Finite Group Databasebeta

← #10848712

Subgroups — #10848712

Order 768.

1158 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic
2#82076C211yesmaximal, characteristic
3C2×#4699C211yesmaximal, characteristic, complemented
4C2×#4681C211yesmaximal, characteristic, complemented
5C3×C4×C22⋊C8C211yesmaximal, characteristic
6#69964C214yesmaximal
7#69777C211yesmaximal, characteristic, complemented
8#69771C211yesmaximal, characteristic, complemented
9#68685C214yesmaximal, complemented
10#25600—31nomaximal, Sylow
11C22×C3×C42C2211yescharacteristic
12C2×#926—21no—
13C2×#924—21no—
14C2×#924C2212yes—
15C2×#924C2212yes—
16C22×Dic12C2214yes—
17#5470—24no—
18#5470—24no—
19C2×#870—21no—
20C2×#869C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6—21no—
24C2×C4×Dic6C2212yes—
25C2×C3×C22⋊C8C2212yes—
26C2×C3×C4×C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28#5212—21no—
29#5212—21no—
30C2×#816C2214yes—
31C2×#816C2214yes—
32#4935C2211yescharacteristic
33#4929C2212yes—
34#4928—21no—
35#4928—21no—
36#4699—24no—
37#4681—24no—
38C3×#336C2212yes—
39#4551C2214yes—
40C2×#533—31no—
41C2×#494—31no—
42C2×#487—31no—
43#2857—31no—
44#2001—34no—
45#1990—31no—
46#1956—31no—
47#1878—34no—
48C4×C22⋊C8D311yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C22×Dic6—21no—
53C22×Dic6C2312yes—
54C22×Dic6C2312yes—
55C22×C3×C8C2312yes—
56C22×C3×C8C2312yes—
57C2×C3×C42—22no—
58C2×C3×C42—22no—
59C2×C3×C42D411yescharacteristic
60C2×C3×C42D411yescharacteristic
61C2×C3×C42C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×#217—22no—
64C2×#217—22no—
65#926—28no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3C2312yes—
74C2×C4⋊Dic3C2312yes—
75#924—28no—
76#924—24no—
77#924—24no—
78#924—44no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3—22no—
83C2×Dic12—28no—
84C2×Dic12—28no—
85C2×Dic12—44no—
86#870—28no—
87#869—22no—
88#869—22no—
89#869—22no—
90#869—22no—
91#869D414yes—
92#869D414yes—
93C4×Dic6—28no—
94C4×Dic6—24no—
95C4×Dic6—24no—
96C3×C22⋊C8—24no—
97C3×C4×C8—22no—
98C3×C4×C8—22no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#816—28no—
102#816—44no—
103C22×Q16—34no—
104#533—64no—
105#533—64no—
106C2×C4⋊Q8—31no—
107C2×C4⋊Q8—31no—
108C2×C4⋊Q8—31no—
109C2×#125—61no—
110C2×C22⋊Q8—32no—
111C2×C22⋊Q8—32no—
112C2×C22⋊Q8—61no—
113C2×C4×Q8—32no—
114C2×C4×Q8—61no—
115C2×#117—61no—
116C22×C42C2×D311yescharacteristic
117#494—64no—
118#487—64no—
119C2×#103—34no—
120C2×#103—34no—
121C2×C22⋊C8C2×D312yes—
122C2×C4×C8C2×D311yescharacteristic
123C2×C4×C8C2×D311yescharacteristic
124#391—32no—
125#389—61no—
126#389—61no—
127#384—31no—
128#380—61no—
129#380—61no—
130#340—34no—
131#336C2×D312yes—
132C24×C3C2×D411yescharacteristic
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4Q1614yescomplemented
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—22no—
153C2×Dic6—24no—
154C2×Dic6—24no—
155C2×Dic6—24no—
156C2×Dic6—22no—
157C2×Dic6—22no—
158C2×Dic6—22no—
159C2×Dic6—24no—
160C2×Dic6—24no—
161C2×Dic6—42no—
162C2×Dic6—42no—
163C2×Dic6C2×D414yes—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8#4114yes—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42—21no—
173C3×C42—21no—
174C3×C42C2×D412yes—
175C3×C42C2×D412yes—
176#217—28no—
177#217—44no—
178C4⋊Dic3—24no—
179C4⋊Dic3—24no—
180C4⋊Dic3—24no—
181C4⋊Dic3—22no—
182C4⋊Dic3—24no—
183C4⋊Dic3—42no—
184C4⋊Dic3—42no—
185C4⋊Dic3—42no—
186C4⋊Dic3—42no—
187C4⋊Dic3C2×D414yes—
188Dic3⋊C4—28no—
189Dic3⋊C4—28no—
190Dic3⋊C4—44no—
191C4×Dic3—28no—
192Dic12—416no—
193C22×Q8—32no—
194C22×Q8—32no—
195C22×Q8—61no—
196C23×C4D1211yescharacteristic
197C23×C4D1211yescharacteristic
198C23×C4C22×D311yescharacteristic
199C2×Q16—68no—
200C2×Q16—68no—
201C2×Q16—124no—
202C22×C8C22×D312yes—
203C22×C8C22×D312yes—
204C4⋊Q8—34no—
205C4⋊Q8—34no—
206C4⋊Q8—62no—
207C4⋊Q8—62no—
208C4⋊Q8—62no—
209C4⋊Q8—62no—
210#125—68no—
211C22⋊Q8—64no—
212C22⋊Q8—64no—
213C22⋊Q8—68no—
214C22⋊Q8—124no—
215C4×Q8—64no—
216C4×Q8—64no—
217C4×Q8—68no—
218#117—68no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—61no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—62no—
236C2×C42D1211yescharacteristic
237C2×C42D1211yescharacteristic
238C2×C42C22×D311yescharacteristic
239C2×C42C22×D311yescharacteristic
240#103—68no—
241#103—124no—
242C22⋊C8—24no—
