Finite Group Databasebeta

← #117694

Subgroups — #117694

Order 512.

776 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21400C211yesmaximal, characteristic
2C2×#1986C211yesmaximal, characteristic
3C2×C4×#107C211yesmaximal, characteristic
4C2×#1503C211yesmaximal, characteristic
5#10851C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7#7848C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C22×#107—21no—
11C2×C4×M16C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1990C412yescomplemented
17#1986—22no—
18#1986—22no—
19#1956C412yescomplemented
20#1892—21no—
21#1892—21no—
22C4×#107—22no—
23C4×#107—22no—
24C4×#103—22no—
25C2×#340—21no—
26C2×#339—22no—
27C2×#339—22no—
28#1503—22no—
29#1503—22no—
30C2×#328—22no—
31#1294—22no—
32#1273—22no—
33C22×M16D412yes—
34C23×C8D411yescharacteristic
35C23×C8D411yescharacteristic
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533D412yes—
43#533Q812yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C42—21no—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62C2×#107—28no—
63C2×#107—24no—
64C2×C4⋊C8—21no—
65C2×#103—22no—
66C2×#103—22no—
67C2×C22⋊C8D412yes—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8C2×C412yes—
80C2×C4×C8D412yes—
81#384C2×C412yes—
82#382—21no—
83#380—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4D412yes—
94C4×C4⋊C4D412yes—
95C2×#95—21no—
96C2×#95—21no—
97#340—24no—
98#340—22no—
99#340—22no—
100#339—28no—
101#339—44no—
102#328—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4—41no—
106C23×C4—41no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16Q1614yes—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—24no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—42no—
124C22×C8C22⋊C412yes—
125C22×C8C22⋊C412yes—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8C4⋊C412yes—
133C4⋊Q8C4⋊C412yes—
134#125—22no—
135#125—22no—
136#125—41no—
137#125—41no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154#117—22no—
155#117—22no—
156#117—41no—
157#117—41no—
158#117—42no—
159#117—42no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—41no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—42no—
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×Q811yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201#107—216no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204#103—44no—
205#103—44no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208#97—41no—
209#97—41no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—41no—
215C4×C8—41no—
216C4×C8—41no—
217C4×C8—41no—
218#95—22no—
219#95—22no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224C24#12711yescharacteristic
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—82no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12411yescharacteristic
266C22×C4C2×Q1612yes—
267C22×C4#13512yes—
268C22×C4AGL(1,ℤ/8)12yes—
269C22×C4#13712yes—
270M16—24no—
271M16—24no—
272M16—42no—
273M16—42no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—44no—
287C2×C8—42no—
288C2×C8#10414yes—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—82no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C42—42no—
316C42—42no—
317C42—44no—
318C42—44no—
319C42—82no—
320C42—82no—
321C42—82no—
322C42C2×C4⋊C411yescharacteristic
323C42#11711yescharacteristic
324C42C4×D411yescharacteristic
325C42C4×D411yescharacteristic
326C42#12111yescharacteristic
327C42#12111yescharacteristic
328C42#12111yescharacteristic
329C42C22⋊Q811yescharacteristic
330C42C22⋊Q811yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42#12311yescharacteristic
333C42#12311yescharacteristic
334C42#12411yescharacteristic
335C42#12411yescharacteristic
336C42#12611yescharacteristic
337C42#12611yescharacteristic
338C23—22no—
339C23—22no—
340C23#39011yescharacteristic
341C23#39211yescharacteristic
342C23#39911yescharacteristic
343C23#49411yescharacteristic
344C23#49511yescharacteristic
345C23#49611yescharacteristic
346C23#49711yescharacteristic
347Q8—44no—
348Q8—82no—
349Q8—82no—
350Q8—81no—
351Q8—81no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4#38011yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4#38511yescharacteristic
385C2×C4#38611yescharacteristic
386C2×C4#38811yescharacteristic
387C2×C4#38811yescharacteristic
388C2×C4#39311yescharacteristic
389C2×C4#39411yescharacteristic
390C2×C4#39411yescharacteristic
391C2×C4#39611yescharacteristic
392C2×C4#39611yescharacteristic
393C2×C4#39711yescharacteristic
394C2×C4C2×#10412yes—
395C2×C4#42112yes—
396C2×C4C4×Q1612yes—
397C2×C4#44012yes—
398C2×C4#44912yes—
399C2×C4#45112yes—
400C2×C4#46212yes—
401C2×C4#46312yes—
402C2×C4#46712yes—
403C2×C4#46812yes—
404C2×C4#47712yes—
405C2×C4#47912yes—
406C2×C4#48412yes—
407C2×C4#48512yes—
408C8—24no—
409C8—42no—
410C8—44no—
411C8—82no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22#143911yescharacteristic
421C22#196911yescharacteristic
422C22#197311yescharacteristic
423C22#197612yes—
424C22#198012yes—
425C22#201211yescharacteristic
426C22#201611yescharacteristic
427C22#209511yescharacteristic
428C22#209711yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—84no—
436C4—84no—
437C4—82no—
438C4—82no—
439C4—162no—
440C4#180612yes—
441C4#186312yes—
442C4#189611yescharacteristic
443C4#189711yescharacteristic
444C4#192812yes—
445C4#203612yes—
446C4#206812yes—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2#971812yes—
452C2#972212yes—
453C2#1107511yescharacteristic
454C2#1112411yescharacteristic
455C2#1112611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#390
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #1990C4semidirect
19. #1956C4semidirect

Maximal subgroups

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

subgroupindex
#214002
C2×#19862
C2×C4×#1072
C2×#15032
#108512
#80142
#78482