Finite Group Databasebeta

← #10831764

Subgroups — #10831764

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5579C211yesmaximal, characteristic
2#82480C211yesmaximal, characteristic, complemented
3C2×#4851C211yesmaximal, characteristic, complemented
4C2×#4710C211yesmaximal, characteristic
5C3×#1985C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic
7#69828C211yesmaximal, characteristic, complemented
8#68690C212yesmaximal, complemented
9#68672C212yesmaximal
10#68671C212yesmaximal
11#68634C212yesmaximal, complemented
12#24343—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15C22×#204C2211yescharacteristic
16#5579—24no—
17#5579—24no—
18C2×#896—22no—
19C2×D3×C4⋊C4—21no—
20C2×C4×D12C2212yes—
21C2×C4×D12C2211yescharacteristic
22C2×#870C2212yes—
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27#5230—21no—
28C2×#821C2211yescharacteristic
29C2×#821C2212yes—
30C2×#818C2211yescharacteristic
31C2×#817C2212yes—
32C2×#816C2212yes—
33C2×#816C2211yescharacteristic
34#4985—21no—
35#4978—22no—
36#4936C2211yescharacteristic
37#4929C2212yes—
38#4928C2212yes—
39#4851—24no—
40#4710—24no—
41C3×#336C2212yes—
42C3×#336C2212yes—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#554—31no—
47C2×#475—31no—
48C2×#440—31no—
49#2581—31no—
50#2059—32no—
51#2026—32no—
52#1993—31no—
53#1985D311yescharacteristic, complemented
54#1877—32no—
55#1864—32no—
56#1804—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12C2311yescharacteristic
59C22×C4×D3—21no—
60C22×C4×D3—22no—
61C22×Dic6—21no—
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C42C2312yes—
71C2×C3×C42C2311yescharacteristic
72#946—28no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—22no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—21no—
84C2×C4×Dic3—21no—
85C2×#204—22no—
86C2×#204—22no—
87C2×#204Q814yescomplemented
88#896—216no—
89D3×C4⋊C4—28no—
90C4×D12—24no—
91C4×D12—22no—
92C4×D12—24no—
93C4×D12D414yes—
94#870—24no—
95#870—24no—
96#869—22no—
97#869D414yes—
98C4×Dic6—22no—
99C4×Dic6—22no—
100C3×C4⋊C8—22no—
101C3×#97—22no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107#821—22no—
108#821—24no—
109#818—22no—
110#817—24no—
111#816—22no—
112#816—24no—
113C22×SD16—31no—
114#554—64no—
115#554—64no—
116C2×C4⋊Q8—31no—
117C2×#125—32no—
118C2×C22⋊Q8—61no—
119C2×C22⋊Q8—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4—31no—
123C2×C4×D4—32no—
124C22×C4⋊C4—61no—
125#475—64no—
126#440—64no—
127C2×#107—31no—
128C2×#106—32no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#103—32no—
131C2×#103—31no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#97C2×D311yescharacteristic
135#400—61no—
136#395—61no—
137#391—32no—
138#389—32no—
139#387—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336C2×D312yes—
145#336C2×D312yes—
146C23×D3—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—42no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—21no—
159C2×D12—21no—
160C2×D12—21no—
161C2×D12—21no—
162C2×D12C2×D412yes—
163C2×D12C2×Q812yes—
164C2×D12#4112yes—
165C2×C4×D3—24no—
166C2×C4×D3—28no—
167C2×C4×D3—44no—
168C2×C4×D3—84no—
169C2×Dic6—21no—
170C2×Dic6—21no—
171C2×Dic6—21no—
172C2×Dic6—21no—
173C2×Dic6—21no—
174C2×Dic6—24no—
175C2×Dic6—82no—
176C2×Dic6C2×Q812yes—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—22no—
180C2×C3×C8—21no—
181C2×C3×C8—22no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—22no—
185C2×C3×C8C2×Q812yes—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—24no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42C2×D412yes—
191C3×C42C2×D412yes—
192(C2×D3)⋊C4—28no—
193(C2×D3)⋊C4—24no—
194C4⋊Dic3—24no—
195C4⋊Dic3—22no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4⋊Dic3C2×D414yes—
202C4⋊Dic3C2×D412yes—
203Dic3⋊C4—28no—
204Dic3⋊C4—28no—
205Dic3⋊C4—44no—
206Dic3⋊C4—42no—
207C4×Dic3—42no—
208#204—28no—
209C22×Q8—31no—
210C22×Q8—61no—
211C22×Q8D1211yescharacteristic
212C22×D4—31no—
213C23×C4—61no—
214C23×C4—62no—
215C2×SD16—34no—
216C2×SD16—62no—
217C2×SD16—62no—
218C22×C8C22×D311yescharacteristic
219C22×C8C22×D312yes—
220C22×C8C22×D311yescharacteristic
221C4⋊Q8—34no—
222C4⋊Q8—62no—
223#125—64no—
224#125—64no—
225C22⋊Q8—616no—
226C22⋊Q8—68no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—62no—
230C4×Q8—62no—
231C4×D4—34no—
232C4×D4—64no—
233C4×D4—64no—
234C4×D4—62no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—32no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—68no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4D1211yescharacteristic
246C2×C4⋊C4D1212yes—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—61no—
