Finite Group Databasebeta

← #115925

Subgroups — #115925

Order 512.

744 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21972C211yesmaximal, characteristic
2#21313C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5#10852C211yesmaximal, characteristic
6C2×#1303C211yesmaximal, characteristic
7C2×#1292C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C4×#117C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1988C414yescomplemented
16#1987—22no—
17#1987—22no—
18#1987—22no—
19#1939—22no—
20#1850—22no—
21#1842—21no—
22#1831—21no—
23#1736—21no—
24#1736—21no—
25C2×#340—22no—
26C2×#339—21no—
27C2×#339—21no—
28#1548—22no—
29#1548—22no—
30C2×#333—21no—
31C2×#333—21no—
32#1303—22no—
33#1292—22no—
34C23×Q8—21no—
35C22×M16—21no—
36C22×M16—21no—
37C23×C8D412yes—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8D412yes—
49C2×C4⋊Q8Q812yes—
50C2×#117D411yescharacteristic
51C2×#117D411yescharacteristic
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#423—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423D412yes—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8C2×C412yes—
80C2×C4⋊C8D411yescharacteristic
81C2×C4⋊C8D411yescharacteristic
82C2×C22⋊C8—21no—
83C2×C4×C8—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—21no—
91C4×C22⋊C4—21no—
92#376—21no—
93C2×#95—21no—
94C43—21no—
95#340—28no—
96#340—24no—
97#340—24no—
98#339—42no—
99#339—42no—
100#336—24no—
101#333—24no—
102#333—24no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×Q8—41no—
109C22×Q8—84no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4#4111yescharacteristic
115C2×M16—22no—
116C2×M16—41no—
117C2×M16—41no—
118C2×M16—41no—
119C2×M16—41no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—44no—
134C22×C8—41no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—41no—
145C4⋊Q8—41no—
146C4⋊Q8—41no—
147C4⋊Q8—41no—
148C4⋊Q8—41no—
149C4⋊Q8—41no—
150C4⋊Q8C4⋊C414yes—
151#117—21no—
152#117—21no—
153#117—21no—
154#117—21no—
155#117—21no—
156#117—21no—
157#117SD1612yes—
158#117Q1612yes—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42—41no—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×Q811yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—22no—
209C4⋊C8—41no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—41no—
213C4⋊C8—41no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—84no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220C4×C8—41no—
221C4×C8—41no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—41no—
226C24#12111yescharacteristic
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—84no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×Q8—84no—
235C2×Q8—84no—
236C2×Q8—84no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12411yescharacteristic
268C22×C4#12611yescharacteristic
269C22×C4C2×SD1611yescharacteristic
270C22×C4C2×Q1611yescharacteristic
271C22×C4#13511yescharacteristic
272C22×C4#13511yescharacteristic
273C22×C4AGL(1,ℤ/8)11yescharacteristic
274C22×C4#13711yescharacteristic
275C22×C4#13711yescharacteristic
276C22×C4#13711yescharacteristic
277M16—42no—
278M16—42no—
279M16—81no—
280M16—81no—
281M16—81no—
282M16—81no—
283M16—81no—
284M16—81no—
285C2×C8—44no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—44no—
295C2×C8—42no—
296C2×C8—81no—
297C2×C8—81no—
298C2×C8—82no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4#10314yes—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—22no—
319C42—22no—
320C42—42no—
321C42—42no—
322C42—41no—
323C42—41no—
324C42—42no—
325C42—42no—
326C42—82no—
327C42—81no—
328C42—81no—
329C42C2×C4⋊C411yescharacteristic
330C42#11711yescharacteristic
331C42C4×D411yescharacteristic
332C42C4×D411yescharacteristic
333C42#12011yescharacteristic
334C42#12111yescharacteristic
335C42#12111yescharacteristic
336C42C22⋊Q811yescharacteristic
337C42C22⋊Q811yescharacteristic
338C42C22⋊Q811yescharacteristic
339C42#12311yescharacteristic
340C42#12311yescharacteristic
341C42#12411yescharacteristic
342C42#12411yescharacteristic
343C42#12611yescharacteristic
344C42#12711yescharacteristic
345C23—22no—
346C23—22no—
347C23#38611yescharacteristic
348C23#39411yescharacteristic
349C23#39611yescharacteristic
350C23#46511yescharacteristic
351C23#46711yescharacteristic
352C23#46811yescharacteristic
353C23#47011yescharacteristic
354Q8—84no—
355Q8—84no—
356Q8—84no—
357Q8—84no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—82no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4—84no—
386C2×C4#38011yescharacteristic
387C2×C4#38511yescharacteristic
388C2×C4#38511yescharacteristic
389C2×C4#38811yescharacteristic
390C2×C4#38811yescharacteristic
391C2×C4#39011yescharacteristic
392C2×C4#39211yescharacteristic
393C2×C4#39311yescharacteristic
394C2×C4#39411yescharacteristic
395C2×C4#39611yescharacteristic
396C2×C4#39711yescharacteristic
397C2×C4#39911yescharacteristic
398C2×C4C2×#10311yescharacteristic
399C2×C4#41611yescharacteristic
400C2×C4#41711yescharacteristic
401C2×C4#41911yescharacteristic
402C2×C4#45512yes—
403C2×C4#45612yes—
404C2×C4#46011yescharacteristic
405C2×C4#46111yescharacteristic
406C2×C4#46111yescharacteristic
407C2×C4#46211yescharacteristic
408C2×C4#46211yescharacteristic
409C2×C4#46211yescharacteristic
410C2×C4#46311yescharacteristic
411C2×C4#46311yescharacteristic
412C2×C4#46411yescharacteristic
413C2×C4#46411yescharacteristic
414C2×C4#46411yescharacteristic
415C2×C4#46411yescharacteristic
416C2×C4#46511yescharacteristic
417C2×C4#46711yescharacteristic
418C2×C4#46811yescharacteristic
419C2×C4#47011yescharacteristic
420C2×C4#47511yescharacteristic
421C2×C4#47711yescharacteristic
422C2×C4#47911yescharacteristic
423C2×C4#47911yescharacteristic
424C8—82no—
425C8—82no—
426C8—81no—
427C8—82no—
428C8—81no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22#143911yescharacteristic
438C22#188212yes—
439C22#189211yescharacteristic
440C22#189411yescharacteristic
441C22#190312yes—
442C22#205511yescharacteristic
443C22#205611yescharacteristic
444C22#206611yescharacteristic
445C22#206811yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—44no—
453C4—84no—
454C4—84no—
455C4—82no—
456C4#180311yescharacteristic
457C4#180511yescharacteristic
458C4#198111yescharacteristic
459C4#198311yescharacteristic
460C4#200411yescharacteristic
461C4#200511yescharacteristic
462C4#202812yes—
463C4#205511yescharacteristic
464C4#205611yescharacteristic
465C4#206611yescharacteristic
466C4#206811yescharacteristic
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2#931512yes—
472C2#931912yes—
473C2#1105811yescharacteristic
474C2#1106411yescharacteristic
475C2#1107711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1988C4semidirect

Maximal subgroups

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

subgroupindex
#219722
#213132
C2×#19872
C2×#15482
#108522
C2×#13032
C2×#12922