Finite Group Databasebeta

← #190808

Subgroups — #190808

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#21924C212yesmaximal, complemented
3#8724C212yesmaximal, complemented
4#8592C211yesmaximal, characteristic, complemented
5#8590C212yesmaximal, complemented
6#8585C211yesmaximal, characteristic, complemented
7#8344C211yesmaximal, characteristic
8#8344C211yesmaximal, characteristic
9#8235C212yesmaximal, complemented
10#8231C211yesmaximal, characteristic, complemented
11C4×#331C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520C412yes—
14#2361—22no—
15#2361—22no—
16#2354—22no—
17D4×C4⋊C4—22no—
18#2326C412yes—
19#2326C412yes—
20#2311C412yes—
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C2211yescharacteristic
24C4×#127C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1924—22no—
28#1923—22no—
29#1917—22no—
30#1916—22no—
31#1842C2212yes—
32#1831C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1767C2212yes—
36C4×C4⋊C8C2212yes—
37C2×#333C2212yes—
38C2×#333C2212yes—
39C2×#331C2211yescharacteristic
40C2×#331C2211yescharacteristic
41C2×#331C2211yescharacteristic
42C2×#331C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47C22×#41C2×C412yes—
48#532—24no—
49#532—22no—
50#532—22no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×#127C2311yescharacteristic
53C2×#124C2×C412yes—
54C2×#121—22no—
55#520—48no—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C412yes—
65C2×#117—22no—
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2×C412yes—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81C2×C4×C8—22no—
82#391C2×C412yes—
83#391C2×C412yes—
84#390—22no—
85#390—22no—
86#390C2×C412yes—
87#390C2×C412yes—
88#389—22no—
89#389—22no—
90#388—24no—
91#388—24no—
92#386—24no—
93#385—24no—
94#385—24no—
95#384—22no—
96#384C2×C412yes—
97#384C2311yescharacteristic
98#384C2311yescharacteristic
99#380—24no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C22⋊C4D412yes—
105C4×C22⋊C4D412yes—
106#376C2×C411yescharacteristic
107#376C2×C411yescharacteristic
108C43C2311yescharacteristic
109C43C2311yescharacteristic
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#333—24no—
117#333—24no—
118#331—22no—
119#331—22no—
120#331—22no—
121#331—22no—
122C2×#41—24no—
123C2×#41—24no—
124C2×#41—44no—
125C2×#41—44no—
126C22×Q8C22×C412yes—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4C22×C412yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8#4114yes—
143#127—22no—
144#127#4112yes—
145#127#4112yes—
146#124—24no—
147#124—24no—
148#121—28no—
149#121—28no—
150C4×Q8—44no—
151C4×Q8—44no—
152C4×D4—28no—
153C4×D4—28no—
154C4×D4—44no—
155C4×D4—44no—
156C4×D4—48no—
157C4×D4—48no—
158C4×D4—82no—
159C4×D4—82no—
160C4×D4—82no—
161C4×D4—82no—
162#117—24no—
163#117—42no—
164#117—48no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4C22⋊C411yescharacteristic
187C2×C22⋊C4C22⋊C411yescharacteristic
188C2×C22⋊C4C22⋊C412yes—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2411yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—82no—
214C4⋊C8—82no—
215C22⋊C8—82no—
216C22⋊C8—82no—
217#97—82no—
218C4×C8—82no—
219#95—24no—
220#95—24no—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223#95C22⋊C412yes—
224#95C22⋊C412yes—
225C24—22no—
226C24C2×C22⋊C411yescharacteristic
227#41—416no—
228#41—416no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8#11714yes—
233C2×D4—28no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4#11714yes—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—28no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D412yes—
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—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—84no—
304C4⋊C4#10014yes—
305C22⋊C4—216no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—84no—
312C22⋊C4—84no—
313C42—21no—
314C42—21no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—41no—
328C42—41no—
329C42—44no—
330C42—44no—
331C42C2×#4112yes—
332C42C2×#4111yescharacteristic
333C42C2×#4111yescharacteristic
334C23—21no—
335C23—22no—
336C23—22no—
337C23—21no—
338C23—24no—
339C23—42no—
340C23—42no—
341C23—44no—
342C23—44no—
343C23#41312yes—
344C23C22×C22⋊C411yescharacteristic
345Q8—44no—
346Q8—44no—
347D4—44no—
348D4—44no—
349D4—416no—
350D4—44no—
351D4—44no—
352D4—44no—
353D4—44no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—416no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—164no—
388C2×C4—164no—
389C2×C4—164no—
390C2×C4C2×#10012yes—
391C2×C4C2×#10012yes—
392C2×C4C2×#10012yes—
393C2×C4C2×#11711yescharacteristic
394C2×C4C2×#11711yescharacteristic
395C2×C4#52512yes—
396C2×C4#52512yes—
397C2×C4#57612yes—
398C2×C4#57712yes—
399C2×C4#57712yes—
400C2×C4#57812yes—
401C8—82no—
402C8—82no—
403C8—82no—
404C8—82no—
405C22—24no—
406C22—22no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—48no—
411C22—44no—
412C22—82no—
413C22—82no—
414C22—84no—
415C22#228011yescharacteristic
416C22C22×#10011yescharacteristic
417C22C2×#41311yescharacteristic
418C22#289712yes—
419C22#289812yes—
420C4—22no—
421C4—22no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—42no—
426C4—41no—
427C4—41no—
428C4—42no—
429C4—42no—
430C4—44no—
431C4—82no—
432C4—82no—
433C4#311912yes—
434C4#312012yes—
435C4#312412yes—
436C4#312512yes—
437C2—42no—
438C2—84no—
439C2#1777011yescharacteristic
440C2#1777311yescharacteristic
441C2#1777512yes—
442C2#2072611yescharacteristic
443C2#2074612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21924C2semidirect
3. #8724C2semidirect
4. #8592C2semidirect
5. #8590C2semidirect
6. #8585C2semidirect
9. #8235C2semidirect
10. #8231C2semidirect
11. C4×#331C2semidirect

Maximal subgroups

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

subgroupindex
#342072
#219242
#87242
#85922
#85902
#85852
#83442
#82352
#82312
C4×#3312