Finite Group Databasebeta

← #116231

Subgroups — #116231

Order 512.

698 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21866C211yesmaximal, characteristic
2#21322C211yesmaximal, characteristic
3C2×#1506C211yesmaximal, characteristic
4#14348C211yesmaximal, characteristic
5C2×#1300C211yesmaximal, characteristic
6#8040C211yesmaximal, characteristic
7#8033C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×#384C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1993—21no—
16#1992—22no—
17#1990—21no—
18#1990—21no—
19#1960—21no—
20#1872—21no—
21#1872—21no—
22#1805—22no—
23#1736—22no—
24#1506—22no—
25#1506C414yescomplemented
26C2×#328—22no—
27#1442—22no—
28#1442—22no—
29C4⋊#95—21no—
30C4⋊#95—21no—
31#1300—22no—
32#1296—22no—
33#1295—22no—
34C22×M16—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⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432C2×C412yes—
66C2×C4⋊C8—21no—
67C2×C4⋊C8D412yes—
68C2×#103—22no—
69C2×#103—22no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97D412yes—
79C2×C4×C8—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#382—21no—
85#382—21no—
86#382—21no—
87#382—21no—
88#376—21no—
89#376—21no—
90#376C2×C412yes—
91#376D412yes—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95D412yes—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#328—28no—
103C22×Q8—21no—
104C22×Q8—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C2×M16—24no—
113C2×M16—42no—
114C2×M16—41no—
115C2×M16—41no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—41no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C4⋊C412yes—
126C4⋊Q8C4⋊C412yes—
127#125—22no—
128#125—22no—
129#125—41no—
130#125—41no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147#117—22no—
148#117—22no—
149#117—41no—
150#117—41no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—82no—
172C2×C22⋊C4—41no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—41no—
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×Q811yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—41no—
194#103—82no—
195#103—82no—
196C22⋊C8—22no—
197C22⋊C8—41no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200#97—22no—
201#97—22no—
202#97—22no—
203#97—41no—
204#97—41no—
205#97—41no—
206C4×C8—41no—
207C4×C8—41no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—84no—
216#95—84no—
217#95C22⋊C414yes—
218C24#12011yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—81no—
226C2×Q8—81no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C4×D411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12111yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4#12411yescharacteristic
269C22×C4#12411yescharacteristic
270C22×C4#12411yescharacteristic
271C22×C4#12711yescharacteristic
272M16—44no—
273M16—44no—
274C2×C8—24no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—81no—
282C2×C8—81no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C42—44no—
308C42—44no—
309C42—42no—
310C42—42no—
311C42C2×C4⋊C411yescharacteristic
312C42#11711yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42#12111yescharacteristic
316C42#12111yescharacteristic
317C42#12111yescharacteristic
318C42C22⋊Q811yescharacteristic
319C42C22⋊Q811yescharacteristic
320C42C22⋊Q811yescharacteristic
321C42#12311yescharacteristic
322C42#12311yescharacteristic
323C42#12311yescharacteristic
324C42#12311yescharacteristic
325C42#12611yescharacteristic
326C42#12611yescharacteristic
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23#38611yescharacteristic
332C23#39211yescharacteristic
333C23#39311yescharacteristic
334C23#45512yes—
335C23#45612yes—
336Q8—44no—
337Q8—82no—
338Q8—82no—
339Q8—82no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—88no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4#38011yescharacteristic
388C2×C4#38511yescharacteristic
389C2×C4#38511yescharacteristic
390C2×C4#38811yescharacteristic
391C2×C4#38811yescharacteristic
392C2×C4#39011yescharacteristic
393C2×C4#39411yescharacteristic
394C2×C4#39411yescharacteristic
395C2×C4#39611yescharacteristic
396C2×C4#39611yescharacteristic
397C2×C4#39711yescharacteristic
398C2×C4#39911yescharacteristic
399C2×C4#45412yes—
400C2×C4#45512yes—
401C8—44no—
402C8—82no—
403C8—82no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22#143911yescharacteristic
413C22#188212yes—
414C22#200311yescharacteristic
415C22#200411yescharacteristic
416C22#200411yescharacteristic
417C22#200511yescharacteristic
418C22#200811yescharacteristic
419C22#200911yescharacteristic
420C22#202812yes—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—84no—
432C4—82no—
433C4—82no—
434C4—162no—
435C4—161no—
436C4—161no—
437C4#186012yes—
438C4#204012yes—
439C2—22no—
440C2—22no—
441C2#1009711yescharacteristic
442C2#1009811yescharacteristic
443C2#1010011yescharacteristic
444C2#1010311yescharacteristic
445C2#1106411yescharacteristic
446C2#1107911yescharacteristic
447C2#1109311yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
25. #1506C4semidirect

Maximal subgroups

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

subgroupindex
#218662
#213222
C2×#15062
#143482
C2×#13002
#80402
#80332