Finite Group Databasebeta

← #117794

Subgroups — #117794

Order 512.

718 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21860C211yesmaximal, characteristic
2#21322C211yesmaximal, characteristic
3C2×#1834C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic
5#10857C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13C2×#384C2211yescharacteristic
14C2×#384C2211yescharacteristic
15#1992—22no—
16#1990—21no—
17#1990—21no—
18#1990—21no—
19#1956—21no—
20#1872—22no—
21#1834—22no—
22#1834—22no—
23#1803—22no—
24C2×#343—22no—
25#1736—22no—
26C2×#340—22no—
27#1493—22no—
28#1493C414yescomplemented
29C2×Q8⋊C8—22no—
30#1296—22no—
31#1278—22no—
32C22×M16—21no—
33C22×M16D412yes—
34C23×C8—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533Q812yes—
43C2×C4⋊Q8D411yescharacteristic
44C2×C4⋊Q8D411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D412yes—
57C22×C42C2311yescharacteristic
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#423—21no—
63#423—21no—
64#423—21no—
65#423C2×C412yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8D412yes—
69C2×#103—24no—
70C2×C22⋊C8—21no—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8C2×C412yes—
82C2×C4×C8D412yes—
83C2×C4×C8D412yes—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384D412yes—
92#343—44no—
93#340—24no—
94#340—28no—
95#340—24no—
96#340—24no—
97Q8⋊C8—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4—22no—
101C23×C4—41no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—41no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16Q1614yes—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—42no—
121C22×C8—42no—
122C22×C8—41no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8C4⋊C412yes—
130C4⋊Q8C4⋊C412yes—
131#125—22no—
132#125—22no—
133#125—41no—
134#125—41no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—41no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—41no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—41no—
151#117—22no—
152#117—22no—
153#117—41no—
154#117—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C22⋊C4—41no—
182C2×C42—41no—
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×Q811yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201#103—84no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204#97—41no—
205#97—41no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—41no—
210C4×C8—41no—
211C4×C8—41no—
212C4×C8—41no—
213C4×C8C22⋊C414yes—
214C24#12111yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4#11711yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4#12311yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4#12711yescharacteristic
255C22×C4C2×Q1612yes—
256C22×C4#13512yes—
257C22×C4AGL(1,ℤ/8)12yes—
258C22×C4#13712yes—
259M16—24no—
260M16—24no—
261M16—42no—
262M16—42no—
263M16—44no—
264M16—82no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—81no—
278C2×C8—81no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C42—42no—
308C42—42no—
309C42C2×C22⋊C411yescharacteristic
310C42C2×C4⋊C411yescharacteristic
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42#12111yescharacteristic
314C42#12111yescharacteristic
315C42#12111yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42C22⋊Q811yescharacteristic
318C42C22⋊Q811yescharacteristic
319C42#12311yescharacteristic
320C42#12311yescharacteristic
321C42#12311yescharacteristic
322C42#12311yescharacteristic
323C42#12411yescharacteristic
324C42#12411yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23#38811yescharacteristic
330C23#39211yescharacteristic
331C23#39611yescharacteristic
332C23#46212yes—
333C23#46312yes—
334Q8—44no—
335Q8—44no—
336Q8—82no—
337Q8—82no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—164no—
363C2×C4—164no—
364C2×C4#38011yescharacteristic
365C2×C4#38511yescharacteristic
366C2×C4#38511yescharacteristic
367C2×C4#38611yescharacteristic
368C2×C4#38811yescharacteristic
369C2×C4#39011yescharacteristic
370C2×C4#39311yescharacteristic
371C2×C4#39411yescharacteristic
372C2×C4#39411yescharacteristic
373C2×C4#39611yescharacteristic
374C2×C4#39711yescharacteristic
375C2×C4#39911yescharacteristic
376C2×C4C4×Q1612yes—
377C2×C4#44012yes—
378C2×C4#44912yes—
379C2×C4#45112yes—
380C2×C4#45412yes—
381C2×C4#45512yes—
382C2×C4#46712yes—
383C2×C4#46812yes—
384C2×C4#47112yes—
385C2×C4#47112yes—
386C2×C4#47712yes—
387C2×C4#47912yes—
388C2×C4#48412yes—
389C2×C4#48512yes—
390C2×C4#49411yescharacteristic
391C2×C4#49511yescharacteristic
392C2×C4#49611yescharacteristic
393C2×C4#49711yescharacteristic
394C8—44no—
395C8—44no—
396C8—82no—
397C8—82no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22#143911yescharacteristic
407C22#192812yes—
408C22#200311yescharacteristic
409C22#200411yescharacteristic
410C22#201211yescharacteristic
411C22#201611yescharacteristic
412C22#201911yescharacteristic
413C22#201911yescharacteristic
414C22#206812yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—82no—
420C4—82no—
421C4—162no—
422C4—162no—
423C4#181512yes—
424C4#186012yes—
425C4#186312yes—
426C4#196911yescharacteristic
427C4#197311yescharacteristic
428C4#203612yes—
429C4#204012yes—
430C4#207912yes—
431C4#209511yescharacteristic
432C4#209711yescharacteristic
433C2—22no—
434C2—22no—
435C2#1015111yescharacteristic
436C2#1015711yescharacteristic
437C2#1016311yescharacteristic
438C2#1016911yescharacteristic
439C2#1107911yescharacteristic
440C2#1110611yescharacteristic
441C2#1112411yescharacteristic
subgroupG/H
center, socleC23#392
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
28. #1493C4semidirect

Maximal subgroups

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

subgroupindex
#218602
#213222
C2×#18342
C2×#14932
#108572
#80432
C2×#12782