Finite Group Databasebeta

← #197768

Subgroups — #197768

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C2×#2960C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#9226C214yesmaximal, complemented
7#9218C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7872C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209C418yescomplemented
14C2×#494—21no—
15C2×#487—21no—
16C2×#462C2214yes—
17C2×#451—22no—
18#2960—24no—
19#2960—24no—
20C2×#446—21no—
21C2×#446C2212yes—
22C2×C8×Q8—21no—
23C2×C8×Q8C2212yes—
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#424—21no—
27C2×#423C2211yescharacteristic
28C2×#405—21no—
29C22×C4×C8C2211yescharacteristic
30C2×#384C2211yescharacteristic
31#1990—21no—
32#1990—21no—
33#1990—21no—
34#1990C2212yes—
35#1990C2212yes—
36#1956—21no—
37#1872—22no—
38#1803C2214yes—
39C2×#340—21no—
40C2×#340—21no—
41#1549—24no—
42C2×#332C2214yes—
43#1296C2214yes—
44#1277C2212yes—
45#1277C2212yes—
46C23×Q8—21no—
47C22×Q16—24no—
48C22×M16—21no—
49C23×C8D411yescharacteristic
50C23×C8D411yescharacteristic
51#533—22no—
52#533—22no—
53#533C2×C414yes—
54#533D414yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8D412yes—
68C2×C4×Q8—21no—
69C2×C4×Q8C2312yes—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C42C2311yescharacteristic
77#494—28no—
78#487—28no—
79#462—28no—
80#462C2×C4116yescomplemented
81#451—216no—
82#446—28no—
83#446—24no—
84#446—24no—
85C8×Q8—28no—
86C8×Q8—24no—
87C8×Q8—24no—
88#431—22no—
89#431—22no—
90#431—22no—
91#431C2×C414yes—
92#424—28no—
93#423—22no—
94#423D414yes—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101C2×#103—24no—
102C2×#103—24no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105#405—28no—
106C2×#97—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8D411yescharacteristic
113C2×C4×C8D411yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#384—21no—
117#384—21no—
118#384—21no—
119#384C2312yes—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—24no—
125#332—28no—
126C22×Q8—21no—
127C22×Q8—24no—
128C22×Q8—22no—
129C22×Q8—41no—
130C22×Q8—42no—
131C22×Q8—82no—
132C22×Q8—82no—
133C22×Q8C2×D412yes—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4#4111yescharacteristic
139C2×Q16—216no—
140C2×Q16—84no—
141C2×Q16—84no—
142C2×M16—24no—
143C2×M16—42no—
144C2×M16—42no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8Q1614yescomplemented
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8C22⋊C414yes—
170C4⋊Q8C22⋊C414yes—
171C4⋊Q8C22×C414yes—
172#125—24no—
173#125—24no—
174C22⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8C22⋊C418yes—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8C22×C414yes—
183C4×Q8C2×D414yes—
184#117—24no—
185#117—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C2×D412yes—
210C2×C22⋊C4—22no—
211C2×C42—22no—
212C2×C42C22×C412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42#4111yescharacteristic
218C2×C42#4112yes—
219C2×C42C2411yescharacteristic
220C4⋊C8—28no—
221C4⋊C8—28no—
222C4⋊C8—24no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8C22×C414yes—
228C4⋊C8C2×D414yes—
229#103—216no—
230#103—84no—
231C22⋊C8—24no—
232C22⋊C8—42no—
233#97—28no—
234C4×C8—28no—
235C4×C8—24no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—42no—
243C24#12111yescharacteristic
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—42no—
250C2×Q8—44no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—88no—
256C2×Q8—82no—
257C2×Q8—82no—
258C2×Q8—88no—
259C2×Q8C2×C22⋊C414yes—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C4×D412yes—
276C22×C4C4×D412yes—
277C22×C4#12012yes—
278C22×C4#12112yes—
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12312yes—
283C22×C4C2×Q1612yes—
284C22×C4#13512yes—
285C22×C4#13712yes—
286C22×C4#13712yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290Q16—816no—
291M16—48no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—28no—
297C2×C8—24no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—24no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8C2×Q1612yes—
314C2×C8C2×Q1612yes—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4C2×C22⋊C414yes—
336C22⋊C4—28no—
337C42—24no—
338C42—24no—
339C42C2×C22⋊C412yes—
340C42C4×D412yes—
341C42C4×D412yes—
342C42#12012yes—
343C42#12112yes—
344C42#12312yes—
345C42C23×C411yescharacteristic
346C42C22×D411yescharacteristic
347C42C22×D411yescharacteristic
348C42C2×#4111yescharacteristic
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23#38612yes—
354C23#46212yes—
355C23#46412yes—
356C23C2×#12111yescharacteristic
357Q8—28no—
358Q8—48no—
359Q8—44no—
360Q8—48no—
361Q8—88no—
362Q8—84no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—88no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—164no—
385C2×C4#38612yes—
386C2×C4#38612yes—
387C2×C4#38612yes—
388C2×C4C4×Q1614yes—
389C2×C4#44114yes—
390C2×C4#45114yes—
391C2×C4#45214yes—
392C2×C4#46212yes—
393C2×C4#46412yes—
394C2×C4#47112yes—
395C2×C4#47312yes—
396C2×C4C22×C22⋊C411yescharacteristic
397C2×C4C2×C4×D411yescharacteristic
398C2×C4C2×C4×D411yescharacteristic
399C2×C4C2×#12011yescharacteristic
400C2×C4C2×#12111yescharacteristic
401C2×C4C2×#12311yescharacteristic
402C2×C4C22×Q1611yescharacteristic
403C2×C4C2×#13511yescharacteristic
404C2×C4C2×#13711yescharacteristic
405C2×C4C2×#13711yescharacteristic
406C2×C4#57612yes—
407C2×C4#57712yes—
408C8—22no—
409C8—22no—
410C8—44no—
411C8—41no—
412C8—41no—
413C8—44no—
414C8—82no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22#187814yes—
422C22C2×#38611yescharacteristic
423C22C2×#46211yescharacteristic
424C22C2×#46411yescharacteristic
425C22#305711yescharacteristic
426C22#305811yescharacteristic
427C22C2×#47111yescharacteristic
428C22C2×#47311yescharacteristic
429C4—21no—
430C4—21no—
431C4—22no—
432C4—44no—
433C4—44no—
434C4—84no—
435C4—162no—
436C4#187814yes—
437C4#190814yes—
438C4#289712yes—
439C4C2×C4×Q1611yescharacteristic
440C4C2×#44111yescharacteristic
441C4C2×#45111yescharacteristic
442C4C2×#45211yescharacteristic
443C4#306212yes—
444C2—24no—
445C2#1920711yescharacteristic
446C2#1920811yescharacteristic
447C2#1921311yescharacteristic
448C2#1921411yescharacteristic
449C2C2×#187811yescharacteristic
450C2#2097511yescharacteristic
451C2C2×#190811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9226C2semidirect
7. #9218C2semidirect
13. #3209C4semidirect
80. #462C2×C4semidirect
153. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C2×#29602
#218602
C2×#19902
C2×#15492
#92262
#92182
#80432
#78722