Finite Group Databasebeta

← #10848709

Subgroups — #10848709

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×#5728C211yesmaximal, characteristic
2C2×C3×#494C211yesmaximal, characteristic, complemented
3#83269C211yesmaximal, characteristic
4C2×#5079C211yesmaximal, characteristic, complemented
5#72071C214yesmaximal
6#72065C214yesmaximal, complemented
7#71785C211yesmaximal, characteristic, complemented
8#69777C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C8C211yesmaximal, characteristic
10#25600—31nomaximal, Sylow
11C22×C3×Q16C2214yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946C2212yes—
15C2×#946C2212yes—
16C2×Q8×Dic3C2212yes—
17#5728—24no—
18#5728—24no—
19C2×#897—21no—
20C2×#892—21no—
21C2×#890—21no—
22C2×#887—21no—
23C2×C42×D3C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C3×#494—24no—
26C2×C3×C4×C8C2211yescharacteristic
27#5229C2212yes—
28C2×#835C2214yes—
29C2×#835C2214yes—
30C2×(C2×D3)⋊C8C2212yes—
31#5079—24no—
32#4987—21no—
33#4987—21no—
34#4972—21no—
35#4972—21no—
36#4935C2211yescharacteristic
37C2×C4×C3⋊C8C2211yescharacteristic
38#4551C2214yes—
39#4550C2212yes—
40C2×#533—31no—
41C2×#494D311yescharacteristic, complemented
42C2×#487—31no—
43#2857—31no—
44#2001—34no—
45#1990—31no—
46#1956—31no—
47#1878—34no—
48C4×C22⋊C8—31no—
49C22×C3×Q8C2312yes—
50C22×C3×Q8C2312yes—
51C22×C4×D3D411yescharacteristic
52C22×C4×D3D411yescharacteristic
53C22×C4×D3C2311yescharacteristic
54C22×Dic6—21no—
55C2×C3×Q16—28no—
56C2×C3×Q16—28no—
57C2×C3×Q16—44no—
58C22×C3×C8C2312yes—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8D414yes—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C42C2311yescharacteristic
68#946—24no—
69#946—24no—
70#946—44no—
71Q8×Dic3—24no—
72Q8×Dic3—24no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2312yes—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8C2312yes—
88#897—28no—
89#892—28no—
90#890—28no—
91#887—28no—
92C42×D3—22no—
93C42×D3—22no—
94#869—22no—
95#869D414yes—
96C3×C4×C8—22no—
97#835—28no—
98#835—44no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101(C2×D3)⋊C8—24no—
102C4×C3⋊C8—22no—
103C22×Q16C2×D314yes—
104#533—64no—
105#533—64no—
106C2×C4⋊Q8—31no—
107C2×C4⋊Q8C2×D311yescharacteristic
108C2×C4⋊Q8C2×D311yescharacteristic
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×C42—31no—
117#494—24no—
118#487—64no—
119C2×#103—34no—
120C2×#103—34no—
121C2×C22⋊C8—32no—
122C2×C4×C8—31no—
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#336—32no—
132C23×D3C2×D411yescharacteristic
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—24no—
137C2×C3×Q8—24no—
138C2×C3×Q8—24no—
139C2×C3×Q8—24no—
140C2×C3×Q8C2×D414yes—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3C2×D411yescharacteristic
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C2×C4×D3—22no—
156C2×C4×D3—21no—
157C2×C4×D3—21no—
158C2×C4×D3—22no—
159C2×C4×D3—22no—
160C2×C4×D3Q1614yescomplemented
161C2×Dic6—24no—
162C2×Dic6—42no—
163C2×Dic6—42no—
164C3×Q16—416no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8#4114yes—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C42C2×D412yes—
173C3×C42C2×D412yes—
174(C2×D3)⋊C4—28no—
175(C2×D3)⋊C4—44no—
176C4⋊Dic3—28no—
177C4⋊Dic3—24no—
178C4⋊Dic3—22no—
179C4⋊Dic3—24no—
180C4⋊Dic3—42no—
181C4⋊Dic3—42no—
182C4⋊Dic3C2×D414yes—
183Dic3⋊C4—28no—
184Dic3⋊C4—44no—
185C4×Dic3—28no—
186C4×Dic3—22no—
187C4×Dic3—22no—
188C4×Dic3—21no—
189C4×Dic3—21no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C22×Q8—61no—
194C22×Q8C22×D312yes—
195C22×Q8C22×D312yes—
196C23×C4—31no—
197C23×C4—31no—
198C23×C4—31no—
199C2×Q16—28no—
200C2×Q16—28no—
201C2×Q16—44no—
202C22×C8—32no—
203C22×C8C22×D312yes—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—34no—
208C4⋊Q8—62no—
209C4⋊Q8#6714yes—
210#125—68no—
211C22⋊Q8—68no—
212C22⋊Q8—64no—
213C22⋊Q8—64no—
214C22⋊Q8—124no—
215C4×Q8—68no—
216C4×Q8—64no—
217C4×Q8—64no—
218#117—68no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—32no—
225C2×C4⋊C4—62no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C22⋊C4—62no—
232C2×C22⋊C4—62no—
233C2×C42—31no—
234C2×C42—31no—
235C2×C42—31no—
236C2×C42—62no—
237C2×C42—62no—
238C2×C42—62no—
239C2×C42C22×D311yescharacteristic
240#103—68no—
241#103—124no—
242C22⋊C8—64no—
243C4×C8—22no—
244C4×C8—62no—
245#95—62no—
246#95—62no—
247C23×C3C22×D411yescharacteristic
248C22×D3—21no—
249C22×D3—21no—
250C22×D3—21no—
251C22×D3—21no—
252C22×D3—21no—
253C22×D3—21no—
254C22×D3C2×Q1612yes—
