Finite Group Databasebeta

← #10900908

Subgroups — #10900908

Order 768.

1502 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 171 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
1#85735C211yesmaximal, characteristic, complemented
2C3×#2482C211yesmaximal, characteristic, complemented
3#82125C212yesmaximal, complemented
4#71594C211yesmaximal, characteristic, complemented
5#69596C214yesmaximal, complemented
6#69537C212yesmaximal, complemented
7#69500C214yesmaximal, complemented
8#29636—31nomaximal, Sylow
9C22×#235C2212yes—
10C2×C3×#117C2211yescharacteristic
11C22×C3×C22⋊C4C2211yescharacteristic
12C2×#953C2214yes—
13#5526—22no—
14#5525—22no—
15#5524—24no—
16#5510—21no—
17#5510—21no—
18#5493C2214yes—
19#5477—22no—
20C3×#399—21no—
21C3×#397—21no—
22C3×#388—22no—
23C3×#385C2212yes—
24C3×#381C2211yescharacteristic
25C3×#379C2212yes—
26#5213C2214yes—
27#5211C2212yes—
28#4960—22no—
29#4956—24no—
30C2×#806C2218yes—
31#4947—22no—
32#4526C2214yes—
33#3449—31no—
34#2482D311yescharacteristic, complemented
35#2404—32no—
36#2044—34no—
37#2021—34no—
38#1820—31no—
39#1782—32no—
40C25×C3C2311yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—24no—
44C22×#67C2314yes—
45C2×#235—22no—
46C2×#235—22no—
47C2×#235—22no—
48C2×#235—22no—
49C2×#235—48no—
50C22×D12D412yescomplemented
51C22×C4×D3D414yescomplemented
52C22×Dic6D412yescomplemented
53C3×#117—22no—
54C3×#117—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4D411yescharacteristic, complemented
57C2×C3×C4⋊C4D411yescharacteristic, complemented
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—22no—
64C2×C3×C22⋊C4C2312yes—
65C2×C3×C22⋊C4C2312yes—
66C2×C3×C42D412yescomplemented
67#953—48no—
68C2×#217—24no—
69C2×#217C2314yes—
70#930—24no—
71#930—24no—
72C2×(C2×D3)⋊C4—24no—
73#897—24no—
74#897—24no—
75#895—24no—
76#895—24no—
77#886—28no—
78#882—28no—
79#875—24no—
80#874—22no—
81C4×D12—24no—
82C4×D12—24no—
83#869—22no—
84C3×#95—21no—
85C3×#95—21no—
86C3×#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—24no—
90#806—48no—
91C22×#41—32no—
92#537—61no—
93#537—61no—
94#536—62no—
95#535—62no—
96#534—34no—
97#532—62no—
98#525—64no—
99C2×#120—34no—
100C2×#117C2×D311yescharacteristic
101C22×C22⋊C4C2×D311yescharacteristic
102C2×#99—38no—
103#399—21no—
104#397—21no—
105#393—62no—
106#392—62no—
107#388—22no—
108#386—34no—
109#386—64no—
110#385C2×D312yes—
111#381—32no—
112#381C2×D311yescharacteristic
113#379C2×D312yes—
114#342—34no—
115C24×C3—22no—
116C24×C3—21no—
117C24×C3—24no—
118C24×C3—42no—
119C24×C3—81no—
120C24×C3C2411yescharacteristic
121C23×D3C2×D414yes—
122C22×C3×C4—24no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—81no—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2×D412yes—
133C2×#67—24no—
134C2×#67—24no—
135C2×#67—28no—
136C2×#67—28no—
137C2×#67—48no—
138C2×#67—48no—
139C2×#67—84no—
140C22×Dic3—24no—
141C22×Dic3C2×D414yes—
142#235—48no—
143#235—48no—
144#235—48no—
145#235—48no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—24no—
