Finite Group Databasebeta

← #207644

Subgroups — #207644

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#540C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20473C211yesmaximal, characteristic, complemented
4#9876C211yesmaximal, characteristic, complemented
5#9875C211yesmaximal, characteristic, complemented
6#9648C211yesmaximal, characteristic, complemented
7#9644C211yesmaximal, characteristic, complemented
8#9074C212yesmaximal, complemented
9#9072C212yesmaximal, complemented
10#8174C212yesmaximal, complemented
11#8172C212yesmaximal, complemented
12C2×#540C411yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2335—22no—
16#2333—22no—
17#2330C411yescharacteristic
18#2328C411yescharacteristic
19#2324C412yes—
20#2319C412yes—
21#2314C411yescharacteristic
22C4×#127C2211yescharacteristic
23C4×#125C2211yescharacteristic
24C4×#123—22no—
25C4×#121—22no—
26C4×#121C2212yes—
27C42×D4C2212yes—
28C4×#117C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34#1770—22no—
35#1767C2212yes—
36#1752—22no—
37#1744C2211yescharacteristic
38#1744C2211yescharacteristic
39C4×#97C2211yescharacteristic
40C4×#97C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C23×C8—21no—
50#540—28no—
51C2×#127C2×C411yescharacteristic
52C2×#125C2×C411yescharacteristic
53C2×#123—22no—
54C2×#121—22no—
55C2×#121C2×C412yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2312yes—
61C2×#117C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—22no—
73C2×#97—22no—
74C2×#97—22no—
75C2×#97—22no—
76C2×C4×C8—21no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83#390—22no—
84#390—22no—
85#390C2×C412yes—
86#390C2×C412yes—
87#389—22no—
88#389C2×C412yes—
89#389C2×C412yes—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#385—22no—
103#384C2×C411yescharacteristic
104#382C2×C411yescharacteristic
105#381—22no—
106#380—22no—
107#380C2×C412yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4D412yes—
119C4×C22⋊C4D412yes—
120#376C2×C411yescharacteristic
121#376C2×C411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129D4⋊C8—24no—
130D4⋊C8—24no—
131D4⋊C8—24no—
132D4⋊C8—24no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C22×C412yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C22×C8—21no—
143C22×C8—24no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—44no—
150#127—24no—
151#125—24no—
152#123—216no—
153#121—216no—
154#121—24no—
155#121M1618yes—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—82no—
165C4×D4—82no—
166C4×D4—82no—
167C4×D4—82no—
168#117—24no—
169#117—42no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—82no—
198C2×C42—82no—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—24no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—82no—
217C22⋊C8—24no—
218C22⋊C8—82no—
219#97—44no—
220#97—44no—
221#97—82no—
222C4×C8—24no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8#4112yes—
228C4×C8#4112yes—
229C4×C8#4112yes—
230C4×C8#4112yes—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242C24—22no—
243C24C2×C22⋊C411yescharacteristic
244C2×D4—28no—
245C2×D4—28no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—82no—
252C2×D4—82no—
253C2×D4—82no—
254C2×D4—82no—
255C2×D4C2×M1614yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—28no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×M1614yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C2×C8—24no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—48no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C4⋊C4—28no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4C2×M1614yes—
319C22⋊C4—28no—
320C22⋊C4—24no—
321C22⋊C4—28no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—82no—
350C42—82no—
351C42C2×#4111yescharacteristic
352C42C2×#4111yescharacteristic
353C42C2×#4111yescharacteristic
354C42C2×#4111yescharacteristic
355C23—21no—
356C23—22no—
357C23—22no—
358C23—21no—
359C23—24no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23#40712yes—
367C23C22×C22⋊C411yescharacteristic
368D4—24no—
369D4—24no—
370D4—42no—
371D4—42no—
372D4—88no—
373D4—88no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—88no—
410C2×C4—88no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#40712yes—
416C2×C4C4×#4111yescharacteristic
417C2×C4C4×#4111yescharacteristic
418C2×C4#52511yescharacteristic
419C2×C4#52511yescharacteristic
420C2×C4#53211yescharacteristic
421C2×C4#53211yescharacteristic
422C2×C4C22×M1612yes—
423C2×C4#56812yes—
424C2×C4#57612yes—
425C2×C4#57712yes—
426C8—24no—
427C8—42no—
428C8—42no—
429C8—42no—
430C8—82no—
431C8—82no—
432C22—24no—
433C22—22no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22#229711yescharacteristic
445C22C2×#40711yescharacteristic
446C22#287911yescharacteristic
447C22#289712yes—
448C22#290112yes—
449C4—22no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—44no—
455C4—42no—
456C4—42no—
457C4—41no—
458C4—41no—
459C4—82no—
460C4—82no—
461C4#297012yes—
462C4#297012yes—
463C4#317411yescharacteristic
464C4#317611yescharacteristic
465C4#317912yes—
466C2—42no—
467C2—84no—
468C2#1745312yes—
469C2#1750512yes—
470C2#2137911yescharacteristic
471C2#2141411yescharacteristic
472C2#2141911yescharacteristic
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. #21926C2semidirect
3. #20473C2semidirect
4. #9876C2semidirect
5. #9875C2semidirect
6. #9648C2semidirect
7. #9644C2semidirect
8. #9074C2semidirect
9. #9072C2semidirect
10. #8174C2semidirect
11. #8172C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5402
#219262
#204732
#98762
#98752
#96482
#96442
#90742
#90722
#81742
#81722