Finite Group Databasebeta

← #198274

Subgroups — #198274

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#60192C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4C2×#1548C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic
6#9248C218yesmaximal, complemented
7#8001C211yesmaximal, characteristic
8#7932C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3209—22no—
12#3209—22no—
13#3209—22no—
14#3209—22no—
15C2×#494C412yes—
16C2×#487C412yes—
17C2×#462C2214yes—
18C2×#451C414yes—
19Q8×M16—22no—
20Q8×M16C2214yes—
21#2962—22no—
22#2962—22no—
23#2961—22no—
24#2960—22no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#2905—21no—
28C2×#423C2211yescharacteristic
29C2×#423C2211yescharacteristic
30#2873—21no—
31C2×C4×M16C2211yescharacteristic
32#1997—22no—
33#1997C2214yes—
34C2×#343—21no—
35C2×#343—21no—
36#1548—22no—
37#1548C2214yes—
38C2×#333—22no—
39#1493—22no—
40#1487C2214yes—
41#1481—22no—
42#1292C2214yes—
43#1283C2214yes—
44C23×Q8C2×C412yes—
45C22×Q16C2×C418yes—
46C22×M16—21no—
47C22×M16—21no—
48C22×M16D411yescharacteristic
49C22×M16D411yescharacteristic
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533D412yes—
55#533D412yes—
56#533D412yes—
57#533D412yes—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8C2×C412yes—
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4⋊Q8C2×C411yescharacteristic
71C2×C4⋊Q8C2312yes—
72C2×#125—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8C2×C412yes—
75C2×C4×Q8—21no—
76C2×C4×Q8C2312yes—
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C42C2311yescharacteristic
81#494—24no—
82#494—24no—
83#487—24no—
84#487—24no—
85#462—28no—
86#462—28no—
87#451—28no—
88#451—28no—
89#446—24no—
90#446—28no—
91#446—24no—
92#446—28no—
93C8×Q8—24no—
94C8×Q8—24no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431C2312yes—
103#424—24no—
104#424—24no—
105#423—24no—
106#423—21no—
107#423—21no—
108#423—21no—
109#423—21no—
110#423—21no—
111#423D412yes—
112#423C2312yes—
113#423C2312yes—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8D412yes—
119C2×C4⋊C8C2312yes—
120C2×#103C2×C418yes—
121C2×C22⋊C8—21no—
122C2×C22⋊C8C2×C412yes—
123#405—24no—
124#405—24no—
125C4×M16—24no—
126C4×M16—21no—
127C4×M16—21no—
128C4×M16D412yes—
129C4×M16C2312yes—
130C2×#97—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8C2×C412yes—
134#343—24no—
135#343—24no—
136#333—28no—
137Q8⋊C8—28no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—84no—
145C22×Q8C22×C414yes—
146C22×Q8C22×C412yes—
147C23×C4—22no—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4#4111yescharacteristic
151C2×Q16—216no—
152C2×Q16—88no—
153C2×M16—21no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—41no—
164C2×M16—42no—
165C2×M16—42no—
166C2×M16Q1614yescomplemented
167C2×M16C2×D412yes—
168C2×M16C2×D412yes—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—41no—
173C22×C8—41no—
174C22×C8C22×C414yes—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C4⋊Q8—41no—
191C4⋊Q8—41no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4⋊Q8C22⋊C412yes—
196C4⋊Q8C22⋊C412yes—
197C4⋊Q8C22⋊C412yes—
198C4⋊Q8C22⋊C412yes—
199#125—22no—
200#125—22no—
201#125—22no—
202#125—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8C22⋊C414yes—
210C22⋊Q8C22⋊C414yes—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8C2×D414yes—
218C4×Q8C2×D414yes—
219#117—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4C22×C412yes—
241C2×C22⋊C4—22no—
242C2×C42—22no—
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4111yescharacteristic
252C2×C42#4111yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42C2411yescharacteristic
255C4⋊C8—28no—
256C4⋊C8—28no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—41no—
