Finite Group Databasebeta

← #222898

Subgroups — #222898

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C211yesmaximal, characteristic
2#22599C212yesmaximal, complemented
3#9878C212yesmaximal, complemented
4#9080C214yesmaximal
5#8701C212yesmaximal, complemented
6#8557C211yesmaximal, characteristic
7C4×#323C211yesmaximal, characteristic
8#7908C212yesmaximal, complemented
9C22×C22⋊Q8C411yescharacteristic
10C22×C4×Q8C411yescharacteristic
11C22×#117C411yescharacteristic
12C2×#431C2214yes—
13#2362—24no—
14#2362—22no—
15#2362—22no—
16C2×#391C2211yescharacteristic
17C2×#389C411yescharacteristic
18C2×#388C412yes—
19C2×#388C2212yes—
20C2×#387C412yes—
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1939—22no—
25#1916—24no—
26#1817C2214yes—
27#1797C2212yes—
28C8×C4⋊C4—22no—
29C4×C22⋊C8C2212yes—
30C2×#323C2212yes—
31C2×#323C2212yes—
32#1280C2212yes—
33#1280C2214yes—
34#1280C2214yes—
35#1280C2211yescharacteristic
36#1280C2211yescharacteristic
37C23×Q8C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8C818yescomplemented
43C2×C4×Q8—22no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—44no—
47C2×#117—21no—
48C2×#117—22no—
49C2×#117—21no—
50C2×#117—44no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2311yescharacteristic
61C22×C42C2×C412yes—
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C412yes—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#431—28no—
67#431—44no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—24no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2312yes—
72C2×C22⋊C8C2312yes—
73C2×C4×C8—24no—
74C2×C4×C8—22no—
75#391—22no—
76#391—22no—
77#389—22no—
78#389—22no—
79#388—24no—
80#388—24no—
81#388—24no—
82#388—24no—
83#387—24no—
84#387—24no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4D414yes—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C2×#95C2×C411yescharacteristic
93C2×#95C2×C412yes—
94C2×#95C2×C412yes—
95C2×#95C2×C412yes—
96C2×#95C2311yescharacteristic
97C2×#95C2312yes—
98#336—22no—
99#336—24no—
100#336—24no—
101#336—22no—
102#323—24no—
103#323—42no—
104C25C22×C411yescharacteristic
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8C2×C814yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4C2×C814yes—
117C23×C4C22×C412yes—
118C23×C4C22×C412yes—
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C412yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C412yes—
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—24no—
130C22⋊Q8—216no—
131C22⋊Q8—216no—
132C4×Q8—44no—
133C4×Q8—48no—
134C4×Q8—44no—
135#117—44no—
136#117—48no—
137#117—44no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C2×C818yes—
156C2×C4⋊C4C2×C814yes—
157C2×C4⋊C4M1614yes—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4C2×C818yes—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—42no—
181C2×C42—44no—
182C2×C42—42no—
183C2×C42—44no—
184C2×C42C2×D414yes—
185C2×C42C2×D414yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C4⋊C8—48no—
189C4⋊C8—44no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C22⋊C8—22no—
195C22⋊C8#4114yes—
196C4×C8—48no—
197C4×C8—44no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—24no—
203#95C22⋊C414yes—
204#95C22⋊C414yes—
205C24—21no—
206C24—21no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—41no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24C22×C812yes—
216C24C23×C411yescharacteristic
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—48no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—48no—
223C2×Q8—42no—
224C2×Q8—42no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—48no—
258C22×C4—48no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C414yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C22×C814yes—
278C22×C4C22×C812yes—
279C22×C4C22×C814yes—
280C22×C4C22×C812yes—
281C22×C4C2×M1614yes—
282C22×C4C2×M1612yes—
283C22×C4#13112yes—
284C22×C4C22×D412yes—
285C22×C4C2×#4112yes—
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C42−1+411yescharacteristic
288C2×C8—24no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—44no—
293C2×C8—42no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—42no—
297C4⋊C4—216no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—48no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4C22⋊C818yes—
306C4⋊C4C22⋊C818yes—
307C4⋊C4#10014yes—
308C22⋊C4—216no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—44no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—42no—
320C42—44no—
321C42—42no—
322C42—44no—
323C42—44no—
324C42—44no—
325C42—42no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23C22×C22⋊C411yescharacteristic
354C23C4×#4111yescharacteristic
355C23#52011yescharacteristic
356C23C23×C811yescharacteristic
357C23C22×M1611yescharacteristic
358C23C2×#13111yescharacteristic
359C23#56811yescharacteristic
360Q8—48no—
361Q8—48no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—48no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4C2×C22⋊C818yes—
417C2×C4C2×C22⋊C812yes—
418C2×C4C2×C22⋊C812yes—
419C2×C4#41012yes—
420C2×C4C2×#10014yes—
421C2×C4C2×#10012yes—
422C2×C4#54612yes—
423C2×C4#54712yes—
424C8—44no—
425C8—84no—
426C8—82no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22#233911yescharacteristic
451C22C22×C22⋊C811yescharacteristic
452C22#288011yescharacteristic
453C22C2×#41011yescharacteristic
454C22#288411yescharacteristic
455C22C22×#10011yescharacteristic
456C22#288911yescharacteristic
457C22C8×#4111yescharacteristic
458C22#297011yescharacteristic
459C22#298811yescharacteristic
460C22#299011yescharacteristic
461C4—22no—
462C4—28no—
463C4—22no—
464C4—22no—
465C4—24no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—44no—
470C4—42no—
471C4—82no—
472C4—81no—
473C4#146014yes—
474C2—21no—
475C2—21no—
476C2—21no—
477C2—21no—
478C2—82no—
479C2C2×#146011yescharacteristic
480C2#1735211yescharacteristic
481C2#1742011yescharacteristic
482C2#1742511yescharacteristic
483C2#2246911yescharacteristic
484C2#2247811yescharacteristic
485C2#2247911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22599C2semidirect
3. #9878C2semidirect
5. #8701C2semidirect
8. #7908C2semidirect
42. C2×C22⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
#225992
#98782
#90802
#87012
#85572
C4×#3232
#79082