Finite Group Databasebeta

← #189535

Subgroups — #189535

Order 512.

1055 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2976C211yesmaximal, characteristic
2#21939C211yesmaximal, characteristic, complemented
3#21938C211yesmaximal, characteristic, complemented
4#21874C212yesmaximal
5#8610C214yesmaximal, complemented
6#8608C214yesmaximal, complemented
7#7983C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9#2976—24no—
10#2976—24no—
11C2×#436C412yescomplemented
12C2×#435C412yescomplemented
13C2×#435C2212yes—
14C2×C8×D4C2212yes—
15C2×#433C412yescomplemented
16C2×#408C412yescomplemented
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2280C2212yes—
20C2×#380C2212yes—
21#1996—22no—
22#1964—22no—
23#1898—24no—
24#1854—24no—
25#1853—24no—
26#1808C2212yes—
27#1807C2212yes—
28#1807C2212yes—
29#1807C2212yes—
30#1799C2218yes—
31#1289C2218yes—
32C22×#41C2×C411yescharacteristic
33C24×C4C2311yescharacteristic
34C22×M16C2×C411yescharacteristic
35C22×M16C2×C411yescharacteristic
36C22×M16C2×C412yescomplemented
37C23×C8C2×C412yescomplemented
38C23×C8C2311yescharacteristic
39C23×C8C2311yescharacteristic
40#525—22no—
41#525—22no—
42#525—22no—
43#525—22no—
44C2×#123Q812yes—
45C2×C22⋊Q8D411yescharacteristic
46C2×C22⋊Q8Q811yescharacteristic
47C2×#121D411yescharacteristic
48C2×#121Q811yescharacteristic
49C2×#120D412yes—
50C2×C4×D4C2×C412yes—
51C2×C4×D4C2312yes—
52C2×#117—22no—
53C2×#117C2×C411yescharacteristic
54C22×C4⋊C4—22no—
55C22×C22⋊C4—22no—
56#436—28no—
57#435—24no—
58#435—24no—
59#435—28no—
60C8×D4—24no—
61C8×D4—24no—
62#433—28no—
63C2×#107—22no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—22no—
67C2×C4⋊C8C2×C412yescomplemented
68C2×C4⋊C8C2×C412yescomplemented
69C2×#99—28no—
70#408—28no—
71#407—22no—
72#407—22no—
73C2×C22⋊C8C2×C412yes—
74C2×C22⋊C8C2×C412yes—
75C2×C22⋊C8C2×C412yes—
76C2×C22⋊C8C2×C412yescomplemented
77C2×C22⋊C8C2×C412yescomplemented
78C2×C22⋊C8D414yes—
79C2×C22⋊C8D414yes—
80C2×C22⋊C8C2312yes—
81C2×#97C2×C412yescomplemented
82C2×C4×C8C2×C412yescomplemented
83#381—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380C2314yes—
89#380C2314yes—
90#379—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C2×#95—22no—
94#342—28no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99C25C2×D411yescharacteristic
100C2×#41—22no—
101C2×#41—22no—
102C2×#41—81no—
103C2×#41—81no—
104C22×Q8C4⋊C411yescharacteristic
105C22×D4C4⋊C412yes—
106C22×D4C4⋊C412yes—
107C22×D4C4⋊C411yescharacteristic
108C22×D4C2×D412yes—
109C23×C4—42no—
110C23×C4—81no—
111C23×C4—81no—
112C23×C4C22×C412yes—
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×Q811yescharacteristic
119C23×C4C2×Q811yescharacteristic
120C23×C4#4112yes—
121C23×C4#4112yes—
122C23×C4C2411yescharacteristic
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—24no—
126C2×M16—81no—
127C2×M16—81no—
128C2×M16—82no—
129C22×C8—24no—
130C22×C8—81no—
131C22×C8—81no—
132C22×C8—82no—
133C22×C8C22×C412yes—
134C22×C8C22×C412yes—
135C22×C8C22×C412yes—
136C22×C8C22×C412yes—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141#123—24no—
142#123—24no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147#121—22no—
148#121—22no—
149#121—22no—
150#121—22no—
151#120—24no—
152#120—24no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159#117—22no—
160#117—22no—
161#117—82no—
162#117—82no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C4⋊C411yescharacteristic
169C2×C4⋊C4C4⋊C411yescharacteristic
170C2×C4⋊C4C4⋊C412yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—84no—
178C2×C22⋊C4C4⋊C412yes—
179C2×C22⋊C4C4⋊C412yes—
180C2×C22⋊C4C4⋊C412yes—
