Finite Group Databasebeta

← #194241

Subgroups — #194241

Order 512.

1236 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×#3226C211yesmaximal, characteristic
2#34367C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic, complemented
4#21811C211yesmaximal, characteristic, complemented
5C2×#1496C211yesmaximal, characteristic
6#8984C214yesmaximal, complemented
7#8981C214yesmaximal, complemented
8#8046C211yesmaximal, characteristic, complemented
9#7894C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3226—24no—
13#3226C418yescomplemented
14C2×#500—21no—
15C2×#486—21no—
16C2×#480—22no—
17C2×#474C2214yes—
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2377—22no—
22#2363—22no—
23#2354C2212yes—
24#2352—22no—
25D4×C4⋊C4C2212yes—
26#2337—22no—
27#2331—21no—
28#2328—21no—
29#2311—22no—
30#2285C2211yescharacteristic
31#2283C2211yescharacteristic
32#1990C2214yes—
33#1987—22no—
34#1981C2212yes—
35#1981C2212yes—
36#1960—22no—
37#1824—22no—
38#1787—22no—
39#1496—24no—
40C2×D4⋊C8C2214yes—
41#1296C2214yes—
42#1279C2212yes—
43#1279C2212yes—
44C22×#41—21no—
45C23×Q8—21no—
46#532—22no—
47#532—22no—
48#532C2×C414yes—
49#532D414yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yescomplemented
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8C2312yes—
59C2×#127D411yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121D412yes—
63C2×C4×D4—22no—
64C2×C4×D4—21no—
65C2×C4×D4C2312yes—
66C2×#117—21no—
67C2×#117—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C42C2311yescharacteristic
72#500—28no—
73#486—28no—
74#480—216no—
75#474—28no—
76#474C2×C4116yescomplemented
77#432—22no—
78#432—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431C2×C414yescomplemented
83C2×#107—24no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#103—24no—
88C2×#103—24no—
89C2×#102—24no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×#97—21no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96#391—24no—
97#390—24no—
98#390—22no—
99#389—22no—
100#388—24no—
101#388—24no—
102#387—24no—
103#386—24no—
104#386—24no—
105#386—24no—
106#385—24no—
107#384C2312yes—
108#383—21no—
109#382—21no—
110#382D412yes—
111#381—21no—
112#380—21no—
113#380—22no—
114#380—22no—
115#380D412yes—
116#380D412yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D412yes—
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C22⋊C4—24no—
123C4×C22⋊C4—21no—
124#376—21no—
125#340—28no—
126D4⋊C8—28no—
127C2×#41—24no—
128C2×#41—24no—
129C2×#41—84no—
130C22×Q8—21no—
131C22×Q8—24no—
132C22×Q8—22no—
133C22×Q8—41no—
134C22×Q8—42no—
135C22×Q8—84no—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C2×D412yes—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8C22⋊C414yes—
160C4⋊Q8C22×C414yes—
161#127—22no—
162#127C22⋊C414yes—
163#124—24no—
164#124—24no—
165#121—24no—
166#121—24no—
167#121—24no—
168#121C22⋊C418yes—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—84no—
174C4×D4C22×C414yes—
175C4×D4C2×D414yes—
176#117—84no—
177#117—82no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D414yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42C22×C412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42C2411yescharacteristic
214#107—216no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—42no—
219C4⋊C8C22×C414yes—
220#103—84no—
221#103—84no—
222#102—216no—
223C22⋊C8—24no—
224C22⋊C8—42no—
225C22⋊C8—42no—
226#97—42no—
227C4×C8—24no—
228C4×C8—22no—
229C4×C8—22no—
230#95—24no—
231#95—24no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95—21no—
239C24—22no—
240C24#12011yescharacteristic
241#41—88no—
242#41—88no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—42no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—82no—
250C2×Q8—88no—
251C2×Q8—84no—
252C2×Q8—88no—
253C2×D4—28no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—82no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4C2×C22⋊C414yes—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4C2×C22⋊C412yes—
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4#12312yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—44no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—44no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—84no—
315C4⋊C4C2×C22⋊C414yes—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—28no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C42—42no—
325C42—42no—
326C42—82no—
327C42—84no—
328C42C2×C22⋊C412yes—
329C42C4×D412yes—
330C42C4×D412yes—
331C42#12112yes—
332C42#12112yes—
333C42#12312yes—
334C42C23×C411yescharacteristic
335C42C22×D411yescharacteristic
336C42C2×#4111yescharacteristic
337C42C2×#4111yescharacteristic
338C23—22no—
339C23—21no—
340C23—21no—
341C23—22no—
342C23—24no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23#38612yes—
350C23C2×#12011yescharacteristic
351Q8—48no—
352Q8—84no—
353Q8—88no—
354Q8—82no—
355Q8—82no—
356D4—28no—
357D4—44no—
358D4—84no—
359D4—84no—
360D4—88no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—88no—
388C2×C4—84no—
389C2×C4—168no—
390C2×C4#38612yes—
391C2×C4#38612yes—
392C2×C4#38612yes—
393C2×C4C22×C22⋊C411yescharacteristic
394C2×C4C2×C4×D411yescharacteristic
395C2×C4C2×C4×D411yescharacteristic
396C2×C4C2×#12111yescharacteristic
397C2×C4C2×#12111yescharacteristic
398C2×C4C2×#12311yescharacteristic
399C2×C4#57614yes—
400C2×C4#57712yes—
401C2×C4#57812yes—
402C8—44no—
403C8—82no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—42no—
409C22—42no—
410C22—44no—
411C22—44no—
412C22—44no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22C2×#38611yescharacteristic
418C22#300812yes—
419C22#300911yescharacteristic
420C22#301111yescharacteristic
421C22#302212yes—
422C4—22no—
423C4—22no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—44no—
428C4—84no—
429C4—82no—
430C4—82no—
431C4—84no—
432C4#289812yes—
433C4#294812yes—
434C4#304112yes—
435C4#309712yes—
436C2—42no—
437C2—84no—
438C2#1896712yes—
439C2#1896912yes—
440C2#2091311yescharacteristic
441C2#2093411yescharacteristic
442C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
3. #21834C2semidirect
4. #21811C2semidirect
6. #8984C2semidirect
7. #8981C2semidirect
8. #8046C2semidirect
9. #7894C2semidirect
13. #3226C4semidirect
56. C2×C4⋊Q8C2×C4semidirect
76. #474C2×C4semidirect
82. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#343672
#218342
#218112
C2×#14962
#89842
#89812
#80462
#78942