Finite Group Databasebeta

← #207524

Subgroups — #207524

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#540C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic, complemented
3#20547C211yesmaximal, characteristic, complemented
4#9867C212yesmaximal, complemented
5#9678C212yesmaximal, complemented
6#8940C212yesmaximal, complemented
7#8939C212yesmaximal, complemented
8#8174C212yesmaximal, complemented
9#8172C212yesmaximal, complemented
10C2×#540C411yescharacteristic
11C2×#431C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2335—22no—
14#2333—22no—
15#2330C411yescharacteristic
16#2328C411yescharacteristic
17#2324C412yes—
18#2319C412yes—
19#2314C411yescharacteristic
20C4×#127C2211yescharacteristic
21C4×#125C2211yescharacteristic
22C4×#123—22no—
23C4×#121—22no—
24C4×#121C2212yes—
25C42×D4C2212yes—
26C4×#117C2211yescharacteristic
27#1924—22no—
28#1917—22no—
29#1770—22no—
30#1767C2212yes—
31#1767C2212yes—
32C4×C4⋊C8C2212yes—
33#1752—22no—
34C4×C22⋊C8C2212yes—
35#1744C2211yescharacteristic
36C4×#97C2211yescharacteristic
37C2×D4⋊C8C2212yes—
38C2×D4⋊C8C2212yes—
39C2×D4⋊C8C2212yes—
40C2×D4⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C23×C8—21no—
46#540—28no—
47C2×#127C2×C411yescharacteristic
48C2×#125C2×C411yescharacteristic
49C2×#123—22no—
50C2×#121—22no—
51C2×#121C2×C412yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2312yes—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C22×C42—22no—
60C22×C42C2311yescharacteristic
61#431—24no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—24no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—22no—
69C2×#97—22no—
70C2×#97—22no—
71C2×#97—22no—
72C2×C4×C8—21no—
73C2×C4×C8C2312yes—
74C2×C4×C8C2312yes—
75#390—22no—
76#390—22no—
77#390C2×C412yes—
78#390C2×C412yes—
79#389—22no—
80#389C2×C412yes—
81#389C2×C412yes—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#386—22no—
91#386—22no—
92#386—22no—
93#386—22no—
94#385—22no—
95#384C2×C411yescharacteristic
96#382C2×C411yescharacteristic
97#381—22no—
98#380—22no—
99#380C2×C412yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4C2312yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4D412yes—
112#376C2×C411yescharacteristic
113#376C2×C411yescharacteristic
114C43C2311yescharacteristic
115C43C2311yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121D4⋊C8—24no—
122D4⋊C8—24no—
123D4⋊C8—24no—
124D4⋊C8—24no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4C22×C412yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C22×C411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—24no—
138C22×C8—22no—
139C22×C8—82no—
140#127—24no—
141#125—24no—
142#123—216no—
143#121—216no—
144#121—24no—
145#121M1618yes—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—42no—
153C4×D4—42no—
154C4×D4—82no—
155C4×D4—82no—
156C4×D4—82no—
157C4×D4—82no—
158#117—24no—
159#117—41no—
160#117—41no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C412yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—82no—
187C2×C42—82no—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2411yescharacteristic
200C4⋊C8—24no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—44no—
204C4⋊C8#4114yes—
205C4⋊C8#4114yes—
206C22⋊C8—24no—
207C22⋊C8—44no—
208#97—44no—
209#97—82no—
210#97—82no—
211C4×C8—24no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95C22⋊C412yes—
224#95C22⋊C412yes—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227C24—22no—
228C24C2×C22⋊C411yescharacteristic
229C2×D4—28no—
230C2×D4—28no—
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—82no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4—82no—
240C2×D4C2×M1614yes—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×M1614yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
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—84no—
289C4⋊C4—28no—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—42no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4C2×M1614yes—
301C22⋊C4—28no—
302C22⋊C4—24no—
303C22⋊C4—28no—
304C22⋊C4—42no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C42—22no—
314C42—21no—
315C42—21no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—82no—
332C42—82no—
333C42C2×#4112yes—
334C42C2×#4112yes—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—21no—
339C23—24no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23#40712yes—
347C23C22×C22⋊C411yescharacteristic
348D4—24no—
349D4—24no—
350D4—42no—
351D4—42no—
352D4—88no—
353D4—88no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
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—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—88no—
389C2×C4—88no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#40712yes—
395C2×C4C4×#4112yes—
396C2×C4#52511yescharacteristic
397C2×C4#52511yescharacteristic
398C2×C4#53212yes—
399C2×C4C22×M1612yes—
400C2×C4#56811yescharacteristic
401C2×C4#56811yescharacteristic
402C2×C4#57612yes—
403C2×C4#57712yes—
404C8—44no—
405C8—44no—
406C8—44no—
407C8—82no—
408C22—24no—
409C22—22no—
410C22—42no—
411C22—41no—
412C22—41no—
413C22—44no—
414C22—44no—
415C22—44no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22#229711yescharacteristic
421C22C2×#40711yescharacteristic
422C22#287911yescharacteristic
423C22#289712yes—
424C22#290112yes—
425C4—22no—
426C4—22no—
427C4—42no—
428C4—42no—
429C4—44no—
430C4—42no—
431C4—42no—
432C4—41no—
433C4—41no—
434C4—82no—
435C4—82no—
436C4#297012yes—
437C4#297012yes—
438C4#311912yes—
439C4#312211yescharacteristic
440C4#312611yescharacteristic
441C2—42no—
442C2—84no—
443C2#1745312yes—
444C2#1750512yes—
445C2#2137911yescharacteristic
446C2#2138411yescharacteristic
447C2#2139911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#540C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21431C2semidirect
3. #20547C2semidirect
4. #9867C2semidirect
5. #9678C2semidirect
6. #8940C2semidirect
7. #8939C2semidirect
8. #8174C2semidirect
9. #8172C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5402
#214312
#205472
#98672
#96782
#89402
#89392
#81742
#81722