266C4⋊C8—41no—
267C4⋊C8—41no—
268C4⋊C8C2×D414yes—
269#103—216no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—41no—
274#97—28no—
275#97—22no—
276#97—22no—
277C4×C8—28no—
278C4×C8—22no—
279C4×C8—22no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—22no—
283C4×C8—41no—
284C24C4×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—44no—
291C2×Q8—48no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—82no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8C2×C22⋊C414yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—84no—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C4×D411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12311yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4C2×Q1612yes—
325C22×C4#13512yes—
326C22×C4#13712yes—
327C22×C4#13712yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C2×#4111yescharacteristic
331Q16—816no—
332M16—22no—
333M16—24no—
334M16—22no—
335M16—22no—
336M16—42no—
337M16—42no—
338M16—42no—
339M16—44no—
340M16—42no—
341M16—42no—
342M16—82no—
343M16C2×Q1614yes—
344C2×C8—22no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—42no—
350C2×C8—44no—
351C2×C8—44no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—82no—
355C2×C8—82no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4C2×C22⋊C414yes—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C42—24no—
379C42—24no—
380C42C2×C22⋊C411yescharacteristic
381C42C2×C22⋊C411yescharacteristic
382C42C4×D411yescharacteristic
383C42C4×D411yescharacteristic
384C42#12011yescharacteristic
385C42#12011yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C42#12111yescharacteristic
389C42#12111yescharacteristic
390C42#12311yescharacteristic
391C42#12311yescharacteristic
392C42C22×D411yescharacteristic
393C42C22×D411yescharacteristic
394C42C22×D411yescharacteristic
395C42C2×#4111yescharacteristic
396C23—41no—
397C23—41no—
398C23#38611yescharacteristic
399C23#38611yescharacteristic
400C23C4×Q1612yes—
401C23#44112yes—
402C23C2×C4×D411yescharacteristic
403Q8—28no—
404Q8—44no—
405Q8—48no—
406Q8—82no—
407Q8—82no—
408Q8—84no—
409Q8—84no—
410Q8—84no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4#38611yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4C4×Q1612yes—
438C2×C4#44112yes—
439C2×C4#45112yes—
440C2×C4#45112yes—
441C2×C4#45212yes—
442C2×C4#45212yes—
443C2×C4#46212yes—
444C2×C4#46212yes—
445C2×C4#46412yes—
446C2×C4#46412yes—
447C2×C4C22×C22⋊C411yescharacteristic
448C2×C4C2×C4×D411yescharacteristic
449C2×C4C2×#12011yescharacteristic
450C2×C4C2×#12111yescharacteristic
451C2×C4C2×#12111yescharacteristic
452C2×C4C2×#12311yescharacteristic
453C2×C4C22×Q1611yescharacteristic
454C2×C4C2×#13511yescharacteristic
455C2×C4C2×#13711yescharacteristic
456C2×C4C2×#13711yescharacteristic
457C8—24no—
458C8—44no—
459C8—42no—
460C8—44no—
461C8—82no—
462C8—82no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—42no—
470C22—42no—
471C22#187812yes—
472C22#187812yes—
473C22C2×#38611yescharacteristic
474C22C2×C4×Q1611yescharacteristic
475C22C2×#44111yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—44no—
483C4—44no—
484C4—84no—
485C4#187812yes—
486C4#187812yes—
487C4#190212yes—
488C4#288812yes—
489C4C2×#45111yescharacteristic
490C4C2×#45211yescharacteristic
491C4C2×#46211yescharacteristic
492C4C2×#46411yescharacteristic
493C2—22no—
494C2—22no—
495C2—21no—
496C2—21no—
497C2C2×#187811yescharacteristic
498C2#2099011yescharacteristic
499C2C2×#190211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9248C2semidirect
166. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#601922
C2×#19972
C2×#15482
#174862
#92482
#80012
#79322