255C3×Q8—28no—
256C3×Q8—44no—
257C3×Q8—44no—
258C3×Q8—44no—
259C2×C3×C4—24no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4#12012yes—
266C2×C3×C4#12012yes—
267C2×C3×C4C2×Q1612yes—
268C2×C3×C4#13512yes—
269C2×C3×C4#13712yes—
270C2×C3×C4#13712yes—
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C22×D411yescharacteristic
273C2×C3×C4C2×#4112yes—
274C2×C3×C4H(5,𝔽2)12yes—
275C2×Dic3—21no—
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—21no—
279C2×Dic3—24no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3C2×Q1612yes—
290C4×D3—28no—
291C4×D3—42no—
292C4×D3—42no—
293C4×D3—44no—
294Dic6—48no—
295C3×C8—24no—
296C3⋊C8—42no—
297C24—31no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—64no—
306C2×Q8—122no—
307C2×Q8—122no—
308C2×Q8C2×#6714yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—31no—
312C22×C4—34no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—62no—
316C22×C4—62no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—62no—
320C22×C4—62no—
321C22×C4—61no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—61no—
325C22×C4—62no—
326C22×C4D3×D411yescharacteristic
327C22×C4D3×D411yescharacteristic
328C22×C4C23×D311yescharacteristic
329Q16—416no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8#23714yes—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—34no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—68no—
342C4⋊C4—68no—
343C4⋊C4—64no—
344C4⋊C4—62no—
345C4⋊C4—64no—
346C4⋊C4—124no—
347C4⋊C4—122no—
348C4⋊C4—122no—
349C22⋊C4—68no—
350C22⋊C4—124no—
351C42—68no—
352C42—62no—
353C42—62no—
354C42—61no—
355C42—61no—
356C42D3×D412yes—
357C42C2×#6712yes—
358C22×C3C2×#12011yescharacteristic
359C22×C3#53211yescharacteristic
360C22×C3#53511yescharacteristic
361C22×C3C22×Q1611yescharacteristic
362C22×C3C2×#13511yescharacteristic
363C22×C3C2×#13711yescharacteristic
364C22×C3C2×#13711yescharacteristic
365C2×D3—22no—
366C2×D3—22no—
367C2×D3—22no—
368C2×D3—41no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—41no—
372C3×C4—44no—
373C3×C4—82no—
374C3×C4#45114yes—
375C3×C4#45214yes—
376C3×C4#54612yes—
377C3×C4#55012yes—
378Dic3—22no—
379Dic3—41no—
380Dic3—44no—
381Dic3—41no—
382Dic3—82no—
383C23—32no—
384C23—61no—
385C23—61no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23C2×D3×D411yescharacteristic
391Q8—28no—
392Q8—44no—
393Q8—44no—
394Q8—44no—
395Q8—128no—
396C2×C4—24no—
397C2×C4—32no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—68no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—64no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—124no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4D3×Q1612yes—
422C2×C4#91812yes—
423C2×C4#91812yes—
424C2×C4#91912yes—
425C2×C4#93712yes—
426C2×C4#93712yes—
427C2×C4C2×D3×D411yescharacteristic
428C2×C4C2×#23712yes—
429C2×C4#100412yes—
430C2×C4C22×#6711yescharacteristic
431C8—24no—
432C8—122no—
433C2×C3#243111yescharacteristic
434C2×C3C2×#45111yescharacteristic
435C2×C3C2×#45211yescharacteristic
436C2×C3#317011yescharacteristic
437C2×C3#318511yescharacteristic
438C2×C3#320911yescharacteristic
439C2×C3#321011yescharacteristic
440D3—42no—
441C22—62no—
442C22—62no—
443C22—62no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22—121no—
448C22#571511yescharacteristic
449C22#571611yescharacteristic
450C22C2×D3×Q1611yescharacteristic
451C22C2×#91811yescharacteristic
452C22C2×#91811yescharacteristic
453C22C2×#91911yescharacteristic
454C22C2×#93711yescharacteristic
455C4—44no—
456C4—62no—
457C4—82no—
458C4—121no—
459C4—124no—
460C4—121no—
461C4—242no—
462C4#518614yes—
463C4#518714yes—
464C4#571812yes—
465C4#580612yes—
466C3#2560011yescharacteristic, complemented, Sylow
467C2—122no—
468C2#8179211yescharacteristic
469C2#8179411yescharacteristic
470C2#8180211yescharacteristic
471C2#8180311yescharacteristic
472C2#8318711yescharacteristic
473C2C2×#518611yescharacteristic
474C2C2×#518711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#494C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#494D3
hypocenterC3#25600
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
2. C2×C3×#494C2semidirect
4. C2×#5079C2semidirect
6. #72065C2semidirect
7. #71785C2semidirect
8. #69777C2semidirect
41. C2×#494D3semidirect
160. C2×C4×D3Q16semidirect
466. C3#25600semidirect

Maximal subgroups

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

subgroupindex
C2×#57282
C2×C3×#4942
#832692
C2×#50792
#720712
#720652
#717852
#697772
C4×(C2×D3)⋊C82
#256003