Finite Group Databasebeta

← #10900904

Subgroups — #10900904

Order 768.

1502 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C3×#3449C211yesmaximal, characteristic, complemented
2#84096C212yesmaximal, complemented
3#82282C211yesmaximal, characteristic, complemented
4#72139C212yesmaximal, complemented
5#71851C214yesmaximal, complemented
6#71850C214yesmaximal, complemented
7#71594C211yesmaximal, characteristic, complemented
8#29636—31nomaximal, Sylow
9C22×C3×#41C2212yes—
10C3×#537—21no—
11C3×#537—21no—
12C3×#536—22no—
13C3×#535—22no—
14C3×#534C2214yes—
15C3×#532—22no—
16C3×#525—24no—
17C2×C3×#117C2211yescharacteristic
18C2×#937C2214yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20#5231—22no—
21#5223—22no—
22#5220C2214yes—
23C2×#834C2218yes—
24#5213—24no—
25#5211C2211yescharacteristic
26#5211C2212yes—
27#4993—21no—
28#4992—21no—
29#4955C2212yes—
30#4946C2212yes—
31#4940—22no—
32#4526C2214yes—
33#3449D311yescharacteristic, complemented
34#2482—31no—
35#2404—32no—
36#2044—34no—
37#2021—34no—
38#1820—31no—
39#1782—32no—
40C24×D3C2311yescharacteristic
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—22no—
44C2×C3×#41—22no—
45C2×C3×#41—48no—
46C22×C3×Q8D412yescomplemented
47C22×C3×D4D412yescomplemented
48C22×C3×D4C2314yes—
49C23×C3×C4D414yescomplemented
50C23×C3×C4C2311yescharacteristic
51C22×#67—24no—
52C23×Dic3C2311yescharacteristic
53C3×C4⋊Q8—22no—
54C3×#127—22no—
55C3×#124—24no—
56C3×#123—28no—
57C3×C22⋊Q8—24no—
58C3×C22⋊Q8—24no—
59C3×#121—24no—
60C3×#121—24no—
61C3×#121—24no—
62C3×#121—24no—
63C3×#120—28no—
64C3×C4×D4—24no—
65C3×C4×D4—24no—
66C3×#117—22no—
67C3×#117—22no—
68C2×C3×C4⋊C4D411yescharacteristic, complemented
69C2×C3×C4⋊C4D411yescharacteristic, complemented
70C2×C3×C22⋊C4C2312yes—
71C2×C3×C42D412yescomplemented
72C2×#217—24no—
73C2×#217C2312yes—
74C2×#217C2314yes—
75#937—48no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—22no—
83C2×Dic3⋊C4—22no—
84#834—48no—
85C3⋊#95—21no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—21no—
90C3⋊#95—24no—
91C22×#41C2×D312yes—
92#537—21no—
93#537—21no—
94#536—22no—
95#535—22no—
96#534C2×D314yes—
97#532—22no—
98#525—24no—
99C2×#120—34no—
100C2×#117C2×D311yescharacteristic
101C22×C22⋊C4—31no—
102C2×#99—38no—
103#399—61no—
104#397—61no—
105#393—62no—
106#392—62no—
107#388—62no—
108#386—34no—
109#386—64no—
110#385—32no—
111#381—32no—
112#381—31no—
113#379—32no—
114#342—34no—
115C24×C3C2×D414yes—
116C24×C3C2411yescharacteristic
117C23×D3—21no—
118C23×D3—22no—
119C23×D3—24no—
120C23×D3—42no—
121C23×D3—81no—
122C3×#41—48no—
123C3×#41—48no—
124C3×#41—48no—
125C3×#41—48no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—44no—
129C2×C3×D4—24no—
130C2×C3×D4—24no—
131C2×C3×D4—28no—
132C2×C3×D4—28no—
133C2×C3×D4—24no—
134C2×C3×D4—24no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—44no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—24no—
141C22×C3×C4—44no—
142C22×C3×C4—44no—
143C22×C3×C4C2×D414yes—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C2×#67—48no—
151C2×#67—48no—
152C2×#67—84no—
153C22×Dic3—22no—
154C22×Dic3—24no—
155C22×Dic3—21no—
156C22×Dic3—22no—
157C22×Dic3—81no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C22⋊C4—28no—
163C3×C22⋊C4—28no—
164C3×C22⋊C4—24no—
165C3×C42—22no—
166C3×C42—22no—
167#217—24no—
168#217—28no—
169#217—84no—
170(C2×D3)⋊C4—48no—
171(C2×D3)⋊C4—44no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—42no—
175(C2×D3)⋊C4—82no—
176Dic3⋊C4—82no—
177C25—31no—
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—22no—
182C2×#41—48no—
183C22×Q8#6712yescomplemented
184C22×D4—64no—
185C22×D4#6712yescomplemented
186C22×D4C22×D314yes—
187C23×C4—31no—
188C23×C4#6714yescomplemented
189C23×C4C22×D311yescharacteristic
190C4⋊Q8—22no—
191#127—22no—
192#124—24no—
193#123—28no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196#121—24no—
197#121—24no—
198#121—24no—
199#121—24no—
200#120—28no—
201#120—128no—
202C4×D4—24no—
203C4×D4—24no—
204#117—22no—
205#117—22no—
206C2×C4⋊C4—62no—
207C2×C4⋊C4#6711yescharacteristic, complemented
208C2×C4⋊C4#6711yescharacteristic, complemented
209C2×C22⋊C4—34no—
210C2×C22⋊C4—32no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—64no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—64no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4C22×D312yes—
220C2×C42#6712yescomplemented
221#99—128no—
222#95—62no—
223#95—61no—
224#95—62no—
225#95—61no—
226#95—62no—
