Finite Group Databasebeta

← #493979

Subgroups — #493979

Order 512.

1022 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34209C211yesmaximal, characteristic
2#20730C212yesmaximal
3#12862C212yesmaximal
4#12804C212yesmaximal, complemented
5#12802C212yesmaximal, complemented
6#12700C212yesmaximal
7#12631C212yesmaximal, complemented
8C4×#356C212yesmaximal, complemented
9C22×C4⋊Q8C2211yescharacteristic
10C2×#519C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2921—22no—
13#2365—22no—
14#2365—22no—
15#2344—22no—
16#2344—22no—
17#2327C411yescharacteristic
18#2327C411yescharacteristic
19#2327C2211yescharacteristic
20#2327C2211yescharacteristic
21#2310C411yescharacteristic
22#2310C2211yescharacteristic
23C4×C4⋊Q8C411yescharacteristic
24C4×C4⋊Q8C411yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27#2285C411yescharacteristic
28C4×#117C411yescharacteristic
29C2×#356C2214yes—
30C2×#356C2214yes—
31#1822—22no—
32#1801—22no—
33#1801C2214yes—
34#1784C2214yes—
35C4×#101—22no—
36C4×#101C2214yes—
37#1394C2214yes—
38#1394C2214yes—
39C23×Q8C2×C411yescharacteristic
40C23×Q8C2311yescharacteristic
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48#519—44no—
49#519—44no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8D412yes—
60C2×C4×Q8C2311yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×#117—21no—
63C2×#117—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117D412yes—
69C2×#117D412yes—
70C22×C4⋊C4—22no—
71C22×C42C2×C411yescharacteristic
72#433—48no—
73#423—24no—
74C2×#101—24no—
75C2×#101C2318yes—
76#407—24no—
77C4×M16—24no—
78#391—22no—
79#391—22no—
80#391C2×C411yescharacteristic
81#391C2×C411yescharacteristic
82#391C2×C411yescharacteristic
83#391C2×C411yescharacteristic
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2311yescharacteristic
87#391C2311yescharacteristic
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#384C2×C411yescharacteristic
101#384C2×C411yescharacteristic
102#382C2×C411yescharacteristic
103#382C2×C411yescharacteristic
104#381—22no—
105#381—22no—
106#380—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116#376C2×C411yescharacteristic
117#376C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120#356—28no—
121#356—28no—
122#343—24no—
123#343—24no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8C22×C411yescharacteristic
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C411yescharacteristic
136C22×Q8C2×D412yes—
137C22×Q8C2411yescharacteristic
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D412yes—
142C2×M16—28no—
143C4⋊Q8—216no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8#4114yes—
151C4⋊Q8#4114yes—
152C4×Q8—42no—
153C4×Q8—42no—
154C4×Q8—42no—
155C4×Q8—42no—
156C4×Q8—44no—
157C4×Q8—44no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—44no—
163C4×Q8—44no—
164#117—24no—
165#117—28no—
166#117—42no—
167#117—44no—
168#117—44no—
169#117—42no—
170#117—44no—
171#117—42no—
172C2×C4⋊C4—216no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4C22⋊C411yescharacteristic
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C411yescharacteristic
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C22⋊C412yes—
212C2×C42C22⋊C412yes—
213C2×C42C22⋊C412yes—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C4⋊C8—48no—
223#101—28no—
224#101—28no—
225#101—48no—
226C22⋊C8—48no—
227#97—44no—
228C4×C8—44no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C24C2×C22⋊C411yescharacteristic
244C2×Q8—24no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—48no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8#11712yes—
267C2×Q8#11712yes—
268C2×Q8C2×#4112yes—
269C2×Q8C2×#4112yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D412yes—
301M16—48no—
302M16—48no—
303C2×C8—48no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—28no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C22⋊C4—24no—
326C22⋊C4—22no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4#9914yes—
330C42—24no—
331C42—21no—
332C42—21no—
333C42—24no—
334C42—24no—
335C42—22no—
336C42—22no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—41no—
341C42—41no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42#11712yes—
352C42#11712yes—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23C2×#9912yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—48no—
362Q8—42no—
363Q8—42no—
364Q8—42no—
365Q8—42no—
366Q8—42no—
367Q8—42no—
368Q8—48no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
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—41no—
391C2×C4—41no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—48no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4C2×#9912yes—
416C2×C4C2×#9912yes—
417C2×C4#41012yes—
418C2×C4C2×#11711yescharacteristic
419C2×C4C2×#11711yescharacteristic
420C2×C4#52514yes—
421C8—88no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—42no—
428C22—42no—
429C22#228011yescharacteristic
430C22C22×#9911yescharacteristic
431C22C2×#41011yescharacteristic
432C4—24no—
433C4—22no—
434C4—22no—
435C4—24no—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—42no—
440C4—41no—
441C4—41no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—81no—
451C2—21no—
452C2—21no—
453C2—42no—
454C2#2072211yescharacteristic
455C2#3337211yescharacteristic
456C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12804C2semidirect
5. #12802C2semidirect
7. #12631C2semidirect
8. C4×#356C2semidirect

Maximal subgroups

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

subgroupindex
#342092
#207302
#128622
#128042
#128022
#127002
#126312
C4×#3562