Finite Group Databasebeta

← #223258

Subgroups — #223258

Order 512.

943 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic
3#22799C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9140C212yesmaximal, complemented
7#9138C212yesmaximal, complemented
8#8724C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10C4×#332C211yesmaximal, characteristic, complemented
11#8040C212yesmaximal, complemented
12C2×#554—21no—
13C2×#432C2212yes—
14C2×#432C2212yes—
15#2868C2211yescharacteristic
16#2857C2212yes—
17#2567—21no—
18#2552—21no—
19#2552C412yes—
20#2502—22no—
21#2440—22no—
22#2440C414yes—
23D4×C4⋊C4—21no—
24D4×C4⋊C4C412yes—
25C4×C4⋊Q8C2211yescharacteristic
26C2×#384C2212yes—
27C42×D4C2211yescharacteristic
28#1954—21no—
29#1954—21no—
30#1924—22no—
31#1924—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1797C2212yes—
35#1797C2212yes—
36#1770—21no—
37#1770—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42C2×#332C2212yes—
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50#554—44no—
51#554—44no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×C4⋊Q8C2311yescharacteristic
56C2×#125—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—24no—
59C2×C4×Q8—21no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2×C414yes—
67C22×C42C2312yes—
68#432—24no—
69#432—24no—
70#432—42no—
71#432—42no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—22no—
87C2×#97—22no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#398—24no—
91#398—24no—
92#395—24no—
93#391—21no—
94#391—22no—
95#389—22no—
96#389—22no—
97#389—21no—
98#389—22no—
99#389C2×C412yes—
100#389C2×C414yes—
101#385—24no—
102#385—24no—
103#384—22no—
104#384—21no—
105#384—21no—
106#384—22no—
107#384C2×C412yes—
108#384C2×C414yes—
109#384C2311yescharacteristic
110#384C2312yes—
111#380—22no—
112#380C2×C414yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
121C43C2311yescharacteristic
122#336—22no—
123#336—22no—
124#336—21no—
125#336—21no—
126#332—22no—
127#332—22no—
128#332—42no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×D4C22⋊C411yescharacteristic
132C23×C4—24no—
133C23×C4C22×C412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—81no—
149C4⋊Q8—81no—
150C4⋊Q8—82no—
151#125—48no—
152C22⋊Q8—416no—
153C22⋊Q8—44no—
154C22⋊Q8—44no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×D4—24no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—48no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—84no—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C414yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2411yescharacteristic
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—81no—
219C4⋊C8—81no—
220C4⋊C8#4112yes—
221C4⋊C8#4112yes—
222C22⋊C8—44no—
223C22⋊C8—44no—
224C22⋊C8—82no—
225C22⋊C8—82no—
226#97—44no—
227#97—44no—
228C4×C8—42no—
229C4×C8—42no—
230#95—22no—
231#95—22no—
232#95—24no—
233#95—24no—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237C24C2×C22⋊C412yes—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—44no—
245C2×Q8—82no—
246C2×Q8—82no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4#10114yes—
250C22×C4—24no—
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—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—48no—
270C22×C4—84no—
271C22×C4—84no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—42no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—48no—
308C4⋊C4—48no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—48no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—41no—
341C42—44no—
342C42AGL(1,ℤ/8)12yes—
343C42AGL(1,ℤ/8)12yes—
344C42#13712yes—
345C42#13712yes—
346C42C2×#4111yescharacteristic
347C42C2×#4111yescharacteristic
348C42H(5,𝔽2)11yescharacteristic
349C422−1+411yescharacteristic
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23C2×#10114yes—
357C23C22×C22⋊C411yescharacteristic
358Q8—81no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362Q8—164no—
363D4—28no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—168no—
402C2×C4C2×#10112yes—
403C2×C4#41812yes—
404C2×C4#41812yes—
405C2×C4#41912yes—
406C2×C4#41912yes—
407C2×C4C4×#4111yescharacteristic
408C2×C4#52011yescharacteristic
409C2×C4#54611yescharacteristic
410C2×C4#54611yescharacteristic
411C2×C4#54711yescharacteristic
412C2×C4#54711yescharacteristic
413C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
414C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
415C2×C4C2×#13711yescharacteristic
416C2×C4C2×#13711yescharacteristic
417C2×C4#57611yescharacteristic
418C2×C4#57711yescharacteristic
419C2×C4#57711yescharacteristic
420C2×C4#57811yescharacteristic
421C8—42no—
422C8—42no—
423C8—82no—
424C8—82no—
425C8—81no—
426C8—81no—
427C22—24no—
428C22—24no—
429C22—24no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22#233911yescharacteristic
435C22C22×#10111yescharacteristic
436C22#288811yescharacteristic
437C22C2×#41811yescharacteristic
438C22C2×#41911yescharacteristic
439C22#289711yescharacteristic
440C22#289811yescharacteristic
441C4—22no—
442C4—22no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—82no—
450C4—84no—
451C4—82no—
452C4#155614yes—
453C4#333011yescharacteristic
454C4#333111yescharacteristic
455C4#333211yescharacteristic
456C4#333311yescharacteristic
457C4#335311yescharacteristic
458C4#335411yescharacteristic
459C4#335611yescharacteristic
460C4#335711yescharacteristic
461C2—44no—
462C2C2×#155611yescharacteristic
463C2#1775811yescharacteristic
464C2#1776011yescharacteristic
465C2#1777411yescharacteristic
466C2#2248111yescharacteristic
467C2#2250011yescharacteristic
468C2#2250211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9898C2semidirect
5. #9896C2semidirect
6. #9140C2semidirect
7. #9138C2semidirect
8. #8724C2semidirect
9. #8586C2semidirect
10. C4×#332C2semidirect
11. #8040C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#227992
#98982
#98962
#91402
#91382
#87242
#85862
C4×#3322
#80402