Finite Group Databasebeta

← #10826907

Subgroups — #10826907

Order 768.

1694 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic, complemented
2#87274C211yesmaximal, characteristic
3#84059C212yesmaximal, complemented
4C2×#5095C212yesmaximal, complemented
5#72137C211yesmaximal, characteristic, complemented
6#71784C212yesmaximal, complemented
7#71731C214yesmaximal, complemented
8#71591C212yesmaximal, complemented
9#23916—31nomaximal, Sylow
10C22×C3×#41C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic
12C2×#1007C2211yescharacteristic
13C3×#536—28no—
14C2×C3×C4⋊Q8C2212yes—
15C2×C3×#124C2212yes—
16C2×C3×#123—22no—
17C2×C3×C22⋊Q8—22no—
18C2×C3×#117C2211yescharacteristic
19#5834—24no—
20#5831—24no—
21#5812C2211yescharacteristic
22C2×#942C2214yes—
23#5805—22no—
24#5805C2212yes—
25#5791—22no—
26#5791C2212yes—
27#5228—24no—
28#5227—24no—
29C2×#836C2212yes—
30#5225C2214yes—
31C2×#833C2212yes—
32#5207C2212yes—
33#5095—28no—
34C2×#805C2214yes—
35#4523C2214yes—
36#3487—31no—
37C2×#536D311yescharacteristic, complemented
38C2×#456—32no—
39#3004—32no—
40#2004—34no—
41#1897—32no—
42#1886—32no—
43#1785—31no—
44C2×C3×#41—24no—
45C2×C3×#41—24no—
46C2×C3×#41—42no—
47C22×C3×Q8—24no—
48C22×C3×Q8—21no—
49C22×C3×Q8—24no—
50C22×C3×Q8—42no—
51C22×C3×Q8C2312yes—
52C22×C3×Q8C2311yescharacteristic
53C22×C3×D4D412yes—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58#1007—24no—
59#1007—42no—
60C2×#239—22no—
61C2×D3×Q8—22no—
62C2×#235—22no—
63C2×#235C2312yes—
64C3×C4⋊Q8—24no—
65C3×C4⋊Q8D418yescomplemented
66C3×#124—24no—
67C3×#124—24no—
68C3×#123—216no—
69C3×C22⋊Q8—216no—
70C3×#117—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C42C2312yes—
76#953—24no—
77#953—22no—
78Q8×Dic3—24no—
79C2×#216—24no—
80#942—28no—
81#942—44no—
82C2×#215—24no—
83#940—24no—
84#940—22no—
85#938—28no—
86#938—24no—
87#936—24no—
88#936—22no—
89#930—24no—
90#930D414yes—
91#929—22no—
92C4×#67—28no—
93C4×#67—24no—
94#926—22no—
95#926C2312yes—
96#924—24no—
97#924D414yes—
98C2×#208C2314yes—
99#836—24no—
100#835—28no—
101#833—42no—
102#830—24no—
103#805—28no—
104C2×2−1+4—31no—
105C22×#41C2×D311yescharacteristic
106C23×Q8C2×D311yescharacteristic
107C2×#137—34no—
108#550—64no—
109#548—64no—
110#536—28no—
111#535—32no—
112#535—62no—
113C2×C4⋊Q8C2×D312yes—
114C2×#124C2×D312yes—
115#525—32no—
116#525—62no—
117C2×#123—22no—
118C2×C22⋊Q8—22no—
119#519—31no—
120C2×#117C2×D311yescharacteristic
121#456—68no—
122#451—64no—
123#448—64no—
124C2×#104—34no—
125#419—34no—
126C2×#101—32no—
127C2×#100—32no—
128#407—32no—
129#339—34no—
130C24×C3C2×D412yes—
131C24×C3C2×D411yescharacteristic
132C3×#41—416no—
133C2×C3×Q8—24no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—24no—
138C2×C3×Q8—24no—
139C2×C3×Q8—416no—
140C2×C3×Q8—42no—
141C2×C3×Q8—42no—
142C2×C3×Q8C2×D414yes—
143C2×C3×Q8C2×D414yes—
144C2×C3×D4—24no—