149C2×D12—24no—
150C2×D12—44no—
151C2×C4×D3—24no—
152C2×C4×D3—24no—
153C2×C4×D3—44no—
154C2×C4×D3—44no—
155C2×Dic6—22no—
156C2×Dic6—22no—
157C2×Dic6—44no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—82no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—44no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—42no—
167C3×C22⋊C4—82no—
168C3×C42—22no—
169C3×C42—22no—
170#217—28no—
171#217—48no—
172(C2×D3)⋊C4—28no—
173(C2×D3)⋊C4—28no—
174(C2×D3)⋊C4—84no—
175C4⋊Dic3—24no—
176C4⋊Dic3—24no—
177C25C22×D311yescharacteristic
178C2×#41—62no—
179C2×#41—62no—
180C2×#41—62no—
181C2×#41—62no—
182C2×#41—128no—
183C22×Q8—32no—
184C22×D4—34no—
185C22×D4—32no—
186C22×D4—64no—
187C23×C4—34no—
188C23×C4C22×D311yescharacteristic
189C23×C4C22×D311yescharacteristic
190C4⋊Q8—62no—
191#127—62no—
192#124—64no—
193#123—68no—
194C22⋊Q8—64no—
195C22⋊Q8—64no—
196#121—64no—
197#121—64no—
198#121—64no—
199#121—64no—
200#120—68no—
201#120—128no—
202C4×D4—64no—
203C4×D4—64no—
204#117—22no—
205#117—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4#6711yescharacteristic, complemented
208C2×C4⋊C4#6711yescharacteristic, complemented
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—34no—
216C2×C22⋊C4—64no—
217C2×C22⋊C4—64no—
218C2×C22⋊C4C22×D312yes—
219C2×C22⋊C4C22×D312yes—
220C2×C42#6712yescomplemented
221#99—128no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—64no—
226#95—62no—
227#95—62no—
228C23×C3—22no—
229C23×C3—24no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—42no—
233C23×C3—44no—
234C23×C3—42no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—44no—
238C23×C3—42no—
239C23×C3—88no—
240C23×C3C22×D412yes—
241C23×C3C22×D411yescharacteristic
242C23×C3H(5,𝔽2)14yes—
243C22×D3—24no—
244C22×D3—24no—
245C22×D3—24no—
246C22×D3—44no—
247C22×D3—44no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—44no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4#12011yescharacteristic
263C2×C3×C4#12011yescharacteristic
264C2×C3×C4#12012yes—
265#67—48no—
266#67—48no—
267#67—816no—
268C2×Dic3—24no—
269C2×Dic3—24no—
270C2×Dic3—24no—
271C2×Dic3—44no—
272C2×Dic3—44no—
273C2×Dic3—44no—
274D12—44no—
275D12—44no—
276D12—84no—
277D12—84no—
278C4×D3—48no—
279C4×D3—48no—
280Dic6—44no—
281Dic6—44no—
282C24—22no—
283C24—21no—
284C24—24no—
285C24—34no—
286C24—42no—
287C24—81no—
288C24C23×D311yescharacteristic
289#41—128no—
290#41—128no—
291#41—128no—
292#41—128no—
293C2×Q8—62no—
294C2×Q8—62no—
295C2×Q8—124no—
296C2×D4—64no—
297C2×D4—64no—
298C2×D4—64no—
299C2×D4—68no—
300C2×D4—68no—
301C2×D4—64no—
302C2×D4—62no—
303C2×D4—62no—
304C2×D4—124no—
305C2×D4—128no—
306C2×D4—128no—
307C2×D4—244no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—34no—
313C22×C4—64no—
314C22×C4—64no—
315C22×C4—64no—
316C22×C4—81no—
317C22×C4—124no—
318C22×C4—124no—
319C22×C4D3×D412yes—
320C22×C4D3×D412yes—
321C22×C4C2×#6711yescharacteristic
322C22×C4C2×#6711yescharacteristic
323C22×C4C2×#6712yes—
324C22×C4C2×#6712yes—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—64no—
