128no—
439D4—128no—
440D4—128no—
441D4—128no—
442D4—244no—
443D4—244no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—64no—
460C2×C4—64no—
461C2×C4—64no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—124no—
465C2×C4—124no—
466C2×C4—124no—
467C2×C4—124no—
468C2×C4—124no—
469C2×C3—24no—
470C2×C3—22no—
471C2×C3—22no—
472C2×C3—42no—
473C2×C3—41no—
474C2×C3#261311yescharacteristic
475C2×C3C2×#45811yescharacteristic
476C2×C3#302611yescharacteristic
477D3—84no—
478C22—21no—
479C22—22no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—21no—
490C22—44no—
491C22—48no—
492C22—44no—
493C22—82no—
494C22—84no—
495C22—128no—
496C22—124no—
497C22—124no—
498C22—124no—
499C22—244no—
500C22—248no—
501C22—248no—
502C22#505312yes—
503C22#565314yes—
504C22C2×#93711yescharacteristic
505C22#583012yes—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—81no—
510C4—81no—
511C4—124no—
512C4—244no—
513C3#2963511yescharacteristic, complemented, Sylow
514C2—22no—
515C2—24no—
516C2—22no—
517C2—42no—
518C2—41no—
519C2—244no—
520C2C2×#505311yescharacteristic
521C2#8290311yescharacteristic
522C2#8337511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2483C2
socleC22×C3C2×#120
prime radicalC2×#5493C2
hypercenter#2483D3
hypocenterC3#29635
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #2483

Complements

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

NH ≅ G/Ntype
2. C3×#2483C2semidirect
3. #82126C2semidirect
4. #71542C2semidirect
5. #69596C2semidirect
6. #69539C2semidirect
7. #69498C2semidirect
14. C2×#929C22semidirect
22. C3×#385C22semidirect
24. C3×#381C22semidirect
27. #5213C22semidirect
30. C2×#806C22semidirect
32. #4526C22semidirect
34. #2483D3semidirect
109. #385C2×D3semidirect
111. #381C2×D3semidirect
513. C3#29635semidirect

Maximal subgroups

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

subgroupindex
C2×#54932
C3×#24832
#821262
#715422
#695962
#695392
#694982
#296353