243C4×C8—22no—
244C4×C8—22no—
245#95—62no—
246#95—62no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3C2×Q1612yes—
254C23×C3C22×D411yescharacteristic
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—28no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—44no—
267C2×C3×C4#12012yes—
268C2×C3×C4#12012yes—
269C2×C3×C4C2×Q1612yes—
270C2×C3×C4C2×Q1612yes—
271C2×C3×C4#13512yes—
272C2×C3×C4#13712yes—
273C2×C3×C4#13712yes—
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C22×D411yescharacteristic
276C2×C3×C4C2×#4112yes—
277C2×C3×C4H(5,𝔽2)12yes—
278C2×Dic3—24no—
279C2×Dic3—24no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290Dic6—28no—
291Dic6—44no—
292Dic6—48no—
293Dic6—44no—
294Dic6—44no—
295C3×C8—24no—
296C3×C8—42no—
297C24C2×D1211yescharacteristic
298C2×Q8—34no—
299C2×Q8—64no—
300C2×Q8—62no—
301C2×Q8—64no—
302C2×Q8—64no—
303C2×Q8—62no—
304C2×Q8—62no—
305C2×Q8—64no—
306C2×Q8—64no—
307C2×Q8—122no—
308C2×Q8—122no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—62no—
322C22×C4Dic1214yescomplemented
323C22×C4C2×D1211yescharacteristic
324C22×C4C2×D1211yescharacteristic
325C22×C4C2×D1211yescharacteristic
326C22×C4C2×D1211yescharacteristic
327C22×C4C2×D1211yescharacteristic
328C22×C4C23×D311yescharacteristic
329Q16—1216no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8#23514yes—
336C4⋊C4—34no—
337C4⋊C4—68no—
338C4⋊C4—68no—
339C4⋊C4—64no—
340C4⋊C4—64no—
341C4⋊C4—64no—
342C4⋊C4—62no—
343C4⋊C4—64no—
344C4⋊C4—124no—
345C4⋊C4—122no—
346C4⋊C4—122no—
347C4⋊C4—122no—
348C4⋊C4—122no—
349C22⋊C4—68no—
350C22⋊C4—124no—
351C42—22no—
352C42—22no—
353C42—21no—
354C42—21no—
355C42—68no—
356C42D3×D412yes—
357C42D3×D412yes—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3C2×#12011yescharacteristic
366C22×C3#53211yescharacteristic
367C22×C3#53511yescharacteristic
368C22×C3C22×Q1611yescharacteristic
369C22×C3C2×#13511yescharacteristic
370C22×C3C2×#13711yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C3×C4—22no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4#45114yes—
376C3×C4#45214yes—
377C3×C4#54612yes—
378C3×C4#55012yes—
379Dic3—44no—
380Dic3—44no—
381Dic3—82no—
382Dic3—82no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23C2×Dic1212yes—
390C23C22×D1211yescharacteristic
391Q8—68no—
392Q8—128no—
393Q8—124no—
394Q8—124no—
395Q8—124no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—28no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—64no—
409C2×C4—64no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4#88212yes—
421C2×C4#88212yes—
422C2×C4#90412yes—
423C2×C4C2×Dic1212yes—
424C2×C4C2×Dic1212yes—
425C2×C4#90912yes—
426C2×C4#90912yes—
427C2×C4C2×#23512yes—
428C2×C4C2×D3×D411yescharacteristic
429C2×C4C2×D3×D411yescharacteristic
430C2×C4#100912yes—
431C8—24no—
432C8—42no—
433C2×C3—42no—
434C2×C3#243111yescharacteristic
435C2×C3C2×#45111yescharacteristic
436C2×C3C2×#45211yescharacteristic
437C2×C3#317011yescharacteristic
438C2×C3#318511yescharacteristic
439C2×C3#320911yescharacteristic
440C2×C3#321011yescharacteristic
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22#547711yescharacteristic
449C22C2×#88211yescharacteristic
450C22#552511yescharacteristic
451C22C2×#90411yescharacteristic
452C22C22×Dic1211yescharacteristic
453C22C2×#90911yescharacteristic
454C22C2×#90911yescharacteristic
455C4—22no—
456C4—41no—
457C4—41no—
458C4—124no—
459C4—124no—
460C4—242no—
461C4—242no—
462C4#473314yes—
463C4#473914yes—
464C4#555012yes—
465C4#555312yes—
466C3#2560011yescharacteristic, complemented, Sylow
467C2—42no—
468C2#7358611yescharacteristic
469C2#7847511yescharacteristic
470C2#7854011yescharacteristic
471C2C2×#473311yescharacteristic
472C2C2×#473911yescharacteristic
473C2#7930611yescharacteristic
474C2#7931311yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C22⋊C8C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC4×C22⋊C8D3
hypocenterC3#25600
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×#4699C2semidirect
4. C2×#4681C2semidirect
7. #69777C2semidirect
8. #69771C2semidirect
9. #68685C2semidirect
142. C22×C3×C4Q16semidirect
322. C22×C4Dic12semidirect
466. C3#25600semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
#820762
C2×#46992
C2×#46812
C3×C4×C22⋊C82
#699642
#697772
#697712
#686852
#256003