Finite Group Databasebeta

← #10877286

Subgroups — #10877286

Order 768.

798 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
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×#4936C211yesmaximal, characteristic
3#71692C211yesmaximal, characteristic
4#71692C211yesmaximal, characteristic
5#71627C212yesmaximal, complemented
6#71530C211yesmaximal, characteristic
7#69686C212yesmaximal, complemented
8#69613C212yesmaximal, complemented
9#69505C211yesmaximal, characteristic
10#69504C211yesmaximal, characteristic
11#69468C212yesmaximal, complemented
12#27686—31nomaximal, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C3×#391C2211yescharacteristic
19C3×#389C2211yescharacteristic
20C3×#387—21no—
21C3×#385—21no—
22C3×#380C2211yescharacteristic
23C3×C4×C4⋊C4—21no—
24C3×C4×C4⋊C4C2211yescharacteristic
25C3×C4×C22⋊C4—21no—
26C2×#835C2212yes—
27C2×#835C2212yes—
28C2×C3⋊#95—21no—
29C2×C3⋊#95—21no—
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#808C2212yes—
33C2×#807C2212yes—
34#4936—21no—
35#4936—21no—
36#4936C2212yes—
37#4936C2212yes—
38C2×C4⋊(C3⋊C8)C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44C4×C22⋊Q8D311yescharacteristic
45C2×#382—31no—
46#2101—32no—
47#2097—32no—
48#1949—32no—
49#1872—31no—
50#1872—31no—
51#1853—31no—
52#1852—31no—
53#1805—32no—
54#1797—31no—
55C22×C3×Q8C2311yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×Dic3—21no—
60C23×Dic3—21no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8Q814yes—
63C3×C4×Q8—24no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42—24no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42D411yescharacteristic
76C2×C3×C42D411yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—42no—
82C2×C4⋊Dic3—42no—
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2312yes—
85C22×C3⋊C8C2312yes—
86C22×C3⋊C8C2312yes—
87C3×#95—21no—
88C3×#95—21no—
89C3×#95—21no—
90C3×#95—21no—
91C3×#95—21no—
92#835—42no—
93#835—42no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98#830—22no—
99#830—22no—
100#808—42no—
101#807—42no—
102C4⋊(C3⋊C8)—24no—
103C2×C22⋊Q8C2×D311yescharacteristic
104C2×C4×Q8C2×D311yescharacteristic
105C22×C4⋊C4—31no—
106C22×C4⋊C4—31no—
107C22×C42C2×D311yescharacteristic
108C2×#107—32no—
109C2×#106—32no—
110C2×C4⋊C8—32no—
111C2×#103—32no—
112C2×#103—32no—
113C2×C22⋊C8—31no—
114C2×C22⋊C8—31no—
115#391C2×D311yescharacteristic
116#389C2×D311yescharacteristic
117#387—21no—
118#385—21no—
119#382—32no—
120#382—32no—
121#382—61no—
122#382—61no—
123#380C2×D311yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2×D311yescharacteristic
126C4×C22⋊C4—21no—
127C2×#95—61no—
128C2×#95—61no—
129#340—32no—
130#340—32no—
131#340—32no—
132#340—32no—
133#336—32no—
134C24×C3C2×D411yescharacteristic
135C2×C3×Q8—21no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8C2×Q812yes—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—41no—
150C22×C3×C4—41no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×Q812yes—
157C22×C3×C4#4112yes—
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164C22×Dic3—42no—
165C22×Dic3—42no—
166C22×Dic3—42no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4C2×Q812yes—
174C3×C22⋊C4—24no—
175C3×C22⋊C4—42no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—22no—
179C3×C42—42no—
180C3×C42—42no—
181C3×C42#4114yes—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3—42no—
187C4⋊Dic3—42no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194C22×Q8C22×D311yescharacteristic
195C23×C4—61no—
196C23×C4—61no—
197C23×C4#6711yescharacteristic
198C23×C4#6711yescharacteristic
199C23×C4C22×D311yescharacteristic
200C22×C8—32no—
201C22×C8—32no—
202C22⋊Q8—22no—
203C22⋊Q8Dic614yes—
204C4×Q8—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—32no—
209C2×C4⋊C4—32no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—122no—
213C2×C4⋊C4—122no—
214C2×C4⋊C4C22×D311yescharacteristic
215C2×C4⋊C4C22×D311yescharacteristic
216C2×C4⋊C4C22×D311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—24no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42#6711yescharacteristic
223C2×C42#6711yescharacteristic
224C2×C42C22×D311yescharacteristic