227#95—64no—
228C23×C3—24no—
229C23×C3—24no—
230C23×C3—24no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—44no—
234C23×C3—44no—
235C23×C3C22×D412yes—
236C23×C3C22×D411yescharacteristic
237C23×C3H(5,𝔽2)14yes—
238C22×D3—22no—
239C22×D3—24no—
240C22×D3—42no—
241C22×D3—44no—
242C22×D3—41no—
243C22×D3—42no—
244C22×D3—41no—
245C22×D3—44no—
246C22×D3—42no—
247C22×D3—88no—
248C3×Q8—44no—
249C3×Q8—44no—
250C3×D4—44no—
251C3×D4—44no—
252C3×D4—48no—
253C3×D4—48no—
254C3×D4—84no—
255C3×D4—84no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—48no—
266C2×C3×C4—48no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4#12011yescharacteristic
270C2×C3×C4#12011yescharacteristic
271C2×C3×C4#12012yes—
272#67—816no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—44no—
277C2×Dic3—44no—
278C2×Dic3—44no—
279C2×Dic3—41no—
280C2×Dic3—44no—
281C2×Dic3—42no—
282C24—61no—
283C24—62no—
284C24—64no—
285C24—122no—
286C24—241no—
287C24C2×#6714yes—
288C24C23×D311yescharacteristic
289#41—48no—
290#41—48no—
291#41—48no—
292#41—48no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—44no—
296C2×D4—28no—
297C2×D4—28no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—44no—
305C2×D4—128no—
306C2×D4—128no—
307C2×D4—244no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—62no—
314C22×C4—64no—
315C22×C4—62no—
316C22×C4—61no—
317C22×C4—241no—
318C22×C4C2×#6714yes—
319C22×C4C2×#6711yescharacteristic
320C22×C4C2×#6711yescharacteristic
321C22×C4C2×#6712yes—
322C22×C4C2×#6712yes—
323C22×C4C2×#6712yes—
324C22×C4C2×#6712yes—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—242no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—24no—
333C22⋊C4—64no—
334C22⋊C4—68no—
335C22⋊C4—128no—
336C22⋊C4—124no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C22⋊C4—122no—
340C22⋊C4—242no—
341C22⋊C4—244no—
342C42—22no—
343C42—22no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—44no—
349C22×C3—48no—
350C22×C3—44no—
351C22×C3—44no—
352C22×C3—44no—
353C22×C3—84no—
354C22×C3C2×#12011yescharacteristic
355C22×C3#53414yes—
356C22×C3#53512yes—
357C2×D3—48no—
358C2×D3—44no—
359C2×D3—42no—
360C2×D3—84no—
361C2×D3—82no—
362C2×D3—81no—
363C2×D3—81no—
364C3×C4—21no—
365C3×C4—22no—
366C3×C4—21no—
367C3×C4—44no—
368C3×C4—42no—
369C3×C4—42no—
370Dic3—82no—
371Dic3—84no—
372Dic3—81no—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—41no—
377C23—41no—
378C23—44no—
379C23—44no—
380C23—62no—
381C23—64no—
382C23—122no—
383C23—124no—
384C23—121no—
385C23—122no—
386C23—124no—
387C23—122no—
388C23—121no—
389C23—248no—
390C23#100414yes—
391C23C22×#6712yes—
392C23C22×#6711yescharacteristic
393Q8—44no—
394Q8—44no—
395D4—44no—
396D4—44no—
397D4—48no—
398D4—48no—
399D4—84no—
400D4—84no—
401D4—2416no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—124no—
418C2×C4—124no—
419C2×C4—122no—
420C2×C4—124no—
421C2×C4—124no—
422C2×C4—122no—
423C2×C4—121no—
424C2×C4#95311yescharacteristic
425C2×C4#95311yescharacteristic
426C2×C4#95312yes—
427C2×C3—24no—
428C2×C3—22no—
429C2×C3—84no—
430C2×C3#261311yescharacteristic
431C2×C3#302511yescharacteristic
432C2×C3#302711yescharacteristic
433D3—82no—
434C22—21no—
435C22—22no—
436C22—21no—
437C22—22no—
438C22—44no—
439C22—48no—
440C22—44no—
441C22—44no—
442C22—44no—
443C22—84no—
444C22—128no—
445C22—124no—
446C22—122no—
447C22—244no—
448C22—242no—
449C22—241no—
450C22—241no—
451C22#580414yes—
452C22#580512yes—
453C22C2×#95311yescharacteristic
454C4—21no—
455C4—22no—
456C4—21no—
457C4—44no—
458C4—42no—
459C4—42no—
460C4—244no—
461C4—242no—
462C4—241no—
463C3#2963611yescharacteristic, complemented, Sylow
464C2—24no—
465C2—22no—
466C2—84no—
467C2—242no—
468C2#8405411yescharacteristic
469C2#8412511yescharacteristic
470C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC3×#3449C2
socleC22×C3C2×#120
hypercenter#3449D3
hypocenterC3#29636
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. C3×#3449C2semidirect
2. #84096C2semidirect
3. #82282C2semidirect
4. #72139C2semidirect
5. #71851C2semidirect
6. #71850C2semidirect
7. #71594C2semidirect
33. #3449D3semidirect
46. C22×C3×Q8D4semidirect
47. C22×C3×D4D4semidirect
49. C23×C3×C4D4semidirect
68. C2×C3×C4⋊C4D4semidirect
69. C2×C3×C4⋊C4D4semidirect
71. C2×C3×C42D4semidirect
183. C22×Q8#67semidirect
185. C22×D4#67semidirect
188. C23×C4#67semidirect
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
C3×#34492
#840962
#822822
#721392
#718512
#718502
#715942
#296363