145C2×C3×D4—24no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—42no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—24no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—24no—
156C22×C3×C4—24no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4C2×D414yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×#67—22no—
165C2×#67—28no—
166C2×#67—24no—
167#239—48no—
168D3×Q8—48no—
169#235—24no—
170#235—24no—
171#235—44no—
172#235—44no—
173C2×D12—22no—
174C2×D12C2×D412yes—
175C2×C4×D3—24no—
176C2×C4×D3—22no—
177C2×Dic6—22no—
178C2×Dic6C2×D412yes—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—216no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—216no—
183C3×C42—22no—
184C3×C42C2×D414yes—
185#217—28no—
186#217—24no—
187#217—22no—
188#216—48no—
189#215—48no—
190(C2×D3)⋊C4—24no—
191C4⋊Dic3—24no—
192C4⋊Dic3C2×D414yes—
193Dic3⋊C4—24no—
194C4×Dic3—24no—
195C4×Dic3—22no—
196#208—24no—
197#208—24no—
198C2×C3⋊C8—24no—
1992−1+4—64no—
2002−1+4—122no—
201C2×#41—24no—
202C2×#41—24no—
203C2×#41—32no—
204C2×#41—42no—
205C2×#41—62no—
206C2×#41—62no—
207C22×Q8—24no—
208C22×Q8—21no—
209C22×Q8—24no—
210C22×Q8—42no—
211C22×Q8—62no—
212C22×Q8C22×D312yes—
213C22×Q8C22×D311yescharacteristic
214C22×D4#6712yes—
215C22×D4C22×D311yescharacteristic
216C23×C4—21no—
217C23×C4#6712yes—
218C23×C4C22×D311yescharacteristic
219#137—68no—
220#137—124no—
221C2×Q16—64no—
222C2×SD16—64no—
223C2×M16—34no—
224C4⋊Q8—24no—
225C4⋊Q8#6718yescomplemented
226#127—62no—
227#127—64no—
228#124—24no—
229#124—24no—
230#124—62no—
231#124—64no—
232#123—216no—
233#123—62no—
234C22⋊Q8—216no—
235C22⋊Q8—34no—
236C22⋊Q8—64no—
237#121—34no—
238#121—64no—
239#121—64no—
240#121—68no—
241#120—62no—
242#120—64no—
243C4×Q8—64no—
244C4×D4—68no—
245C4×D4—64no—
246#117—24no—
247#117—32no—
248#117—62no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—24no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—24no—
253C2×C42C22×D312yes—
254#104—68no—
255#103—68no—
256#101—64no—
257#100—122no—
258C22⋊C8—64no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—42no—
265C23×C3—44no—
266C23×C3#12012yes—
267C23×C3C22×D411yescharacteristic
268C22×D3—22no—
269C3×Q8—28no—
270C3×Q8—28no—
271C3×Q8—48no—
272C3×Q8—44no—
273C3×D4—48no—
274C3×D4—44no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—28no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—48no—
283C2×C3×C4—44no—
284C2×C3×C4—48no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4#12012yes—
288C2×C3×C4#12014yes—
289C2×C3×C4C22×D412yes—
290C2×C3×C4C22×D412yes—
291C2×C3×C4H(5,𝔽2)12yes—
292#67—42no—
293#67—42no—
294#67—84no—
295#67—88no—
296C2×Dic3—24no—
297C2×Dic3—22no—
298D12—24no—
299D12—44no—
300C4×D3—48no—
301C4×D3—44no—
302Dic6—24no—
303Dic6—44no—
304C3⋊C8—44no—
305C24C2×#6712yes—
306C24C2×#6711yescharacteristic
307#41—416no—
308#41—64no—
309#41—64no—
310#41—124no—
311#41—128no—
312#41—124no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—24no—
318C2×Q8—24no—
