Finite Group Databasebeta

← #224915

Subgroups — #224915

Order 512.

907 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic, complemented
3#22722C211yesmaximal, characteristic, complemented
4#9710C212yesmaximal
5#9704C212yesmaximal
6#9694C211yesmaximal, characteristic, complemented
7#9692C211yesmaximal, characteristic, complemented
8#8591C212yesmaximal, complemented
9#8585C211yesmaximal, characteristic
10#8233C212yesmaximal, complemented
11C4×#331C211yesmaximal, characteristic
12C2×#556—21no—
13C2×#446—21no—
14#2868C2211yescharacteristic
15#2858C2212yes—
16#2547—21no—
17#2545—22no—
18#2538—21no—
19#2538C412yes—
20#2512—22no—
21#2512C414yes—
22#2354—21no—
23#2354C412yes—
24C4×C4⋊Q8C2212yes—
25C4×#127C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#1984—21no—
28#1940—22no—
29#1939—21no—
30#1917—22no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1842C2212yes—
34#1831C2212yes—
35#1770—21no—
36#1767C2212yes—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#332C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#331C2211yescharacteristic
45C2×#331C2211yescharacteristic
46C2×#331C2212yes—
47#1296C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2212yes—
50#556—44no—
51#556—44no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2312yes—
55C2×#127C2311yescharacteristic
56C2×#126—24no—
57C2×#124—22no—
58C2×#124—22no—
59C2×C22⋊Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C412yes—
67C2×#117—22no—
68#446—44no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×#97—22no—
82C2×#97—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#397—24no—
86#396—24no—
87#396—24no—
88#391—22no—
89#390—21no—
90#390—21no—
91#390—22no—
92#390—22no—
93#390C2×C412yes—
94#390C2×C414yes—
95#390C2×C414yes—
96#389—22no—
97#389C2×C414yes—
98#388—24no—
99#388—24no—
100#384—21no—
101#384—21no—
102#384C2×C412yes—
103#384C2×C412yes—
104#384C2311yescharacteristic
105#380—22no—
106#380—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C414yes—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D411yescharacteristic
115C43C2311yescharacteristic
116C43C2312yes—
117#336—22no—
118#336—22no—
119#336—21no—
120#336—22no—
121#336—22no—
122#336—21no—
123#332—24no—
124#332—24no—
125#332—42no—
126#332—42no—
127#331—22no—
128#331—22no—
129#331—42no—
130C22×Q8—22no—
131C22×Q8C22⋊C411yescharacteristic
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C22×C412yes—
135C23×C4—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—81no—
149C4⋊Q8—81no—
150#127—21no—
151#127—21no—
152#127—21no—
153#127—21no—
154#126—416no—
155#124—44no—
156#124—44no—
157#124—82no—
158#124—82no—
159C22⋊Q8—48no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—42no—
165C4×Q8—44no—
166C4×Q8—44no—
167C4×D4—44no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172#117—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C22×C412yes—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C42—24no—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2411yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—44no—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—81no—
224C4⋊C8—82no—
225C4⋊C8—81no—
226C4⋊C8#4112yes—
227C4⋊C8#4112yes—
228#97—44no—
229#97—82no—
230C4×C8—42no—
231C4×C8—42no—
232#95—24no—
233#95—24no—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237C24—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—48no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C22⋊C4—48no—
318C22⋊C4—44no—
319C22⋊C4—48no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—21no—
330C42—21no—
331C42—22no—
332C42—22no—
333C42—21no—
334C42—21no—
335C42—22no—
336C42—22no—
337C42—41no—
338C42—42no—
339C42—42no—
340C42—48no—
341C42AGL(1,ℤ/8)12yes—
342C42AGL(1,ℤ/8)12yes—
343C42AGL(1,ℤ/8)12yes—
344C42#13712yes—
345C42C2×#4111yescharacteristic
346C42C2×#4111yescharacteristic
347C42H(5,𝔽2)11yescharacteristic
348C422−1+411yescharacteristic
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23C22×C22⋊C411yescharacteristic
357Q8—44no—
358Q8—82no—
359Q8—82no—
360Q8—82no—
361Q8—82no—
362D4—82no—
363D4—82no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—81no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—88no—
400C2×C4#41312yes—
401C2×C4#41712yes—
402C2×C4#41712yes—
403C2×C4#41812yes—
404C2×C4#41812yes—
405C2×C4C4×#4111yescharacteristic
406C2×C4#52011yescharacteristic
407C2×C4#54811yescharacteristic
408C2×C4#54811yescharacteristic
409C2×C4#55011yescharacteristic
410C2×C4#55011yescharacteristic
411C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
412C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
413C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
414C2×C4C2×#13711yescharacteristic
415C2×C4#57611yescharacteristic
416C2×C4#57711yescharacteristic
417C2×C4#57711yescharacteristic
418C2×C4#57811yescharacteristic
419C8—42no—
420C8—42no—
421C8—81no—
422C8—82no—
423C8—81no—
424C8—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22#234111yescharacteristic
433C22C2×#41311yescharacteristic
434C22#288911yescharacteristic
435C22C2×#41811yescharacteristic
436C22C2×#41711yescharacteristic
437C22#289711yescharacteristic
438C22#289811yescharacteristic
439C4—21no—
440C4—21no—
441C4—22no—
442C4—22no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—41no—
447C4—42no—
448C4—84no—
449C4—82no—
450C4#155714yes—
451C4#336611yescharacteristic
452C4#336711yescharacteristic
453C4#336811yescharacteristic
454C4#336911yescharacteristic
455C4#339711yescharacteristic
456C4#339711yescharacteristic
457C4#339811yescharacteristic
458C4#339911yescharacteristic
459C2—162no—
460C2C2×#155711yescharacteristic
461C2#1775911yescharacteristic
462C2#1776511yescharacteristic
463C2#1778111yescharacteristic
464C2#2253711yescharacteristic
465C2#2254411yescharacteristic
466C2#2254511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22825C2semidirect
3. #22722C2semidirect
6. #9694C2semidirect
7. #9692C2semidirect
8. #8591C2semidirect
10. #8233C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#228252
#227222
#97102
#97042
#96942
#96922
#85912
#85852
#82332
C4×#3312