Finite Group Databasebeta

← #497596

Subgroups — #497596

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39611C211yesmaximal, characteristic
2#22513C212yesmaximal, complemented
3#13300C212yesmaximal, complemented
4#13300C212yesmaximal, complemented
5#13298C212yesmaximal, complemented
6#12799C211yesmaximal, characteristic
7#12799C211yesmaximal, characteristic
8#12797C211yesmaximal, characteristic
9#12628C211yesmaximal, characteristic
10#12628C211yesmaximal, characteristic
11C4×#353C211yesmaximal, characteristic
12C2×#554C411yescharacteristic, complemented
13C2×#554C2211yescharacteristic
14#2859C411yescharacteristic, complemented
15#2732—22no—
16#2683—22no—
17#2660C411yescharacteristic, complemented
18#2660C411yescharacteristic, complemented
19#2566C411yescharacteristic, complemented
20#2566C411yescharacteristic, complemented
21#2566C2211yescharacteristic
22#2566C2211yescharacteristic
23#2542C411yescharacteristic, complemented
24#2542C2211yescharacteristic
25#2449—22no—
26#2449—22no—
27#2345—22no—
28#2328C2211yescharacteristic
29#2328C2211yescharacteristic
30C4×#127C2211yescharacteristic
31#2295C411yescharacteristic, complemented
32C2×#353C2211yescharacteristic
33C2×#353C2211yescharacteristic
34C2×#353C2211yescharacteristic
35C2×#353C2211yescharacteristic
36#1996—22no—
37#1996—22no—
38#1996—22no—
39#1799C2212yes—
40#1799C2212yes—
41#1799C2212yes—
42C4×#99C2212yes—
43C4×#99C2212yes—
44C4×#99C2212yes—
45#1393C2212yes—
46#1393C2212yes—
47#1393C2212yes—
48#1393C2212yes—
49#1393C2212yes—
50#1393C2212yes—
51#554—42no—
52#554—42no—
53#554—42no—
54#554—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127C2311yescharacteristic
58C2×#125C2×C411yescharacteristic
59C2×#125C2×C411yescharacteristic
60C2×#125C2×C411yescharacteristic
61C2×#125C2×C411yescharacteristic
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8D412yes—
68C2×C22⋊Q8D412yes—
69C2×C22⋊Q8D412yes—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×D4—22no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4D412yes—
80C2×#99C2312yes—
81C2×#99C2312yes—
82C2×#99C2312yes—
83C2×#99C2312yes—
84#399—22no—
85#399—22no—
86#398—22no—
87#398—22no—
88#396—22no—
89#396—22no—
90#396—22no—
91#396—22no—
92#395—22no—
93#395—22no—
94#391—21no—
95#390—22no—
96#390C2×C411yescharacteristic
97#390C2×C411yescharacteristic
98#390C2×C411yescharacteristic
99#390C2311yescharacteristic
100#390C2311yescharacteristic
101#390C2311yescharacteristic
102#389—22no—
103#389—22no—
104#389—21no—
105#389—42no—
106#387—21no—
107#387—21no—
108#387—42no—
109#387—42no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#384C2×C411yescharacteristic
122#384C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#380—22no—
128#380—22no—
129#380—22no—
130C4×C4⋊C4—42no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137#376C2×C411yescharacteristic
138#376C2×C411yescharacteristic
139C43C2×C411yescharacteristic
140#353—22no—
141#353—22no—
142#353—22no—
143#353Q814yescomplemented
144#342—22no—
145#342—22no—
146#342—22no—
147#342—22no—
148#342—22no—
149#342—22no—
150C22×Q8—21no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22⋊C412yes—
153C22×D4—22no—
154C22×D4C22×C411yescharacteristic
155C22×D4C2411yescharacteristic
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164#127—22no—
165#127C2×Q812yes—
166#127#4112yes—
167#125—41no—
168#125—41no—
169#125—42no—
170#125—41no—
171#125—41no—
172#125—42no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—44no—
180C22⋊Q8—44no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—84no—
184C22⋊Q8—84no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—84no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C412yes—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C22⋊C4C2×D412yes—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260#99—22no—
261#99—22no—
262#99—22no—
263#99—22no—
264#99—22no—
265#99—22no—
266#99—22no—
267#99C2×Q814yes—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—42no—
280#95—42no—
281#95—42no—
282#95—42no—
283#95—42no—
284#95—42no—
285C24—21no—
286C24C22×D412yes—
287C2×Q8—22no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—81no—
291C2×Q8—82no—
292C2×Q8—82no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4C4×Q812yes—
303C2×D4C22×Q811yescharacteristic
304C2×D4C2×#4111yescharacteristic
305C2×D42−1+412yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C23×C411yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—82no—
370C4⋊C4—81no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—84no—
375C4⋊C4—162no—
376C4⋊C4—162no—
377C4⋊C4#9912yes—
378C4⋊C4#9912yes—
379C22⋊C4—24no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—22no—
403C42—22no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42—81no—
409C42—84no—
410C42—162no—
411C42—162no—
412C42C4×Q812yes—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23C22×C22⋊C411yescharacteristic
423C23#54812yes—
424C23D4×Q812yes—
425Q8—82no—
426Q8—162no—
427Q8—162no—
428D4—41no—
429D4—41no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—84no—
434D4—84no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—44no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—84no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—161no—
490C2×C4—161no—
491C2×C4—162no—
492C2×C4C2×#9912yes—
493C2×C4C2×#9912yes—
494C2×C4C2×#9911yescharacteristic
495C2×C4C2×#9911yescharacteristic
496C2×C4C2×C4×Q811yescharacteristic
497C2×C4#51911yescharacteristic
498C22—21no—
499C22—21no—
500C22—22no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22—44no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—82no—
509C22—82no—
510C22—81no—
511C22—81no—
512C22Q8×C22⋊C411yescharacteristic
513C22C22×#9911yescharacteristic
514C22#288411yescharacteristic
515C4—22no—
516C4—21no—
517C4—21no—
518C4—22no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—42no—
528C4—41no—
529C4—81no—
530C4—81no—
531C4—84no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—162no—
536C4—162no—
537C4—162no—
538C2—21no—
539C2—21no—
540C2—42no—
541C2—42no—
542C2—82no—
543C2Q8×#9911yescharacteristic
544C2#3335811yescharacteristic
545C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#554 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22513C2semidirect
3. #13300C2semidirect
4. #13300C2semidirect
5. #13298C2semidirect
12. C2×#554C4semidirect
14. #2859C4semidirect
17. #2660C4semidirect
18. #2660C4semidirect
19. #2566C4semidirect
20. #2566C4semidirect
23. #2542C4semidirect
31. #2295C4semidirect
143. #353Q8semidirect

Maximal subgroups

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

subgroupindex
#396112
#225132
#133002
#132982
#127992
#127972
#126282
C4×#3532