Finite Group Databasebeta

← #216771

Subgroups — #216771

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic
3#13964C211yesmaximal, characteristic
4#9885C211yesmaximal, characteristic
5#9884C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic
7#9679C211yesmaximal, characteristic
8#9153C212yesmaximal
9#9145C212yesmaximal
10#8576C212yesmaximal
11#8574C212yesmaximal
12#2857C2211yescharacteristic
13#2857C2211yescharacteristic
14#2855C2212yes—
15#2646—21no—
16#2627—21no—
17#2590—22no—
18#2588—22no—
19#2511C412yescomplemented
20#2507C412yescomplemented
21#2451C412yescomplemented
22#2446C412yescomplemented
23#2329—21no—
24#2313—21no—
25C4×C22⋊Q8C2212yes—
26C2×#383C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1916—22no—
30#1916—22no—
31#1817C2212yes—
32#1797C2212yes—
33C4×C22⋊C8C2211yescharacteristic
34C4×C22⋊C8C2211yescharacteristic
35C2×#328C2212yes—
36C2×#328C2212yes—
37C2×#328C2212yes—
38C2×#328C2212yes—
39#1441—21no—
40#1441—21no—
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45#1434C2211yescharacteristic
46#1434C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8C2×C412yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—22no—
65C2×C22⋊C8C2312yes—
66C2×C22⋊C8C2312yes—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69#400—22no—
70#400—22no—
71#400—22no—
72#398—22no—
73#398—22no—
74#398—22no—
75#398—22no—
76#395—22no—
77#391—21no—
78#391—22no—
79#391C2×C412yes—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2×C412yes—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—22no—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#385—22no—
101#385—22no—
102#384C2×C411yescharacteristic
103#384C2×C411yescharacteristic
104#383—42no—
105#381—22no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380C2312yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4D412yes—
116C4×C22⋊C4—22no—
117#376—42no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95D411yescharacteristic
128C2×#95D411yescharacteristic
129C2×#95C2311yescharacteristic
130C2×#95C2311yescharacteristic
131C2×#95C2311yescharacteristic
132C2×#95C2311yescharacteristic
133#336—22no—
134#336—24no—
135#336—21no—
136#336—21no—
137#328—42no—
138#328—42no—
139#328—42no—
140#328—42no—
141C22×Q8—21no—
142C22×Q8C22×C412yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C4⋊Q8—81no—
157C4⋊Q8—81no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C4×Q8—82no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4—81no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—81no—
205C2×C42—81no—
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C4⋊C8—82no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C4×C8—81no—
220C4×C8—81no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—42no—
245#95—42no—
246#95—81no—
247#95—81no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252C24C22×D411yescharacteristic
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—81no—
259C2×Q8—81no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4#13512yes—
325C22×C4#13512yes—
326C22×C4#13712yes—
327C22×C4#13712yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4H(5,𝔽2)12yes—
331C22×C42−1+412yes—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—84no—
361C4⋊C4—84no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C42—24no—
367C42—24no—
368C42—41no—
369C42—41no—
370C42—42no—
371C42—42no—
372C42—41no—
373C42—41no—
374C42—48no—
375C42—82no—
376C42—82no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23C22×C22⋊C411yescharacteristic
382C23#53511yescharacteristic
383C23#53611yescharacteristic
384C23C2×#13511yescharacteristic
385C23C2×#13511yescharacteristic
386C23C2×#13711yescharacteristic
387C23C2×#13711yescharacteristic
388Q8—82no—
389Q8—82no—
390Q8—162no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4#41612yes—
472C2×C4#41612yes—
473C2×C4#41912yes—
474C2×C4#41912yes—
475C2×C4#52012yes—
476C2×C4#53712yes—
477C8—82no—
478C8—82no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22#231111yescharacteristic
492C22#288311yescharacteristic
493C22#288411yescharacteristic
494C22C2×#41611yescharacteristic
495C22C2×#41911yescharacteristic
496C22#290111yescharacteristic
497C22#290111yescharacteristic
498C22#321011yescharacteristic
499C22#321011yescharacteristic
500C22#323111yescharacteristic
501C22#323111yescharacteristic
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4—161no—
519C4—161no—
520C2—21no—
521C2—21no—
522C2—21no—
523C2—21no—
524C2#1761511yescharacteristic
525C2#1761611yescharacteristic
526C2#1762111yescharacteristic
527C2#1763211yescharacteristic
528C2#2186411yescharacteristic
529C2#2187711yescharacteristic
530C2#2190111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #2511C4semidirect
20. #2507C4semidirect
21. #2451C4semidirect
22. #2446C4semidirect

Maximal subgroups

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

subgroupindex
#350152
#219122
#139642
#98852
#98842
#96792
#91532
#91452
#85762
#85742