Finite Group Databasebeta

← #10769484

Subgroups — #10769484

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic
2#83154C212yesmaximal, complemented
3#82478C211yesmaximal, characteristic, complemented
4#71904C212yesmaximal, complemented
5#71889C214yesmaximal, complemented
6#71880C212yesmaximal
7#71755C212yesmaximal, complemented
8#71625C211yesmaximal, characteristic, complemented
9#18338—31nomaximal, Sylow
10C3×#544C2211yescharacteristic
11C3×#542—22no—
12C3×#541—22no—
13C3×#539—21no—
14C3×#539C2212yes—
15C3×#537C2211yescharacteristic
16C3×#536—21no—
17C3×#533—22no—
18C3×#529—22no—
19C3×#529—21no—
20C3×#529C2212yes—
21C3×C4×#41—21no—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×#869C2211yescharacteristic
24#5229—21no—
25#5069—22no—
26#5060—22no—
27#5050C2214yes—
28#5040C2214yes—
29#5031—22no—
30#5019—22no—
31#4979C2212yes—
32#4979C2212yes—
33#4975—21no—
34C4×C4⋊Dic3C2211yescharacteristic
35#4928—21no—
36#4928—21no—
37#4926C2212yes—
38#4538C2214yes—
39#4537C2214yes—
40#4528C2212yes—
41#4528C2212yes—
42#3461D311yescharacteristic
43#3218—32no—
44#2567—31no—
45#1605—32no—
46#1604—32no—
47#1603—34no—
48#1509—32no—
49#1467—31no—
50C2×C3×#41—21no—
51C22×C3×Q8D411yescharacteristic
52C22×Dic6C2312yes—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8—22no—
55C3×#126—22no—
56C3×#126—22no—
57C3×#126—22no—
58C3×#126—22no—
59C3×#125—21no—
60C3×#125—22no—
61C3×#125—22no—
62C3×#125—22no—
63C3×#125D412yes—
64C3×#124—22no—
65C3×#124—21no—
66C3×#124D412yes—
67C3×#123—22no—
68C3×#123—22no—
69C3×#123—22no—
70C3×#123—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×C22⋊Q8—21no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8C2312yes—
78C3×#121—22no—
79C3×#121—21no—
80C3×#121C2312yes—
81C3×C4×Q8—21no—
82C3×C4×Q8—21no—
83C3×C4×Q8—21no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8D412yes—
87C3×C4×D4—21no—
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4D412yes—
93C3×#117—22no—
94C3×#117—22no—
95C3×#117—21no—
96C3×#117—22no—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C42—22no—
101C2×C3×C42D411yescharacteristic
102C2×C3×C42D411yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#216—24no—
105C2×#214—24no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×Dic3⋊C4—22no—
110C2×Dic3⋊C4—22no—
111C2×C4×Dic3—21no—
112#869—22no—
113#869—22no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116C3⋊#95C2312yes—
117#830C2314yes—
118#810—24no—
119#810—24no—
120C4⋊(C3⋊C8)—22no—
121#803—22no—
122#544C2×D311yescharacteristic
123#542—22no—
124#541—22no—
125#539—21no—
126#539C2×D312yes—
127#537C2×D311yescharacteristic
128#536—21no—
129#533—22no—
130C2×C4⋊Q8—31no—
131#529—22no—
132#529—21no—
133#529C2×D312yes—
134C4×#41—21no—
135C2×C4×Q8C2×D311yescharacteristic
136#497—62no—
137#491—62no—
138#462—62no—
139#461—62no—
140#451—34no—
141#448—34no—
142#432—32no—
143#395—32no—
144#395—32no—
145#391—61no—
146#389—61no—
147#389—61no—
148#384—61no—
149C4×C4⋊C4—31no—
150#328—34no—
151#327—34no—
152#324—32no—
153#324—32no—
154C3×#41—42no—
155C3×#41—42no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—41no—
161C2×C3×Q8C2×D412yes—
162C2×C3×D4—21no—
163C2×C3×D4—22no—
164C2×C3×D4—21no—
165C2×C3×D4C2×D412yes—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—22no—
177C22×Dic3—22no—
178C2×Dic6—24no—
179C2×Dic6—82no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4C2×D412yes—
196C3×C4⋊C4C2×D412yes—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—22no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—22no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—21no—
209C3×C42—22no—
210C3×C42—21no—
211C3×C42—21no—
212C3×C42—21no—
213C3×C42C2×D412yes—
214C3×C42C2×D412yes—
215#216—84no—
216#214—84no—
217C4⋊Dic3—24no—
218C4⋊Dic3—41no—
219C4⋊Dic3—42no—
220C4⋊Dic3—41no—
221Dic3⋊C4—82no—
222Dic3⋊C4—82no—
223C4×Dic3—42no—
224C2×C3⋊C8—24no—
225C2×#41—21no—
226C22×Q8—32no—
227C22×Q8#6711yescharacteristic
228C2×Q16—64no—
229C2×SD16—64no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—62no—
233C4⋊Q8—62no—
234#126—22no—
235#126—22no—
236#126—22no—
237#126—22no—
238#125—21no—
239#125—22no—
240#125—22no—
241#125—22no—
242#125#6712yes—
243#124—22no—
244#124—21no—
