Finite Group Databasebeta

← #235778

Subgroups — #235778

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic
2C22×#487C211yesmaximal, characteristic, complemented
3#35289C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic, complemented
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1492C211yesmaximal, characteristic
7#10058C218yesmaximal, complemented
8#7890C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3227—24no—
13#3227—24no—
14C2×#498—21no—
15C2×#487—24no—
16C2×#487—24no—
17C2×#487—22no—
18C2×#487C2214yes—
19C2×#486—21no—
20C2×#481—22no—
21C2×#476C2214yes—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C22×#103—22no—
25C22×C4×C8C2211yescharacteristic
26#2862—21no—
27#2858—21no—
28#2640—22no—
29#2563—22no—
30#2547C2214yes—
31#2504—22no—
32#2436—22no—
33#2311—22no—
34C4×#117C2211yescharacteristic
35#1988—22no—
36#1987C2214yes—
37#1736—22no—
38#1549—24no—
39C2×#332C2214yes—
40#1492—24no—
41C2×D4⋊C8C2214yes—
42#1279C2214yes—
43C22×#41—21no—
44C23×Q8—21no—
45C23×Q8—21no—
46C22×M16—21no—
47C23×C8—21no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8D414yes—
65C2×C4⋊Q8D411yescharacteristic
66C2×C4⋊Q8D412yes—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×#127D411yescharacteristic
70C2×#124—24no—
71C2×#124—21no—
72C2×#121—21no—
73C2×#121D412yes—
74C2×C4×D4—21no—
75C2×C4×D4C2312yes—
76C2×#117D412yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C42C2311yescharacteristic
82#498—28no—
83#487—216no—
84#487—28no—
85#487—28no—
86#486—28no—
87#481—216no—
88#476—28no—
89#476—28no—
90#431—22no—
91#431D414yes—
92C2×#106—24no—
93#423—22no—
94#423—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×#103—216no—
99C2×#103—28no—
100C2×#102—24no—
101C2×C22⋊C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—22no—
104C2×C4×C8—21no—
105C2×C4×C8D414yes—
106C2×C4×C8C2312yes—
107#397—28no—
108#393—28no—
109#391—24no—
110#391—24no—
111#390—24no—
112#390—24no—
113#386—28no—
114#382—24no—
115#380—28no—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4C2312yes—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376—21no—
121C43—21no—
122#340—28no—
123#332—28no—
124D4⋊C8—28no—
125C2×#41—42no—
126C2×#41—42no—
127C2×#41—44no—
128C2×#41—44no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—24no—
133C22×Q8—22no—
134C22×Q8—48no—
135C22×Q8—41no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—84no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C2×D412yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—41no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—42no—
151C2×M16—44no—
152C22×C8—24no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—44no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8C2×D414yes—
179#127—22no—
180#127—22no—
181#124—42no—
182#124—42no—
183#124—88no—
184#121—24no—
185#121—24no—
186#121—42no—
187#121—42no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—42no—
193C4×D4—42no—
194#117—24no—
195#117—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—216no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230#106—216no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—42no—
235C4⋊C8C2×D414yes—
236#103—232no—
237#102—216no—
238C22⋊C8—24no—
239C4×C8—24no—
240C4×C8—24no—
241C4×C8—22no—
242C4×C8C2×D414yes—
243#95—216no—
244#95—22no—
245#95—22no—
246C24—22no—
247C24#12011yescharacteristic
248#41—88no—
249#41—84no—
250#41—84no—
251C2×Q8—24no—
252C2×Q8—48no—
253C2×Q8—416no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—48no—
258C2×Q8—42no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—84no—
263C2×Q8—88no—
264C2×Q8—88no—
265C2×Q8—88no—
266C2×Q8—84no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—28no—
270C2×D4—44no—
271C2×D4—48no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—44no—
275C2×D4—44no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4#12012yes—
291C22×C4#12012yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12712yes—
299C22×C4#13514yes—
300C22×C4#13714yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304M16—84no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—44no—
308C2×C8—48no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—44no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—816no—
334C42—44no—
335C42—44no—
336C42—44no—
337C42#12012yes—
338C42#12012yes—
339C42#12112yes—
340C42#12112yes—
341C42#12112yes—
342C42#12712yes—
343C42C22×D411yescharacteristic
344C42C22×D411yescharacteristic
345C42C22×D411yescharacteristic
346C42C2×#4111yescharacteristic
347C23—22no—
348C23—22no—
349C23—42no—
350C23—44no—
351C23—44no—
352C23—44no—
353C23#39212yes—
354C23#45214yes—
355C23C2×#12011yescharacteristic
356Q8—48no—
357Q8—84no—
358Q8—88no—
359Q8—84no—
360Q8—88no—
361Q8—84no—
362Q8—84no—
363D4—44no—
364D4—44no—
365D4—88no—
366D4—84no—
367D4—84no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—88no—
383C2×C4—88no—
384C2×C4—84no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4#39212yes—
389C2×C4#39212yes—
390C2×C4#39212yes—
391C2×C4#45214yes—
392C2×C4#45414yes—
393C2×C4#46418yes—
394C2×C4#49514yes—
395C2×C4#49714yes—
396C2×C4C2×#12011yescharacteristic
397C2×C4C2×#12011yescharacteristic
398C2×C4C2×#12111yescharacteristic
399C2×C4C2×#12111yescharacteristic
400C2×C4C2×#12111yescharacteristic
401C2×C4C2×#12711yescharacteristic
402C2×C4C2×#13512yes—
403C2×C4C2×#13712yes—
404C2×C4#57714yes—
405C8—44no—
406C8—44no—
407C8—82no—
408C22—22no—
409C22—22no—
410C22—28no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—88no—
415C22#201514yes—
416C22C2×#39211yescharacteristic
417C22C2×#45212yes—
418C22#300912yes—
419C22C2×#45412yes—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—84no—
424C4—84no—
425C4—82no—
426C4—84no—
427C4#200314yes—
428C4#201514yes—
429C4#301012yes—
430C4C2×#46412yes—
431C4#305712yes—
432C4C2×#49511yescharacteristic
433C4C2×#49711yescharacteristic
434C2—24no—
435C2—162no—
436C2#1864714yes—
437C2#2338811yescharacteristic
438C2C2×#200311yescharacteristic
439C2C2×#201511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×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
2. C22×#487C2semidirect
4. #21313C2semidirect
5. C2×#1549C2semidirect
7. #10058C2semidirect
8. #7890C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
C22×#4872
#352892
#213132
C2×#15492
C2×#14922
#100582
#78902