Finite Group Databasebeta

← #110684

Subgroups — #110684

Order 512.

738 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21866C211yesmaximal, characteristic
2C2×#1986C211yesmaximal, characteristic
3C2×#1515C211yesmaximal, characteristic
4#13781C211yesmaximal, characteristic
5#8033C211yesmaximal, characteristic
6C2×#1288C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×#383C2211yescharacteristic
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1993C412yescomplemented
16#1986—22no—
17#1986C414yescomplemented
18#1960C412yescomplemented
19#1872—22no—
20#1805—22no—
21C2×#339—21no—
22C2×#339—21no—
23#1515—22no—
24#1515C414yescomplemented
25C2×#328—22no—
26#1436—22no—
27#1436—22no—
28#1434—21no—
29C4×#95—21no—
30#1295—22no—
31#1288—22no—
32#1285—22no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D412yes—
52C22×C42—22no—
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432C2×C412yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×#103—24no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16C2×C412yes—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97C2×C412yes—
76C2×C4×C8—21no—
77#383—21no—
78#383—21no—
79#383C2×C412yes—
80#383D412yes—
81#382—21no—
82#382—21no—
83#382C2×C412yes—
84#382D412yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4D412yes—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98#340—24no—
99#339—24no—
100#339—24no—
101#328—28no—
102C22×Q8—21no—
103C22×Q8—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C2×M16—24no—
112C2×M16—22no—
113C2×M16—42no—
114C2×M16—41no—
115C22×C8—22no—
116C22×C8—41no—
117C22×C8—41no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8C4212yescomplemented
125C4⋊Q8C4212yescomplemented
126#125—22no—
127#125—22no—
128#125—41no—
129#125—41no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—41no—
135C22⋊Q8—41no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—41no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146#117—22no—
147#117—22no—
148#117—41no—
149#117—41no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C22⋊C414yes—
162C2×C22⋊C4—41no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—41no—
166C2×C42—42no—
167C2×C42—42no—
168C2×C42—44no—
169C2×C42—82no—
170C2×C42—82no—
171C2×C42C22⋊C412yes—
172C2×C42C22⋊C412yes—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—41no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189#103—84no—
190C22⋊C8—22no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—41no—
199#97—41no—
200C4×C8—22no—
201C4×C8—41no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—44no—
214#95—82no—
215#95—82no—
216#95C22⋊C414yes—
217C24#12011yescharacteristic
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—82no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C4×D411yescharacteristic
260C22×C4C4×D411yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12311yescharacteristic
272C22×C4#12311yescharacteristic
273C22×C4#12311yescharacteristic
274M16—24no—
275M16—44no—
276M16—42no—
277C2×C8—44no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—82no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—82no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42—81no—
307C42—81no—
308C42—88no—
309C42C2×C4211yescharacteristic
310C42#11711yescharacteristic
311C42#11711yescharacteristic
312C42#11711yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42C4×D411yescharacteristic
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42#12411yescharacteristic
320C42#12411yescharacteristic
321C42#12411yescharacteristic
322C42#12611yescharacteristic
323C42#12611yescharacteristic
324C42#12711yescharacteristic
325C23—22no—
326C23—21no—
327C23—21no—
328C23#38611yescharacteristic
329C23#38611yescharacteristic
330C23#38611yescharacteristic
331C23#45512yes—
332C23#45612yes—
333Q8—44no—
334Q8—82no—
335Q8—82no—
336Q8—82no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4C4×C22⋊C411yescharacteristic
389C2×C4C4×C22⋊C411yescharacteristic
390C2×C4C4×C22⋊C411yescharacteristic
391C2×C4#38811yescharacteristic
392C2×C4#38811yescharacteristic
393C2×C4#38811yescharacteristic
394C2×C4#38811yescharacteristic
395C2×C4#38811yescharacteristic
396C2×C4#38811yescharacteristic
397C2×C4#39011yescharacteristic
398C2×C4#39011yescharacteristic
399C2×C4#39011yescharacteristic
400C8—44no—
401C8—82no—
402C8—82no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22#143711yescharacteristic
412C22#188212yes—
413C22#188212yes—
414C22#188511yescharacteristic
415C22#188611yescharacteristic
416C22#190612yes—
417C22#191111yescharacteristic
418C22#191211yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—44no—
432C4—82no—
433C4—84no—
434C4—84no—
435C4—82no—
436C4—82no—
437C2—21no—
438C2—22no—
439C2—21no—
440C2#937912yes—
441C2#938012yes—
442C2#1094511yescharacteristic
443C2#1094911yescharacteristic
444C2#1095711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ 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. #1993C4semidirect
17. #1986C4semidirect
18. #1960C4semidirect
24. #1515C4semidirect
124. C4⋊Q8C42semidirect
125. C4⋊Q8C42semidirect

Maximal subgroups

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

subgroupindex
#218662
C2×#19862
C2×#15152
#137812
#80332
C2×#12882
C2×#12852