249C2×C42—61no—
250C2×C42C22×D312yes—
251C2×C42C22×D311yescharacteristic
252#107—62no—
253#106—64no—
254C4⋊C8—22no—
255#103—62no—
256#103—64no—
257#102—64no—
258#102—62no—
259#97—22no—
260#95—61no—
261#95—61no—
262#95—62no—
263#95—62no—
264#95—62no—
265C23×C3C22×D411yescharacteristic
266C22×D3—22no—
267C22×D3—22no—
268C22×D3—22no—
269C22×D3—42no—
270C22×D3—42no—
271C3×Q8—44no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4#12012yes—
280C2×C3×C4#12012yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C22×Q811yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C42−1+411yescharacteristic
286C2×C3×C42−1+411yescharacteristic
287C2×Dic3—22no—
288C2×Dic3—24no—
289C2×Dic3—22no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—42no—
293C2×Dic3—44no—
294C2×Dic3—41no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299D12—22no—
300D12—22no—
301D12—22no—
302D12C22⋊Q814yes—
303C4×D3—48no—
304C4×D3—88no—
305Dic6—22no—
306Dic6—22no—
307Dic6—41no—
308Dic6—41no—
309Dic6—84no—
310C3×C8—22no—
311C3×C8—41no—
312C3×C8—42no—
313C24—61no—
314C2×Q8—22no—
315C2×Q8—32no—
316C2×Q8—42no—
317C2×Q8—64no—
318C2×Q8—61no—
319C2×Q8—61no—
320C2×Q8—61no—
321C2×Q8—61no—
322C2×Q8—61no—
323C2×Q8—242no—
324C2×D4—32no—
325C2×D4—32no—
326C2×D4—32no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C22×C4—68no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—62no—
334C22×C4—64no—
335C22×C4—62no—
336C22×C4—61no—
337C22×C4—124no—
338C22×C4—244no—
339C22×C4C2×D1212yes—
340C22×C4C2×D1212yes—
341C22×C4C2×D1211yescharacteristic
342C22×C4C2×D1211yescharacteristic
343C22×C4C23×D311yescharacteristic
344SD16—68no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8D3×Q812yes—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—34no—
357C4⋊C4—32no—
358C4⋊C4—68no—
359C4⋊C4—68no—
360C4⋊C4—64no—
361C4⋊C4—61no—
362C4⋊C4—62no—
363C4⋊C4—61no—
364C4⋊C4—61no—
365C4⋊C4—62no—
366C4⋊C4—62no—
367C4⋊C4—124no—
368C4⋊C4—122no—
369C22⋊C4—68no—
370C22⋊C4—64no—
371C42—22no—
372C42—22no—
373C42—122no—
374C42D3×D412yes—
375C42D3×D412yes—
376C22×C3C2×#12011yescharacteristic
377C22×C3#54811yescharacteristic
378C22×C3D4×Q811yescharacteristic
379C22×C3#57512yes—
380C22×C3#57712yes—
381C2×D3—24no—
382C2×D3—44no—
383C2×D3—44no—
384C2×D3—42no—
385C2×D3—42no—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4C2×C22⋊Q811yescharacteristic
391C3×C4#53611yescharacteristic
392C3×C4#54811yescharacteristic
393C3×C4D4×Q811yescharacteristic
394Dic3—42no—
395Dic3—42no—
396Dic3—44no—
397Dic3—82no—
398Dic3—81no—
399C23—62no—
400C23—62no—
401C23—62no—
402C23—122no—
403C23—122no—
404C23C22×D1211yescharacteristic
405Q8—44no—
406Q8—62no—
407Q8—62no—
408Q8—121no—
409Q8—121no—
410Q8—244no—
411D4—34no—
412D4—62no—
413D4—62no—
414D4—62no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—62no—
423C2×C4—64no—
424C2×C4—62no—
425C2×C4—128no—
426C2×C4—121no—
427C2×C4—122no—
428C2×C4—124no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—248no—
436C2×C4#88212yes—
437C2×C4#88212yes—
438C2×C4C2×D3×D411yescharacteristic
439C2×C4C2×D3×D411yescharacteristic
440C2×C4C2×D3×Q811yescharacteristic
441C2×C4#100711yescharacteristic
442C2×C4D3×#4111yescharacteristic
443C2×C4#101011yescharacteristic
444C8—22no—
445C8—42no—
446C8—41no—
447C2×C3#240911yescharacteristic
448C2×C3#300511yescharacteristic
449C2×C3#300911yescharacteristic
450C2×C3#337612yes—
451C2×C3#338812yes—
452D3—44no—
453C22—64no—
454C22—122no—
455C22—122no—
456C22—124no—
457C22—124no—
458C22C2×#88211yescharacteristic
459C22Q8×D1211yescharacteristic
460C22#557611yescharacteristic
461C22#574812yes—
462C22#575112yes—
463C4—22no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—122no—
468C4—122no—
469C4—124no—
470C4—241no—
471C4—242no—
472C4D3×C22⋊Q811yescharacteristic
473C4#562811yescharacteristic
474C4#567711yescharacteristic
475C4#572711yescharacteristic
476C3#2434311yescharacteristic, complemented, Sylow
477C2—124no—
478C2#7661911yescharacteristic
479C2#7863811yescharacteristic
480C2#7867111yescharacteristic
481C2#7997412yes—
482C2#7997812yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1985D3
hypocenterC3#24343
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #82480C2semidirect
3. C2×#4851C2semidirect
5. C3×#1985C2semidirect
7. #69828C2semidirect
8. #68690C2semidirect
11. #68634C2semidirect
53. #1985D3semidirect
87. C2×#204Q8semidirect
476. C3#24343semidirect

Maximal subgroups

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

subgroupindex
C2×#55792
#824802
C2×#48512
C2×#47102
C3×#19852
#704802
#698282
#686902
#686722
#686712
#686342
#243433