Finite Group Databasebeta

← #365512

Subgroups — #365512

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21322C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic
6#11340C214yesmaximal
7#10899C214yesmaximal
8#10851C211yesmaximal, characteristic
9C4×#343C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#428C2214yes—
12C2×C4×M16C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2370—22no—
15#2369—21no—
16#2365—21no—
17#2355—21no—
18#2355—22no—
19Q8×C4⋊C4—21no—
20#2329—22no—
21#2317C2212yes—
22C4×C22⋊Q8—22no—
23C2×#389—22no—
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#384C2212yes—
27#2274C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1992—22no—
31#1990—21no—
32#1990—21no—
33#1986C418yescomplemented
34#1847—22no—
35#1845—21no—
36#1844—21no—
37C2×#343C2212yes—
38C2×#343C2212yes—
39#1736C2214yes—
40C2×#339C2214yes—
41C2×#339C2214yes—
42#1444—22no—
43C4⋊#95C2212yes—
44C22×M16C2312yes—
45C22×M16C2312yes—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533Q814yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125D412yes—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×#117C2312yes—
59C2×#117C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42D412yes—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#428—28no—
71#428—44no—
72C2×#107—24no—
73C2×#106—24no—
74C4×M16—22no—
75C4×M16—22no—
76C4×M16—22no—
77C4×M16C2×C414yes—
78C2×#97—21no—
79C2×#97—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#391—24no—
83#391—22no—
84#389—24no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—48no—
89#387—24no—
90#387—24no—
91#387—24no—
92#385—24no—
93#384—22no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—44no—
99#384D411yescharacteristic
100#384D411yescharacteristic
101#382—22no—
102#382D412yes—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C414yes—
119C4×C22⋊C4—24no—
120C2×#95—21no—
121C2×#95—24no—
122C2×#95D412yes—
123#343—24no—
124#343Q818yescomplemented
125#340—24no—
126#340—24no—
127#339—44no—
128#339C2×C4116yes—
129C22×Q8—21no—
130C22×Q8—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16C22×C414yes—
147C2×M16C2×Q814yes—
148C2×M16C2×Q814yes—
149C2×M16#4114yes—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8C4⋊C414yescomplemented
155C4⋊Q8C4⋊C414yescomplemented
156#125—24no—
157#125C4⋊C418yescomplemented
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—84no—
167#117—22no—
168#117—22no—
169#117—24no—
170#117—24no—
171#117C22×C414yes—
172#117C2×Q814yes—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—84no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×Q814yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—44no—
223C2×C42C22×C414yes—
224C2×C42C22×C412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×Q812yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232#107—48no—
233#106—48no—
234#97—24no—
235#97—42no—
236C4×C8—24no—
237C4×C8—42no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—48no—
248C24C22×D411yescharacteristic
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—81no—
254C2×Q8—81no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4C22⋊Q812yes—
291C22×C4C22⋊Q812yes—
292C22×C4C4⋊Q814yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C22×Q812yes—
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4112yes—
300C22×C4C2×#4111yescharacteristic
301C22×C4H(5,𝔽2)12yes—
302C22×C42−1+412yes—
303M16—24no—
304M16—24no—
305M16—24no—
306M16—24no—
307M16—24no—
308M16—42no—
309M16—42no—
310M16C4×Q818yescomplemented
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—42no—
316C2×C8—42no—
317C4⋊C4—216no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4—816no—
342C4⋊C4—82no—
343C4⋊C4C2×C4⋊C414yes—
344C4⋊C4C2×C4⋊C414yes—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C42—24no—
350C42—24no—
351C42—28no—
352C42—44no—
353C42—44no—
354C42—44no—
355C42—82no—
356C42—84no—
357C42—84no—
358C42C2×C4⋊C412yes—
359C42C2×C4⋊C412yes—
360C42C4×D412yes—
361C42C4×D412yes—
362C42C22⋊Q812yes—
363C42C22⋊Q812yes—
364C42#12312yes—
365C42#12312yes—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23C2×C4×D411yescharacteristic
371C23C2×#12011yescharacteristic
372C23C2×C22⋊Q811yescharacteristic
373C23C2×C4⋊Q811yescharacteristic
374C23#53211yescharacteristic
375C23#53511yescharacteristic
376C23#53611yescharacteristic
377Q8—48no—
378Q8—84no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—28no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4#38512yes—
424C2×C4#38512yes—
425C2×C4#38512yes—
426C2×C4#38512yes—
427C2×C4#45312yes—
428C2×C4#45612yes—
429C2×C4C22×C4⋊C411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C4×Q812yes—
432C2×C4#52012yes—
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4#53312yes—
436C2×C4#54212yes—
437C2×C4#54612yes—
438C2×C4D4×Q812yes—
439C2×C4D4×Q812yes—
440C2×C4#55012yes—
441C2×C4#55212yes—
442C2×C4#55412yes—
443C2×C4#55712yes—
444C2×C4Q8212yes—
445C8—28no—
446C8—44no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22C2×#38511yescharacteristic
460C22C4×C4⋊Q811yescharacteristic
461C22#232611yescharacteristic
462C22#243111yescharacteristic
463C22#243411yescharacteristic
464C22#256511yescharacteristic
465C22#256711yescharacteristic
466C22C2×#45311yescharacteristic
467C22C2×#45611yescharacteristic
468C22#302111yescharacteristic
469C22#302211yescharacteristic
470C4—22no—
471C4—22no—
472C4—22no—
473C4—48no—
474C4—44no—
475C4—44no—
476C4—44no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—162no—
481C4#185914yes—
482C4Q8×C4⋊C412yes—
483C4#236012yes—
484C4#242312yes—
485C4#245112yes—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2#1397611yescharacteristic
491C2C2×#185911yescharacteristic
492C2#2086511yescharacteristic
493C2#2562211yescharacteristic
494C2#2562311yescharacteristic
495C2#2582411yescharacteristic
496C2#2582511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
33. #1986C4semidirect
124. #343Q8semidirect
154. C4⋊Q8C4⋊C4semidirect
155. C4⋊Q8C4⋊C4semidirect
157. #125C4⋊C4semidirect
310. M16C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#342132
#213222
C2×#19862
#208002
#139102
#113402
#108992
#108512
C4×#3432