Finite Group Databasebeta

← #206984

Subgroups — #206984

Order 512.

926 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20463C211yesmaximal, characteristic
3#14688C211yesmaximal, characteristic
4#9889C212yesmaximal
5#9888C212yesmaximal
6#9887C214yesmaximal
7#8025C214yesmaximal
8C2×#431C2212yes—
9#2864C412yescomplemented
10#2863—21no—
11#2863—21no—
12#2862C411yescharacteristic, complemented
13#2859C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2857C2211yescharacteristic
16#2855—21no—
17#2855—21no—
18#2851C2212yes—
19C4×C4⋊Q8C411yescharacteristic, complemented
20C4×C4⋊Q8C411yescharacteristic, complemented
21C4×#125C412yescomplemented
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24#2295—21no—
25#2295—21no—
26#2283C2212yes—
27C2×#383C2212yes—
28C2×C43C2211yescharacteristic
29#1767—21no—
30#1767—21no—
31#1749C2212yes—
32#1749C2212yes—
33#1744—21no—
34C4×#97—21no—
35C4⋊#95—21no—
36C4⋊#95—21no—
37C4⋊#95C2212yes—
38#1434—21no—
39#1434—21no—
40#1434C2212yes—
41#1294C2218yes—
42#1294C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125C2×C412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C22×C4⋊C4—22no—
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#431—24no—
59#431Q818yescomplemented
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2314yes—
62C2×#97—22no—
63C2×#97—22no—
64C2×C4×C8—22no—
65#391—21no—
66#391—21no—
67#391—21no—
68#391—21no—
69#391—21no—
70#391—21no—
71#391—21no—
72#391—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C412yes—
84#389C2×C412yes—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#385—22no—
95#384C2×C411yescharacteristic
96#384C2×C411yescharacteristic
97#383—22no—
98#383—28no—
99#382C2×C412yes—
100#382C2×C412yes—
101#382C2×C412yes—
102#381—22no—
103#380—21no—
104#380—21no—
105#380C2312yes—
106#380C2312yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C22⋊C4—22no—
114#376—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95C2314yes—
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43D411yescharacteristic
129#336—22no—
130#336—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C23×C4—24no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—24no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144#125—44no—
145C22⋊Q8—44no—
146C22⋊Q8—44no—
147C4×Q8—44no—
148C4×Q8—44no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×Q814yes—
207C2×C42#4114yes—
208C4⋊C8—28no—
209C22⋊C8—24no—
210C22⋊C8C2×Q818yes—
211#97—44no—
212#97—44no—
213C4×C8—28no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—28no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234C24C22×D411yescharacteristic
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4#12412yes—
283C22×C4#12412yes—
284C22×C4C4⋊Q812yes—
285C22×C4C4⋊Q812yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×Q812yes—
289C22×C4C2×#4112yes—
290C2×C8—28no—
291C2×C8—44no—
292C2×C8—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—88no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—44no—
337C42—44no—
338C42C4×Q814yes—
339C42C4×Q814yes—
340C42C22⋊Q814yes—
341C42C22⋊Q814yes—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23C22×C22⋊C411yescharacteristic
347C23C2×#12411yescharacteristic
348C23C2×C4⋊Q811yescharacteristic
349C23#57512yes—
350C23#57512yes—
351Q8—84no—
352Q8—84no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4#39112yes—
415C2×C4#39112yes—
416C2×C4#39112yes—
417C2×C4#39112yes—
418C2×C4#41012yes—
419C2×C4#41012yes—
420C2×C4#42112yes—
421C2×C4#42112yes—
422C2×C4C2×C4×Q812yes—
423C2×C4C2×C22⋊Q812yes—
424C8—48no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22C2×#39111yescharacteristic
434C22C2×#41011yescharacteristic
435C22C2×#41011yescharacteristic
436C22#289611yescharacteristic
437C22#289611yescharacteristic
438C22C2×#42111yescharacteristic
439C22C2×#42111yescharacteristic
440C22#313512yes—
441C22#316212yes—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—164no—
461C2—22no—
462C2—22no—
463C2#1757911yescharacteristic
464C2#1758011yescharacteristic
465C2#1759311yescharacteristic
466C2#1759411yescharacteristic
467C2#2130611yescharacteristic
468C2#2132111yescharacteristic
469C2#2133211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2864C4semidirect
12. #2862C4semidirect
13. #2859C4semidirect
19. C4×C4⋊Q8C4semidirect
20. C4×C4⋊Q8C4semidirect
21. C4×#125C4semidirect
59. #431Q8semidirect

Maximal subgroups

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

subgroupindex
#366352
#204632
#146882
#98892
#98882
#98872
#80252