Finite Group Databasebeta

← #10849641

Subgroups — #10849641

Order 768.

988 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3510C211yesmaximal, characteristic
2#83342C211yesmaximal, characteristic, complemented
3#82994C211yesmaximal, characteristic, complemented
4#82509C211yesmaximal, characteristic, complemented
5#71863C212yesmaximal, complemented
6#71852C212yesmaximal
7#71839C211yesmaximal, characteristic, complemented
8#71727C211yesmaximal, characteristic, complemented
9#71658C212yesmaximal, complemented
10#71647C212yesmaximal, complemented
11C4×#837C211yesmaximal, characteristic, complemented
12#25689—31nomaximal, Sylow
13C3×#559—22no—
14C3×#556—22no—
15C3×#551—24no—
16C3×#542—22no—
17C3×#536C2211yescharacteristic
18C3×#535C2211yescharacteristic
19C3×#533C2211yescharacteristic
20C3×#529—22no—
21C2×C3×#124C2211yescharacteristic
22C3×#519C2211yescharacteristic
23C3×#518C2211yescharacteristic
24C3×C4×#41C2211yescharacteristic
25C2×#870C2211yescharacteristic
26C2×#837C2212yes—
27#5232C2212yes—
28#5229C2212yes—
29C2×#836C2212yes—
30#5221C2212yes—
31C2×#833C2212yes—
32#5081—22no—
33#5065—22no—
34#5022—22no—
35#5009—22no—
36#4995C2212yes—
37#4995C2212yes—
38#4995C2212yes—
39#4945—24no—
40C42×Dic3C2211yescharacteristic
41#4924C2211yescharacteristic
42C4×#208C2211yescharacteristic
43#4532C2212yes—
44#4532C2212yes—
45#4532C2212yes—
46#3510D311yescharacteristic
47#3166—31no—
48#3080—31no—
49#2861—31no—
50#2038—32no—
51#2037—32no—
52#1994—31no—
53#1967—31no—
54#1857—32no—
55#1856—32no—
56C4×#104—31no—
57C2×C3×#41C2311yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8D412yes—
62C3×#127—21no—
63C3×#126—24no—
64C3×#126—24no—
65C3×#125D412yes—
66C3×#124—24no—
67C3×#124—21no—
68C3×#124—21no—
69C3×#124—22no—
70C3×#124—21no—
71C3×#124—21no—
72C3×#123—24no—
73C3×#123—24no—
74C3×#123—22no—
75C3×C22⋊Q8—24no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#120—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8D412yes—
84C3×C4×D4—24no—
85C3×C4×D4—22no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×C4×D4D412yes—
89C3×#117—24no—
90C3×#117—22no—
91C3×#117—21no—
92C3×#117C2311yescharacteristic
93C3×#117C2311yescharacteristic
94C3×#117C2311yescharacteristic
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4D414yes—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#217—24no—
100C2×C4⋊Dic3C2312yes—
101C2×Dic3⋊C4—22no—
102C2×C4×Dic3—22no—
103C2×C4×Dic3C2312yes—
104C2×#208C2312yes—
105C2×#208C2312yes—
106#870—22no—
107#870—21no—
108#870—21no—
109#837—24no—
110#836—24no—
111#835—22no—
112#833—24no—
113#832—22no—
114C3⋊#95—24no—
115#808—22no—
116#808—22no—
117#808—22no—
118#807—22no—
119#807—22no—
120#807—22no—
121C4⋊(C3⋊C8)—22no—
122#803—21no—
123C4×C3⋊C8—21no—
124#559—22no—
125#556—22no—
126#551—24no—
127#542—22no—
128#536C2×D311yescharacteristic
129#535C2×D311yescharacteristic
130#533C2×D311yescharacteristic
131#529—22no—
132C2×#125—31no—
133C2×#124C2×D311yescharacteristic
134#519C2×D311yescharacteristic
135#518C2×D311yescharacteristic
136C4×#41C2×D311yescharacteristic
137#501—62no—
138#499—62no—
139#479—62no—
140#478—62no—
141#428—32no—
142#428—32no—
143#428—32no—
144#423—31no—
145C2×#104—32no—
146#417—32no—
147C2×#101—32no—
148C2×#100—32no—
149C4×M16—31no—
150#391—32no—
151#390—32no—
152#388—64no—
153C43—31no—
154#339—32no—
155#339—32no—
156#339—32no—
157C24×C3C2×D412yes—
158C3×#41—22no—
159C3×#41—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8C2×D411yescharacteristic
166C2×C3×D4—21no—
167C2×C3×D4—24no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4C2×D411yescharacteristic
173C22×C3×C4—24no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2411yescharacteristic
179C22×Dic3—22no—
180C22×Dic3—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4C2×D412yes—
188C3×C4⋊C4C2×D412yes—
189C3×C4⋊C4C2×D412yes—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—21no—
194C3×C22⋊C4—21no—
195C3×C22⋊C4—21no—
196C3×C22⋊C4—24no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—22no—
199C3×C42—24no—
200C3×C42—22no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—21no—
204C3×C42C2×D411yescharacteristic
205C3×C42C2×D411yescharacteristic
206C3×C42C2×D412yes—
207#217—48no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210Dic3⋊C4—42no—
211Dic3⋊C4—42no—
212C4×Dic3—22no—
213C4×Dic3—22no—
214C4×Dic3—44no—
215#208—22no—
216#208—22no—
217#208—22no—
218#208#4114yes—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×#41C22×D311yescharacteristic
222C22×Q8C22×D311yescharacteristic
223C22×D4C22×D311yescharacteristic
224C2×M16—32no—
225C2×M16—32no—
226C4⋊Q8—21no—
227C4⋊Q8#6712yes—
228#127—21no—
229#126—24no—
230#126—24no—
231#125—62no—
232#125—61no—
233#125—61no—
234#125#6712yes—
235#124—24no—
236#124—21no—
