Finite Group Databasebeta

← #387106

Subgroups — #387106

Order 512.

1169 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35391C211yesmaximal, characteristic
2#21333C212yesmaximal
3#11381C211yesmaximal, characteristic
4#11381C211yesmaximal, characteristic
5#11222C212yesmaximal
6#11205C212yesmaximal
7#10918C212yesmaximal
8#10915C212yesmaximal
9#10829C212yesmaximal
10C2×#543C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12#2583C412yescomplemented
13#2583C412yescomplemented
14#2549—24no—
15#2502C412yescomplemented
16#2440—24no—
17#2434C412yescomplemented
18#2315C2212yes—
19#2315C2212yes—
20C2×#385C2212yes—
21#2273C2211yescharacteristic
22#2273C2212yes—
23#1996—24no—
24#1996—24no—
25C2×#342C2212yes—
26C2×#342C2212yes—
27C2×#342C2212yes—
28C2×#342C2212yes—
29C2×#342C2212yes—
30C2×#342C2212yes—
31C2×#342C2212yes—
32C2×#342C2211yescharacteristic
33C2×#342C2212yes—
34C2×#342C2211yescharacteristic
35C2×#342C2212yes—
36C22⋊#95C2211yescharacteristic
37C22⋊#95C2212yes—
38C22⋊#95C2211yescharacteristic
39C23×Q8C2×C411yescharacteristic
40C24×C4C2311yescharacteristic
41#543—41no—
42#543—42no—
43#543—41no—
44C2×C4⋊Q8—22no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—42no—
47C2×C22⋊Q8—41no—
48C2×C22⋊Q8—41no—
49C2×C22⋊Q8C2×C412yes—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8D414yes—
52C2×C22⋊Q8D414yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2312yes—
58C22×C22⋊C4C2312yes—
59C22×C22⋊C4C2311yescharacteristic
60C22×C22⋊C4C2312yes—
61C22×C22⋊C4C2312yes—
62C22×C22⋊C4C2312yes—
63C22×C22⋊C4C2311yescharacteristic
64C22×C22⋊C4C2312yes—
65#398—22no—
66#398—22no—
67#398—24no—
68#395—22no—
69#395—22no—
70#395—24no—
71#391—22no—
72#391—22no—
73#389—24no—
74#389—24no—
75#385—22no—
76#385—22no—
77#385—24no—
78#385—42no—
79#385—42no—
80#385C2×C414yes—
81#385C2×C414yes—
82#385C2×C414yes—
83#385C2×C414yes—
84#384—22no—
85#380—24no—
86#380—22no—
87#380—22no—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—42no—
92C4×C22⋊C4—42no—
93C4×C22⋊C4C2×C412yes—
94C4×C22⋊C4C2×C412yes—
95C2×#95—22no—
96C2×#95—41no—
97C2×#95—42no—
98C2×#95—41no—
99#342—24no—
100#342—24no—
101#342—24no—
102#342—24no—
103#342—24no—
104#342—24no—
105#342—24no—
106#342—41no—
107#342—42no—
108#342—41no—
109#342Q818yescomplemented
110C25C2411yescharacteristic
111C22×Q8—22no—
112C22×Q8—41no—
113C22×Q8—41no—
114C22×Q8—82no—
115C22×Q8C22⋊C412yes—
116C22×Q8C22⋊C412yes—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—82no—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C4⋊Q8—84no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138C22⋊Q8—42no—
139C22⋊Q8—42no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144C22⋊Q8—48no—
145C22⋊Q8—48no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C414yes—
164C2×C4⋊C4C22⋊C414yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
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⋊C4C22×C414yes—
203C2×C22⋊C4C22×C414yes—
204C2×C22⋊C4C2×D414yes—
205C2×C22⋊C4C2×D414yes—
206C2×C22⋊C4C2×Q814yes—
207C2×C22⋊C4C2×Q814yes—
208C2×C22⋊C4C2×Q814yes—
209C2×C22⋊C4#4114yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—42no—
213C2×C42—42no—
214#95—24no—
215#95—24no—
216#95—22no—
217#95—22no—
218#95—24no—
219#95—24no—
220#95—42no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—82no—
225#95—84no—
226#95—82no—
227C24—81no—
228C24—81no—
229C24C23×C411yescharacteristic
230C24C22×D412yes—
231C24C22×Q812yes—
232C24C2×#4112yes—
233C24H(5,𝔽2)12yes—
234C24H(5,𝔽2)11yescharacteristic
235C24H(5,𝔽2)11yescharacteristic
236C242−1+412yes—
237C242−1+412yes—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×Q8—82no—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
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—24no—
273C22×C4—24no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—81no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—82no—
299C22×C4—81no—
300C22×C4—81no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4C2×C22⋊C414yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C414yes—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4—162no—
328C4⋊C4—162no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4C4×Q818yes—
357C42—82no—
358C42—82no—
359C42—162no—
360C42—162no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23C22×C22⋊C411yescharacteristic
376C23C2×C4×Q812yes—
377C23#51811yescharacteristic
378C23#51811yescharacteristic
379C23#51912yes—
380C23#53412yes—
381C23#53412yes—
382C23#54212yes—
383C23#54212yes—
384C23#54412yes—
385C23#54412yes—
386C23#54812yes—
387C23#54812yes—
388C23D4×Q812yes—
389C23D4×Q812yes—
390C23#55412yes—
391C23#55511yescharacteristic
392C23Q8211yescharacteristic
393C23#56311yescharacteristic
394C23#56311yescharacteristic
395C23#56312yes—
396Q8—88no—
397Q8—162no—
398Q8—162no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—28no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—162no—
459C2×C4—162no—
460C2×C4—162no—
461C2×C4—162no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—22no—
467C22—21no—
468C22—22no—
469C22—22no—
470C22—21no—
471C22—21no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—41no—
480C22—41no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—41no—
487C22—41no—
488C22#230811yescharacteristic
489C22#233112yes—
490C22Q8×C22⋊C412yes—
491C22#236911yescharacteristic
492C22#238111yescharacteristic
493C22#272912yes—
494C22#272912yes—
495C22#273112yes—
496C22#273412yes—
497C4—48no—
498C4—42no—
499C4—42no—
500C4—44no—
501C4—42no—
502C4—44no—
503C4—84no—
504C4—84no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—162no—
509C4—162no—
510C2—22no—
511C2—22no—
512C2—22no—
513C2—22no—
514C2—22no—
515C2—21no—
516C2—21no—
517C2—22no—
518C2#1445011yescharacteristic
519C2#2177611yescharacteristic
520C2#2177711yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2583C4semidirect
13. #2583C4semidirect
15. #2502C4semidirect
17. #2434C4semidirect
109. #342Q8semidirect

Maximal subgroups

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

subgroupindex
#353912
#213332
#113812
#112222
#112052
#109182
#109152
#108292