181C2×C22⋊C4C2×D412yes—
182C2×C22⋊C4C2×Q812yes—
183C2×C42—42no—
184C2×C42C22×C412yes—
185#107—82no—
186#106—82no—
187#106—82no—
188#106—82no—
189C4⋊C8—24no—
190C4⋊C8—24no—
191#99—88no—
192C22⋊C8—24no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C22⋊C8C22⋊C418yes—
198C22⋊C8C22⋊C418yes—
199#97—24no—
200C4×C8—24no—
201#95—24no—
202#95—24no—
203#95—42no—
204#95—42no—
205C24—24no—
206C24—81no—
207C24—81no—
208C24C2×C4⋊C412yes—
209C24C2×C4⋊C411yescharacteristic
210C24C2×C4⋊C411yescharacteristic
211C24#12112yes—
212C24#12112yes—
213C24C22×D411yescharacteristic
214#41—82no—
215#41—82no—
216#41—84no—
217C2×Q8—22no—
218C2×Q8—81no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—28no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—22no—
225C2×D4—82no—
226C2×D4—81no—
227C2×D4—82no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—84no—
250C22×C4—84no—
251C22×C4—84no—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C4⋊C412yes—
255C22×C4C2×C4⋊C412yes—
256C22×C4C2×C4⋊C412yes—
257C22×C4C2×C4⋊C411yescharacteristic
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4#11712yes—
260C22×C4#11712yes—
261C22×C4#12112yes—
262C22×C4#12112yes—
263C22×C4C22⋊Q812yes—
264C22×C4C22⋊Q812yes—
265C22×C4C22⋊Q812yes—
266C22×C4C22⋊Q812yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273M16—84no—
274M16—82no—
275M16—82no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—84no—
287C2×C8C2×C22⋊C414yes—
288C2×C8C2×C22⋊C414yes—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—84no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—24no—
306C22⋊C4—44no—
307C22⋊C4—48no—
308C22⋊C4—44no—
309C22⋊C4—84no—
310C22⋊C4—84no—
311C22⋊C4—84no—
312C42—22no—
313C42—22no—
314C42—84no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—24no—
320C23—22no—
321C23—24no—
322C23—42no—
323C23—41no—
324C23—42no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—88no—
335C23#38012yes—
336C23#38012yes—
337C23C22×C4⋊C411yescharacteristic
338C23C2×#12111yescharacteristic
339C23C2×#12111yescharacteristic
340C23#57714yes—
341Q8—82no—
342D4—44no—
343D4—44no—
344D4—82no—
345D4—84no—
346D4—84no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—168no—
386C2×C4#38012yes—
387C2×C4#38012yes—
388C2×C4C22×C22⋊C411yescharacteristic
389C2×C4C2×#11711yescharacteristic
390C2×C4C2×C22⋊Q811yescharacteristic
391C2×C4C2×C22⋊Q811yescharacteristic
392C8—24no—
393C8—24no—
394C8—42no—
395C8—42no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—44no—
409C22—44no—
410C22—44no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—84no—
415C22—82no—
416C22—82no—
417C22—84no—
418C22C2×#38011yescharacteristic
419C22#291212yes—
420C22#305712yes—
421C22#306012yes—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—22no—
426C4—22no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—164no—
431C4—164no—
432C4#288311yescharacteristic
433C4#288311yescharacteristic
434C2—22no—
435C2—22no—
436C2—22no—
437C2—42no—
438C2—42no—
439C2—82no—
440C2#2063311yescharacteristic
441C2#2066211yescharacteristic
442C2#2067111yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21939C2semidirect
3. #21938C2semidirect
5. #8610C2semidirect
6. #8608C2semidirect
7. #7983C2semidirect
11. C2×#436C4semidirect
12. C2×#435C4semidirect
15. C2×#433C4semidirect
16. C2×#408C4semidirect
36. C22×M16C2×C4semidirect
37. C23×C8C2×C4semidirect
67. C2×C4⋊C8C2×C4semidirect
68. C2×C4⋊C8C2×C4semidirect
76. C2×C22⋊C8C2×C4semidirect
77. C2×C22⋊C8C2×C4semidirect
81. C2×#97C2×C4semidirect
82. C2×C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29762
#219392
#219382
#218742
#86102
#86082
#79832