237#124—21no—
238#124—21no—
239#124—21no—
240#124—22no—
241#123—24no—
242#123—24no—
243#123—22no—
244C22⋊Q8—24no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247#121—22no—
248#120—22no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8#6712yes—
253C4×D4—24no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4#6712yes—
258#117—24no—
259#117—22no—
260#117—21no—
261#117C22×D311yescharacteristic
262#117C22×D311yescharacteristic
263#117C22×D311yescharacteristic
264C2×C4⋊C4—32no—
265C2×C4⋊C4—62no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4#6714yes—
269C2×C42—21no—
270C2×C42—32no—
271C2×C42—62no—
272C2×C42C22×D311yescharacteristic
273#107—62no—
274#107—62no—
275#107—62no—
276#106—62no—
277#106—62no—
278#106—62no—
279C4⋊C8—62no—
280#104—64no—
281#103—62no—
282#102—62no—
283#101—64no—
284#100—64no—
285#97—61no—
286C4×C8—61no—
287#95—64no—
288C23×C3—22no—
289C23×C3—21no—
290C23×C3—24no—
291C23×C3—44no—
292C23×C3C22×D411yescharacteristic
293C3×Q8—21no—
294C3×Q8—21no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—42no—
298C3×Q8—82no—
299C3×D4—21no—
300C3×D4—21no—
301C3×D4—42no—
302C3×D4—41no—
303C3×D4—41no—
304C3×D4—42no—
305C3×D4—84no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—24no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—82no—
315C2×C3×C4—84no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12012yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4112yes—
321C2×C3×C42−1+412yes—
322C2×Dic3—24no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—44no—
326C2×Dic3—42no—
327C3⋊C8—24no—
328C3⋊C8—42no—
329C24C2×#6712yes—
330#41—22no—
331#41—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8C2×#6711yescharacteristic
338C2×D4—21no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4C2×#6711yescharacteristic
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4C2×#6712yes—
351C22×C4C2×#6712yes—
352C22×C4C23×D311yescharacteristic
353M16—34no—
354M16—62no—
355M16—62no—
356M16—62no—
357C2×C8—62no—
358C2×C8—62no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—62no—
366C4⋊C4—62no—
367C4⋊C4—122no—
368C4⋊C4—122no—
369C4⋊C4C2×#6712yes—
370C4⋊C4C2×#6712yes—
371C4⋊C4C2×#6712yes—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—21no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—22no—
381C22⋊C4—128no—
382C42—24no—
383C42—22no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—62no—
388C42—62no—
389C42—124no—
390C42C2×#6711yescharacteristic
391C42C2×#6711yescharacteristic
392C42C2×#6712yes—
393C22×C3—22no—
394C22×C3—41no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—44no—
398C22×C3—84no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#54712yes—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—42no—
404C3×C4—41no—
405C3×C4—42no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4#53211yescharacteristic
409C3×C4#53611yescharacteristic
410C3×C4#54812yes—
411Dic3—44no—
412Dic3—82no—
413C23—22no—
414C23—21no—
415C23—24no—
416C23—44no—
417C23C22×#6711yescharacteristic
418Q8—21no—
419Q8—21no—
420Q8—41no—
421Q8—42no—
422Q8—41no—
423Q8—82no—
424D4—21no—
425D4—21no—
426D4—42no—
427D4—41no—
428D4—42no—
429D4—41no—
430D4—84no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—64no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4—124no—
445C2×C4—122no—
446C2×C4#95312yes—
447C2×C4#95312yes—
448C2×C4D3×#4112yes—
449C2×C4#101012yes—
450C2×C4C22×#6711yescharacteristic
451C2×C4C22×#6711yescharacteristic
452C8—64no—
453C8—122no—
454C2×C3—22no—
455C2×C3—81no—
456C2×C3—82no—
457C2×C3#243211yescharacteristic
458C2×C3#301812yes—
459C22—22no—
460C22—41no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22—84no—
465C22#582412yes—
466C22C2×#95311yescharacteristic
467C4—22no—
468C4—22no—
469C4—42no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—124no—
475C4—242no—
476C4#582511yescharacteristic
477C4#583112yes—
478C4#583211yescharacteristic
479C3#2568911yescharacteristic, complemented, Sylow
480C2—22no—
481C2—82no—
482C2—81no—
483C2#8409311yescharacteristic
484C2#8412112yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3510C2
socleC22×C3C2×#120
prime radicalC3×#535C22
hypercenter#3510D3
hypocenterC3#25689
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3510

Complements

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

NH ≅ G/Ntype
2. #83342C2semidirect
3. #82994C2semidirect
4. #82509C2semidirect
5. #71863C2semidirect
7. #71839C2semidirect
8. #71727C2semidirect
9. #71658C2semidirect
10. #71647C2semidirect
11. C4×#837C2semidirect
479. C3#25689semidirect

Maximal subgroups

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

subgroupindex
C3×#35102
#833422
#829942
#825092
#718632
#718522
#718392
#717272
#716582
#716472
C4×#8372
#256893