Finite Group Databasebeta

← #493748

Subgroups — #493748

Order 512.

1195 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2C2×#2129C212yesmaximal
3#20632C212yesmaximal
4#12800C214yesmaximal, complemented
5#12798C214yesmaximal, complemented
6#12583C212yesmaximal
7C2×#536C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C2×#410C2214yes—
11#2327C412yescomplemented
12#2326C2212yes—
13#2323—22no—
14#2322—22no—
15#2317C411yescharacteristic, complemented
16#2311C2211yescharacteristic
17#2310C411yescharacteristic, complemented
18C4×C4⋊Q8C412yescomplemented
19C4×#124C2212yes—
20C4×C22⋊Q8—22no—
21C4×#123—22no—
22#2285—22no—
23C4×#117C411yescharacteristic, complemented
24#2129—28no—
25C2×#354C2218yes—
26#1799—22no—
27#1799—22no—
28#1799C2214yes—
29#1799C2214yes—
30#1740—22no—
31#1393C2218yes—
32C22×#41C2311yescharacteristic
33C23×Q8C2×C411yescharacteristic
34#536—24no—
35#536D414yes—
36#536Q814yes—
37C2×C4⋊Q8C2×C412yes—
38C2×#124C2312yes—
39C2×#123—22no—
40C2×C22⋊Q8—22no—
41C4×#41—42no—
42C4×#41—42no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—42no—
46C2×C4×Q8—42no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2311yescharacteristic
54C2×C4×D4C2311yescharacteristic
55C2×#117—24no—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C2×#117D412yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C42—21no—
63C22×C42C2×C411yescharacteristic
64C22×C42D412yes—
65#410—28no—
66#410—28no—
67C2×#99—24no—
68C2×#99C2318yes—
69#391—22no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#390C2312yes—
74#389—24no—
75#389—22no—
76#388—24no—
77#388—24no—
78#387—24no—
79#387—24no—
80#386—22no—
81#385—24no—
82#385—22no—
83#384—42no—
84#384C2×C412yes—
85#382—42no—
86#382C2×C412yes—
87#381—22no—
88#380—22no—
89#380—24no—
90#380—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—42no—
95C4×C4⋊C4—42no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—24no—
99#376C2×C412yes—
100C43C2×C412yes—
101#354—216no—
102#342—24no—
103#342—24no—
104C2×#41—42no—
105C2×#41—42no—
106C2×#41C2×D412yes—
107C2×#41C2×Q812yes—
108C2×#41#4114yes—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—42no—
112C22×Q8—42no—
113C22×Q8C4⋊C414yescomplemented
114C22×Q8C22×C412yes—
115C22×Q8C22×C411yescharacteristic
116C22×D4C2×D412yes—
117C22×D4C2411yescharacteristic
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C4⋊Q8—24no—
127C4⋊Q8C4⋊C418yescomplemented
128#124—24no—
129#124C2×D414yes—
130#124C2×Q814yes—
131#123—28no—
132#123—28no—
133C22⋊Q8—28no—
134C22⋊Q8—28no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—42no—
138C4×Q8—44no—
139C4×Q8—44no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×D4—42no—
145C4×D4—42no—
146C4×D4—42no—
147C4×D4—42no—
148C4×D4—41no—
149C4×D4—42no—
150C4×D4—41no—
151#117—22no—
152#117—42no—
153#117—42no—
154#117—42no—
155#117—42no—
156#117—48no—
157#117—48no—
158#117C4⋊C414yescomplemented
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C22⋊C4—21no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C2×D412yes—
183C2×C22⋊C4C2×D412yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—44no—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C412yes—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206#99—28no—
207#99—28no—
208#99—48no—
209#95—22no—
210#95—22no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—44no—
216#95—44no—
217C24C2×C22⋊C411yescharacteristic
218C24C22×D412yes—
219#41—44no—
220#41—44no—
221#41—44no—
222#41—44no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—48no—
228C2×Q8—48no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8C2×C4⋊C414yes—
234C2×Q8C2×C4⋊C412yes—
235C2×Q8#11712yes—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—41no—
240C2×D4#12114yes—
241C2×D4C22⋊Q814yes—
242C2×D4C22×D411yescharacteristic
243C2×D4C22×Q811yescharacteristic
244C2×D4C2×#4112yes—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—48no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—81no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C4⋊C412yes—
280C22×C4#11712yes—
281C22×C4#12114yes—
282C22×C4C22⋊Q814yes—
283C22×C4C23×C411yescharacteristic
284C4⋊C4—24no—
285C4⋊C4—216no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—88no—
296C4⋊C4—84no—
297C4⋊C4—88no—
298C4⋊C4—88no—
299C22⋊C4—28no—
300C22⋊C4—28no—
301C22⋊C4—24no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—48no—
307C22⋊C4—48no—
308C22⋊C4—84no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—21no—
319C42—21no—
320C42—42no—
321C42—44no—
322C42—44no—
323C42—42no—
324C42—42no—
325C42—88no—
326C42—82no—
327C42#9912yes—
328C42#9912yes—
329C42C2×C4⋊C414yes—
330C23—21no—
331C23—22no—
332C23—21no—
333C23—22no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23#38012yes—
339C23C22×C22⋊C411yescharacteristic
340C23C2×#12112yes—
341C23C2×C22⋊Q812yes—
342Q8—42no—
343Q8—42no—
344Q8—44no—
345Q8—44no—
346Q8—48no—
347D4—44no—
348D4—44no—
349D4—41no—
350D4—41no—
351D4—42no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—88no—
393C2×C4—88no—
394C2×C4—88no—
395C2×C4#38014yes—
396C2×C4#38012yes—
397C2×C4C2×#9911yescharacteristic
398C2×C4C2×#9912yes—
399C2×C4C2×#9912yes—
400C2×C4C2×#9911yescharacteristic
401C2×C4#41012yes—
402C2×C4C22×C4⋊C411yescharacteristic
403C2×C4C2×#11711yescharacteristic
404C22—21no—
405C22—21no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—42no—
416C22—41no—
417C22—82no—
418C22C2×#38011yescharacteristic
419C22C22×#9911yescharacteristic
420C22C2×#41011yescharacteristic
421C4—21no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—24no—
426C4—44no—
427C4—41no—
428C4—42no—
429C4—42no—
430C4—41no—
431C4—44no—
432C4—82no—
433C4—82no—
434C4—168no—
435C4#179914yes—
436C4#212912yescomplemented
437C2—21no—
438C2—21no—
439C2—42no—
440C2—42no—
441C2—42no—
442C2C2×#179911yescharacteristic
443C2C2×#212911yescharacteristic
444C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12800C2semidirect
5. #12798C2semidirect
9. C22×C4×Q8C4semidirect
11. #2327C4semidirect
15. #2317C4semidirect
17. #2310C4semidirect
18. C4×C4⋊Q8C4semidirect
23. C4×#117C4semidirect
113. C22×Q8C4⋊C4semidirect
127. C4⋊Q8C4⋊C4semidirect
158. #117C4⋊C4semidirect
436. C4#2129semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
C2×#21292
#206322
#128002
#127982
#125832