319C2×Q8—32no—
320C2×Q8—416no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—62no—
324C2×Q8—128no—
325C2×Q8C2×#6714yes—
326C2×Q8C2×#6714yes—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—32no—
333C2×D4—42no—
334C2×D4—62no—
335C2×D4—62no—
336C2×D4—68no—
337C2×D4—64no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—64no—
349C22×C4—62no—
350C22×C4D3×D414yes—
351C22×C4C2×#6711yescharacteristic
352C22×C4C2×#6712yes—
353C22×C4C23×D311yescharacteristic
354Q16—128no—
355SD16—128no—
356M16—64no—
357M16—64no—
358C2×C8—64no—
359C4⋊C4—24no—
360C4⋊C4—216no—
361C4⋊C4—34no—
362C4⋊C4—64no—
363C4⋊C4—64no—
364C22⋊C4—24no—
365C22⋊C4—216no—
366C22⋊C4—68no—
367C22⋊C4—64no—
368C22⋊C4—62no—
369C22⋊C4—64no—
370C42—22no—
371C42—64no—
372C42—62no—
373C42C2×#6714yes—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—24no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—44no—
380C22×C3—42no—
381C22×C3—44no—
382C22×C3—82no—
383C22×C3C2×#12011yescharacteristic
384C22×C3C2×#12011yescharacteristic
385C22×C3#53411yescharacteristic
386C2×D3—42no—
387C2×D3—44no—
388C3×C4—24no—
389C3×C4—22no—
390C3×C4—44no—
391C3×C4—44no—
392C3×C4D4214yes—
393Dic3—42no—
394Dic3—44no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—42no—
401C23—44no—
402C23—62no—
403C23#93712yes—
404C23C22×#6711yescharacteristic
405Q8—28no—
406Q8—28no—
407Q8—48no—
408Q8—44no—
409Q8—64no—
410Q8—124no—
411D4—48no—
412D4—44no—
413D4—64no—
414D4—124no—
415D4—122no—
416D4—122no—
417D4—244no—
418D4—248no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—64no—
432C2×C4—62no—
433C2×C4—128no—
434C2×C4—124no—
435C2×C4#93712yes—
436C2×C4#95314yes—
437C2×C4C2×D3×D412yes—
438C2×C4#100412yes—
439C2×C4C22×#6712yes—
440C8—124no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—42no—
444C2×C3—82no—
445C2×C3#240311yescharacteristic
446C2×C3#302212yes—
447D3—82no—
448C22—22no—
449C22—22no—
450C22—24no—
451C22—42no—
452C22—42no—
453C22—44no—
454C22—42no—
455C22—44no—
456C22—82no—
457C22—122no—
458C22—124no—
459C22C2×#93711yescharacteristic
460C22#580411yescharacteristic
461C22C2×#95311yescharacteristic
462C4—22no—
463C4—24no—
464C4—44no—
465C4—44no—
466C4—122no—
467C4—124no—
468C4D4×#6714yes—
469C3#2391611yescharacteristic, complemented, Sylow
470C2—21no—
471C2—21no—
472C2—42no—
473C2—82no—
474C2—242no—
475C2#8341612yes—
476C2#8405211yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#536C2
socleC22×C3C2×#120
prime radical#87274C2
hypercenterC2×#536D3
hypocenterC3#23916
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×C3×#536C2semidirect
3. #84059C2semidirect
4. C2×#5095C2semidirect
5. #72137C2semidirect
6. #71784C2semidirect
7. #71731C2semidirect
8. #71591C2semidirect
37. C2×#536D3semidirect
65. C3×C4⋊Q8D4semidirect
225. C4⋊Q8#67semidirect
469. C3#23916semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
#872742
#840592
C2×#50952
#721372
#717842
#717312
#715912
#239163