328C4⋊C4—64no—
329C4⋊C4—82no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—44no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—68no—
337C22⋊C4—68no—
338C22⋊C4—68no—
339C22⋊C4—82no—
340C22⋊C4—128no—
341C22⋊C4—244no—
342C42—22no—
343C42—22no—
344C22×C3—22no—
345C22×C3—21no—
346C22×C3—22no—
347C22×C3—21no—
348C22×C3—44no—
349C22×C3—48no—
350C22×C3—44no—
351C22×C3—42no—
352C22×C3—84no—
353C22×C3—82no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3C2×#12011yescharacteristic
357C22×C3#53414yes—
358C22×C3#53512yes—
359C2×D3—48no—
360C2×D3—44no—
361C2×D3—44no—
362C2×D3—44no—
363C2×D3—84no—
364C3×C4—21no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—42no—
368C3×C4—42no—
369C3×C4—82no—
370C3×C4—81no—
371Dic3—44no—
372Dic3—84no—
373C23—22no—
374C23—24no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—44no—
379C23—41no—
380C23—42no—
381C23—44no—
382C23—41no—
383C23—42no—
384C23—64no—
385C23—64no—
386C23—64no—
387C23—88no—
388C23—124no—
389C23—124no—
390C23C2×D3×D412yes—
391C23#100914yes—
392C23C22×#6711yescharacteristic
393Q8—124no—
394Q8—124no—
395D4—124no—
396D4—124no—
397D4—128no—
398D4—128no—
399D4—244no—
400D4—244no—
401D4—2416no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—64no—
417C2×C4—64no—
418C2×C4—64no—
419C2×C4—128no—
420C2×C4—128no—
421C2×C4—124no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4#93711yescharacteristic
425C2×C4#93711yescharacteristic
426C2×C4#93712yes—
427C2×C3—24no—
428C2×C3—22no—
429C2×C3—82no—
430C2×C3#261311yescharacteristic
431C2×C3#302511yescharacteristic
432C2×C3#302711yescharacteristic
433D3—84no—
434C22—21no—
435C22—22no—
436C22—21no—
437C22—22no—
438C22—44no—
439C22—48no—
440C22—44no—
441C22—42no—
442C22—82no—
443C22—84no—
444C22—81no—
445C22—81no—
446C22—128no—
447C22—124no—
448C22—124no—
449C22—124no—
450C22—244no—
451C22#565314yes—
452C22C2×#93711yescharacteristic
453C22#583012yes—
454C4—21no—
455C4—22no—
456C4—21no—
457C4—42no—
458C4—42no—
459C4—82no—
460C4—81no—
461C4—124no—
462C4—244no—
463C3#2963611yescharacteristic, complemented, Sylow
464C2—22no—
465C2—24no—
466C2—82no—
467C2—244no—
468C2#8290311yescharacteristic
469C2#8293611yescharacteristic
470C2#8339611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#2482C2
socleC22×C3C2×#120
hypercenter#2482D3
hypocenterC3#29636
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
1. #85735C2semidirect
2. C3×#2482C2semidirect
3. #82125C2semidirect
4. #71594C2semidirect
5. #69596C2semidirect
6. #69537C2semidirect
7. #69500C2semidirect
34. #2482D3semidirect
50. C22×D12D4semidirect
51. C22×C4×D3D4semidirect
52. C22×Dic6D4semidirect
56. C2×C3×C4⋊C4D4semidirect
57. C2×C3×C4⋊C4D4semidirect
66. C2×C3×C42D4semidirect
207. C2×C4⋊C4#67semidirect
208. C2×C4⋊C4#67semidirect
220. C2×C42#67semidirect
463. C3#29636semidirect

Maximal subgroups

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

subgroupindex
#857352
C3×#24822
#821252
#715942
#695962
#695372
#695002
#296363