Finite Group Databasebeta

← #107563

Subgroups — #107563

Order 512.

929 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34380C211yesmaximal, characteristic
2#17578C211yesmaximal, characteristic
3#17578C211yesmaximal, characteristic
4#8108C211yesmaximal, characteristic
5#8100C211yesmaximal, characteristic
6#8016C211yesmaximal, characteristic
7#7983C211yesmaximal, characteristic
8C22×C22⋊Q8C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11#2378—22no—
12#2360—22no—
13#2344C412yescomplemented
14#2342—22no—
15Q8×C22⋊C4C412yescomplemented
16#2329—22no—
17#2323—22no—
18#2316—21no—
19#2313—21no—
20#2280C2211yescharacteristic
21#2280C2211yescharacteristic
22#2280C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#1520—22no—
25#1520C414yescomplemented
26#1511—22no—
27#1511C414yescomplemented
28#1508—22no—
29#1508—22no—
30#1305—21no—
31#1305—21no—
32#1304—22no—
33#1294—22no—
34#1289—22no—
35C23×Q8C2×C411yescharacteristic
36C24×C4C2311yescharacteristic
37C22×M16—21no—
38C22×M16—21no—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×C22⋊Q8C2×C412yes—
47C2×C22⋊Q8D412yes—
48C2×C22⋊Q8D412yes—
49C2×C22⋊Q8D412yes—
50C2×C22⋊Q8Q812yes—
51C2×C4×Q8—22no—
52C2×#117—21no—
53C2×#117—22no—
54C2×#117—21no—
55C2×#117—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61#407—21no—
62#407—21no—
63#407—21no—
64#407—21no—
65#407—21no—
66#407—21no—
67#407—21no—
68#407C2×C412yes—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8C2×C412yes—
73#391—22no—
74#391—24no—
75#389—24no—
76#389—24no—
77#389—22no—
78#388—24no—
79#388—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#385—24no—
84#381—21no—
85#381—22no—
86#381—21no—
87#381C2×C412yes—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—22no—
92#380—21no—
93#380—21no—
94#380—21no—
95#380C2×C412yes—
96#380C2×C412yes—
97#380D411yescharacteristic
98#380D411yescharacteristic
99#380D412yes—
100#379—21no—
101#379—21no—
102#379D412yes—
103C4×C4⋊C4—22no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4C2×C412yes—
112C2×#95D412yes—
113#328—28no—
114#328—28no—
115C25C2×D411yescharacteristic
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—82no—
120C22×Q8C4212yes—
121C22×Q8C22⋊C412yes—
122C22×Q8C4⋊C412yes—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—82no—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×Q811yescharacteristic
137C23×C4#4111yescharacteristic
138C23×C4#4111yescharacteristic
139C23×C4#4111yescharacteristic
140C23×C4#4111yescharacteristic
141C2×M16—22no—
142C2×M16—41no—
143C2×M16—41no—
144C2×M16—41no—
145C2×M16—82no—
146C2×M16—82no—
147C22×C8—22no—
148C22×C8—41no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8C22⋊C414yes—
162C22⋊Q8C22⋊C414yes—
163C22⋊Q8C4⋊C414yes—
164C22⋊Q8C4⋊C414yes—
165C4×Q8—84no—
166#117—82no—
167#117—84no—
168#117—82no—
169#117—82no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C4212yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C4⋊C412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—41no—
214C2×C42—41no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—41no—
220C22⋊C8—41no—
221C22⋊C8—41no—
222C22⋊C8—41no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—41no—
235#95C22⋊C414yes—
236C24—41no—
237C24—41no—
238C24—42no—
239C24—81no—
240C24C2×C22⋊C411yescharacteristic
241C24C4×D411yescharacteristic
242C24C4×D411yescharacteristic
243C24#12011yescharacteristic
244C24#12011yescharacteristic
245C24#12111yescharacteristic
246C24C22⋊Q811yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—44no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—82no—
254C2×Q8—82no—
255C2×Q8—82no—
256C2×Q8#9514yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4C2×C4211yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4#11711yescharacteristic
297C22×C4#11711yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4C22⋊Q811yescharacteristic
310C22×C4C22⋊Q811yescharacteristic
311C22×C4C22⋊Q811yescharacteristic
312C22×C4#12311yescharacteristic
313C22×C4#12311yescharacteristic
314C22×C4#12311yescharacteristic
315C22×C4#12311yescharacteristic
316M16—82no—
317M16—82no—
318M16—82no—
319M16—82no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—82no—
323C2×C8—82no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—44no—
332C4⋊C4—82no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4#9514yes—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—81no—
351C22⋊C4—81no—
352C22⋊C4—81no—
353C22⋊C4—81no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C42—44no—
359C42—82no—
360C42—82no—
361C42—84no—
362C42—84no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—22no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—84no—
388C23—84no—
389C23C4×C22⋊C411yescharacteristic
390C23#37911yescharacteristic
391C23#38011yescharacteristic
392C23#38511yescharacteristic
393C23#38511yescharacteristic
394C23#38611yescharacteristic
395C23#38611yescharacteristic
396Q8—44no—
397Q8—44no—
398Q8—82no—
399Q8—82no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
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—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—164no—
453C2×C4C2×#9511yescharacteristic
454C2×C4C4×C22⋊C411yescharacteristic
455C2×C4#38011yescharacteristic
456C2×C4#38111yescharacteristic
457C2×C4#38511yescharacteristic
458C2×C4#38511yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C8—82no—
462C8—82no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—81no—
487C22—81no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22C22⋊#9511yescharacteristic
493C22#179511yescharacteristic
494C22#179611yescharacteristic
495C22#190512yes—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—44no—
504C4—44no—
505C4—82no—
506C4—82no—
507C4—84no—
508C4—162no—
509C4—162no—
510C4#182812yes—
511C2—21no—
512C2—21no—
513C2—22no—
514C2—41no—
515C2—41no—
516C2—41no—
517C2—41no—
518C2—41no—
519C2#1076811yescharacteristic
520C2#1077511yescharacteristic
521C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2344C4semidirect
15. Q8×C22⋊C4C4semidirect
25. #1520C4semidirect
27. #1511C4semidirect

Maximal subgroups

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

subgroupindex
#343802
#175782
#81082
#81002
#80162
#79832