245#124#6712yes—
246#123—22no—
247#123—22no—
248#123—22no—
249#123—22no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—21no—
255C22⋊Q8—22no—
256C22⋊Q8C22×D312yes—
257#121—22no—
258#121—21no—
259#121C22×D312yes—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8#6712yes—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4#6712yes—
272#117—22no—
273#117—22no—
274#117—21no—
275#117—22no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4#6711yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C42—22no—
285C2×C42—61no—
286C2×C42#6711yescharacteristic
287C2×C42#6711yescharacteristic
288C2×C42C22×D311yescharacteristic
289C4⋊C8—62no—
290#103—64no—
291#103—64no—
292C22⋊C8—34no—
293#97—62no—
294#95—32no—
295#95—62no—
296#95—62no—
297C23×C3—22no—
298C23×C3C22×D411yescharacteristic
299C3×Q8—44no—
300C3×Q8—42no—
301C3×Q8—42no—
302C3×Q8—81no—
303C3×D4—42no—
304C3×D4—42no—
305C3×D4—42no—
306C3×D4—42no—
307C3×D4—81no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—41no—
322C2×C3×C4—44no—
323C2×C3×C4—44no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—82no—
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4#12012yes—
329C2×C3×C4#12011yescharacteristic
330C2×C3×C4#13712yes—
331C2×C3×C4C22×D412yes—
332C2×Dic3—24no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338Dic6—44no—
339Dic6—44no—
340C3⋊C8—44no—
341#41—42no—
342#41—42no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—64no—
349C2×Q8—242no—
350C2×Q8C2×#6712yes—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4C2×#6712yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4C2×#6711yescharacteristic
361C22×C4C2×#6711yescharacteristic
362C22×C4C2×#6711yescharacteristic
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365C22×C4C2×#6711yescharacteristic
366C22×C4C23×D311yescharacteristic
367Q16—244no—
368SD16—244no—
369C2×C8—64no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—64no—
386C4⋊C4—121no—
387C4⋊C4—122no—
388C4⋊C4—121no—
389C4⋊C4—242no—
390C4⋊C4—242no—
391C4⋊C4C2×#6712yes—
392C4⋊C4C2×#6712yes—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C42—22no—
403C42—22no—
404C42—21no—
405C42—22no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—122no—
410C42C2×#6712yes—
411C42C2×#6712yes—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3C2×#12011yescharacteristic
418C22×C3C2×#13711yescharacteristic
419C22×C3#57511yescharacteristic
420C3×C4—22no—
421C3×C4—22no—
422C3×C4—21no—
423C3×C4—22no—
424C3×C4—41no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—42no—
430C3×C4#45612yes—
431Dic3—44no—
432Dic3—82no—
433C23—22no—
434C23C22×#6711yescharacteristic
435Q8—44no—
436Q8—42no—
437Q8—42no—
438Q8—81no—
439Q8—124no—
440Q8—124no—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—81no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—64no—
465C2×C4—82no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4#95112yes—
472C2×C4#95311yescharacteristic
473C2×C4#95312yes—
474C2×C4#95311yescharacteristic
475C2×C4C22×#6712yes—
476C8—124no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—82no—
480C2×C3#300711yescharacteristic
481C2×C3C2×#45611yescharacteristic
482C2×C3#302711yescharacteristic
483C22—21no—
484C22—21no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22#582211yescharacteristic
489C22C2×#95111yescharacteristic
490C22C2×#95311yescharacteristic
491C4—22no—
492C4—22no—
493C4—21no—
494C4—22no—
495C4—41no—
496C4—42no—
497C4—42no—
498C4—41no—
499C4—41no—
500C4—42no—
501C4—124no—
502C4—242no—
503C4#524612yes—
504C3#1833811yescharacteristic, complemented, Sylow
505C2—21no—
506C2—21no—
507C2—82no—
508C2#8408411yescharacteristic
509C2C2×#524611yescharacteristic
510C2#8412611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3461C2
socleC22×C3C2×#120
prime radicalC3×#537C22
hypercenter#3461D3
hypocenterC3#18338
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
2. #83154C2semidirect
3. #82478C2semidirect
4. #71904C2semidirect
5. #71889C2semidirect
7. #71755C2semidirect
8. #71625C2semidirect
504. C3#18338semidirect

Maximal subgroups

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

subgroupindex
C3×#34612
#831542
#824782
#719042
#718892
#718802
#717552
#716252
#183383