Finite Group Databasebeta

← #496407

Subgroups — #496407

Order 512.

847 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#13303C214yesmaximal, complemented
5#13302C212yesmaximal, complemented
6#13301C212yesmaximal, complemented
7#12862C214yesmaximal
8C2×#533C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2920—21no—
11#2920—21no—
12#2855—21no—
13#2855—21no—
14#2627—21no—
15#2627—21no—
16#2582—21no—
17#2582—21no—
18#2581—21no—
19#2581—21no—
20#2568C411yescharacteristic
21#2568C2211yescharacteristic
22#2567C411yescharacteristic
23#2567C2211yescharacteristic
24#2565C412yes—
25#2565C2212yes—
26#2317C412yes—
27#2310C2212yes—
28#2285C411yescharacteristic
29#1821—21no—
30#1821—21no—
31#1801—21no—
32#1801—21no—
33#1801C2212yes—
34#1801C2212yes—
35#1784C2212yes—
36#1784C2212yes—
37C4×#101—21no—
38C4×#101—21no—
39C4×#101C2212yes—
40C4×#101C2212yes—
41#1394C2218yes—
42#1394C2218yes—
43C23×Q8C2311yescharacteristic
44#533—41no—
45#533—41no—
46#533—41no—
47#533—41no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125C2×C412yes—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—41no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×#117C2×C412yes—
68C2×#117C2×C412yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75#432—41no—
76#432—41no—
77#432—41no—
78#432—41no—
79#431—41no—
80#431—41no—
81#431—41no—
82#431—41no—
83#423—22no—
84#423—22no—
85C2×#101—22no—
86C2×#101—22no—
87C2×#101C2314yes—
88C2×#101C2314yes—
89#407—22no—
90#407—22no—
91C4×M16—22no—
92C4×M16—22no—
93#400—21no—
94#400—21no—
95#400—21no—
96#400—21no—
97#398—22no—
98#398—22no—
99#398—22no—
100#398—22no—
101#395—21no—
102#395—21no—
103#395—21no—
104#395—21no—
105#391—22no—
106#391—22no—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391C2×C412yes—
112#391C2×C412yes—
113#391C2312yes—
114#391C2312yes—
115#389—22no—
116#389—22no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#387—22no—
122#387—22no—
123#387—22no—
124#387—22no—
125#385—22no—
126#385—22no—
127#384—21no—
128#384C2×C412yes—
129#384C2×C412yes—
130#382—21no—
131#382C2×C412yes—
132#382C2×C412yes—
133#381—22no—
134#381—22no—
135#380—21no—
136#380—21no—
137C4×C4⋊C4C2×C412yes—
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140#343—22no—
141#343—22no—
142#343—22no—
143#343—22no—
144C22×Q8—24no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—41no—
148C22×Q8—41no—
149C22×Q8—42no—
150C22×Q8C22×C411yescharacteristic
151C22×Q8C2×D411yescharacteristic
152C22×Q8C2×D411yescharacteristic
153C22×Q8C2411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—24no—
162C2×M16—24no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167#125—42no—
168#125—42no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C22⋊Q8—41no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C22⋊Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8—42no—
194C4×Q8—42no—
195#117—44no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C412yes—
254C2×C42C22×C412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C4⋊C8—42no—
263#101—28no—
264#101—28no—
265#101—44no—
266#101—44no—
267C22⋊C8—42no—
268C22⋊C8—42no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271#97—42no—
272#97—42no—
273C4×C8—42no—
274C4×C8—42no—
275#95—22no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290C24C2×C22⋊C411yescharacteristic
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—42no—
305C2×Q8—44no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—82no—
309C2×Q82−1+414yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—81no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4#9911yescharacteristic
337C22×C4#9911yescharacteristic
338C22×C4#9911yescharacteristic
339C22×C4#9911yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354M16—44no—
355M16—44no—
356M16—44no—
357M16—44no—
358C2×C8—44no—
359C2×C8—44no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—44no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C22⋊C4—44no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—42no—
416C42—42no—
417C42—44no—
418C23—21no—
419C23—21no—
420C23—41no—
421C23—41no—
422C23C2×#9911yescharacteristic
423C23C2×#9911yescharacteristic
424C23C22×C22⋊C411yescharacteristic
425Q8—44no—
426Q8—44no—
427Q8—44no—
428Q8—44no—
429Q8—82no—
430Q8—84no—
431Q8—82no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4C2×#9911yescharacteristic
488C2×C4C2×#9911yescharacteristic
489C2×C4C2×#9911yescharacteristic
490C2×C4C2×#9911yescharacteristic
491C2×C4#41012yes—
492C2×C4#51912yes—
493C2×C4#53612yes—
494C2×C4#53612yes—
495C8—84no—
496C8—84no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—42no—
506C22—42no—
507C22#231011yescharacteristic
508C22C22×#9911yescharacteristic
509C22C2×#41011yescharacteristic
510C4—22no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—82no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4#213312yes—
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2#2182511yescharacteristic
535C2C2×#213311yescharacteristic
536C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13303C2semidirect
5. #13302C2semidirect
6. #13301C2semidirect

Maximal subgroups

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

subgroupindex
#351792
#207272
#133032
#133022
#133012
#128622