225C2×C42C22×D311yescharacteristic
226#107—122no—
227#106—122no—
228C4⋊C8—64no—
229#103—122no—
230#103—122no—
231C22⋊C8—62no—
232C22⋊C8—62no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—62no—
239#95—62no—
240#95—62no—
241#95—62no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3C22⋊Q812yes—
247C23×C3#13512yes—
248C23×C3#13712yes—
249C23×C3C22×D411yescharacteristic
250C3×Q8—24no—
251C3×Q8—42no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—44no—
269C2×C3×C4—44no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4C22⋊Q812yes—
281C2×C3×C4#12312yes—
282C2×C3×C4#12312yes—
283C2×C3×C4C22×Q811yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4112yes—
286C2×C3×C4H(5,𝔽2)12yes—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—84no—
294C2×Dic3—84no—
295C3⋊C8—42no—
296C3⋊C8—42no—
297C24C2×#6711yescharacteristic
298C2×Q8—21no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8C2×Dic612yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—122no—
317C22×C4—122no—
318C22×C4—122no—
319C22×C4—122no—
320C22×C4—122no—
321C22×C4—122no—
322C22×C4C2×Dic612yes—
323C22×C4#23512yes—
324C22×C4C2×#6711yescharacteristic
325C22×C4C2×#6711yescharacteristic
326C22×C4C2×#6711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328C22×C4C2×#6711yescharacteristic
329C22×C4C23×D311yescharacteristic
330C2×C8—62no—
331C2×C8—62no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—62no—
335C2×C8—62no—
336C4⋊C4—24no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—62no—
343C4⋊C4—62no—
344C4⋊C4—62no—
345C4⋊C4—62no—
346C4⋊C4—122no—
347C4⋊C4—122no—
348C4⋊C4C2×Dic612yes—
349C22⋊C4—24no—
350C22⋊C4—42no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—42no—
356C42#23714yes—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3#47914yes—
364C22×C3C2×C22⋊Q811yescharacteristic
365C22×C3C2×#12311yescharacteristic
366C22×C3#53511yescharacteristic
367C22×C3C2×#13511yescharacteristic
368C22×C3C2×#13711yescharacteristic
369C22×C3#57511yescharacteristic
370C22×C3#57511yescharacteristic
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—42no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—42no—
378C3×C4—81no—
379C3×C4—81no—
380C3×C4#55312yes—
381C3×C4#55412yes—
382Dic3—82no—
383Dic3—82no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23#92412yes—
389C23#94212yes—
390C23#95012yes—
391C23C22×#6711yescharacteristic
392Q8—24no—
393Q8—42no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—244no—
429C2×C4—244no—
430C2×C4#92412yes—
431C2×C4#93512yes—
432C2×C4#93512yes—
433C2×C4C22×Dic611yescharacteristic
434C2×C4C2×#23511yescharacteristic
435C2×C4C2×#23712yes—
436C2×C4#100912yes—
437C8—122no—
438C8—122no—
439C2×C3—24no—
440C2×C3#248711yescharacteristic
441C2×C3C2×#47911yescharacteristic
442C2×C3#308011yescharacteristic
443C2×C3#309111yescharacteristic
444C2×C3#309511yescharacteristic
445C2×C3#321111yescharacteristic
446C2×C3#321511yescharacteristic
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22#502214yes—
454C22C2×#92411yescharacteristic
455C22C2×#93511yescharacteristic
456C22C2×#94211yescharacteristic
457C22C2×#95011yescharacteristic
458C22#582211yescharacteristic
459C22#582211yescharacteristic
460C22#583011yescharacteristic
461C4—22no—
462C4—22no—
463C4—22no—
464C4—41no—
465C4—42no—
466C4—42no—
467C4—41no—
468C4—81no—
469C4—81no—
470C4—242no—
471C4—242no—
472C4#554312yes—
473C4#556512yes—
474C3#2768611yescharacteristic, complemented, Sylow
475C2—24no—
476C2#8237411yescharacteristic
477C2C2×#502211yescharacteristic
478C2#8250911yescharacteristic
479C2#8272711yescharacteristic
480C2#8291311yescharacteristic
481C2#8314411yescharacteristic
482C2#8335711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C22⋊Q8C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC4×C22⋊Q8D3
hypocenterC3#27686
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
5. #71627C2semidirect
7. #69686C2semidirect
8. #69613C2semidirect
11. #69468C2semidirect
474. C3#27686semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
C2×#49362
#716922
#716272
#715302
#696862
#696132
#695052
#695042
#694682
#276863