Finite Group Databasebeta

← #107789

Subgroups — #107789

Order 512.

926 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2320C211yesmaximal, characteristic
2#20740C211yesmaximal, characteristic
3C2×#1491C211yesmaximal, characteristic
4#10851C211yesmaximal, characteristic
5C2×#1300C211yesmaximal, characteristic
6#8022C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2320—22no—
13#2320—22no—
14#2320—22no—
15#2320C414yescomplemented
16C2×#391—21no—
17C2×#389—21no—
18C2×#387—22no—
19#2285C2211yescharacteristic
20C2×#383—21no—
21C2×#382—21no—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1823—21no—
25#1803C414yescomplemented
26#1787—21no—
27#1763—22no—
28C2×#339—22no—
29#1491—22no—
30#1491C414yescomplemented
31C2×Q8⋊C8—22no—
32#1300—22no—
33#1294—21no—
34#1294—21no—
35#1285—22no—
36C23×Q8D411yescharacteristic
37C22×M16—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2×C412yes—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8D412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8Q812yes—
50C2×#117—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C42—21no—
56C22×C42—21no—
57C22×C42D412yes—
58C22×C42C2311yescharacteristic
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432C2×C412yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8C2×C412yes—
69C2×#103—24no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C4×M16—21no—
73C4×M16—21no—
74C4×M16—21no—
75C4×M16C2×C412yes—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×C4×C8—21no—
80#391—24no—
81#391—24no—
82#389—24no—
83#389—24no—
84#387—28no—
85#387—28no—
86#384C2×C412yes—
87#383—24no—
88#383—24no—
89#382—24no—
90#382—24no—
91#382—21no—
92#380—21no—
93#380—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4D412yes—
100#376—21no—
101#376—21no—
102#376C2×C412yes—
103#376D412yes—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111#340—24no—
112#339—44no—
113Q8⋊C8—28no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—44no—
118C22×Q8C22⋊C412yes—
119C22×Q8C22⋊C412yes—
120C22×Q8C4⋊C412yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—41no—
133C2×M16—41no—
134C2×M16—41no—
135C22×C8—22no—
136C22×C8—41no—
137C22×C8—41no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8C4214yes—
151C4×Q8C22⋊C414yes—
152C4×Q8C4⋊C414yes—
153C4×Q8C4⋊C414yes—
154#117—42no—
155#117—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C4⋊C412yes—
177C2×C22⋊C4—41no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—44no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—41no—
191C2×C42C22⋊C414yes—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×Q811yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—41no—
208#103—48no—
209C22⋊C8—22no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213#97—22no—
214#97—22no—
215#97—41no—
216#97—41no—
217#97—41no—
218#97—41no—
219C4×C8—22no—
220C4×C8—41no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—41no—
231#95—41no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C24#12011yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8#9514yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
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—24no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C4⋊C411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C4×D411yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4C22⋊Q811yescharacteristic
294C22×C4C22⋊Q811yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4#12311yescharacteristic
297M16—24no—
298M16—42no—
299M16—42no—
300M16—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—44no—
307C2×C8—82no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—82no—
320C4⋊C4—88no—
321C4⋊C4#9514yes—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—44no—
329C42—44no—
330C42—44no—
331C42—82no—
332C42—82no—
333C42C2×C4211yescharacteristic
334C42C2×C22⋊C411yescharacteristic
335C42C2×C4⋊C411yescharacteristic
336C42C2×C4⋊C411yescharacteristic
337C42#11711yescharacteristic
338C42#11711yescharacteristic
339C42C4×D411yescharacteristic
340C42C4×D411yescharacteristic
341C42C4×D411yescharacteristic
342C42C4×D411yescharacteristic
343C42#12111yescharacteristic
344C42#12111yescharacteristic
345C42C22⋊Q811yescharacteristic
346C42C22⋊Q811yescharacteristic
347C42#12311yescharacteristic
348C42#12311yescharacteristic
349C23—22no—
350C23—22no—
351C23#37911yescharacteristic
352C23#38511yescharacteristic
353C23#38611yescharacteristic
354C23#45311yescharacteristic
355C23#45411yescharacteristic
356C23#45511yescharacteristic
357C23#45611yescharacteristic
358Q8—24no—
359Q8—24no—
360Q8—44no—
361Q8—44no—
362Q8—44no—
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—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—84no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—84no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4C2×#9511yescharacteristic
416C2×C4C4×C22⋊C411yescharacteristic
417C2×C4C4×C22⋊C411yescharacteristic
418C2×C4#38011yescharacteristic
419C2×C4#38011yescharacteristic
420C2×C4#38111yescharacteristic
421C2×C4#38511yescharacteristic
422C2×C4#38511yescharacteristic
423C2×C4#38511yescharacteristic
424C2×C4#38611yescharacteristic
425C2×C4#38611yescharacteristic
426C2×C4#38611yescharacteristic
427C2×C4#42114yes—
428C8—44no—
429C8—82no—
430C8—82no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22C22⋊#9511yescharacteristic
440C22#179211yescharacteristic
441C22#179311yescharacteristic
442C22#185911yescharacteristic
443C22#186011yescharacteristic
444C22#188211yescharacteristic
445C22#188311yescharacteristic
446C22#189712yes—
447C22#190512yes—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—44no—
459C4—44no—
460C4—82no—
461C4—84no—
462C4—82no—
463C4—162no—
464C4#173312yes—
465C4#189712yes—
466C2—21no—
467C2—21no—
468C2—21no—
469C2—21no—
470C2#932812yes—
471C2#933012yes—
472C2#1072011yescharacteristic
473C2#1078111yescharacteristic
474C2#1078611yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2320C4semidirect
25. #1803C4semidirect
30. #1491C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23202
#207402
C2×#14912
#108512
C2×#13002
#80222
C2×#12852