Finite Group Databasebeta

← #198611

Subgroups — #198611

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 503 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
4#17708C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#9268C218yesmaximal, complemented
7#8079C211yesmaximal, characteristic
8#7932C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3209—22no—
12#3209—22no—
13#3209C414yes—
14#3209C2214yes—
15C2×#494—21no—
16C2×#487—21no—
17C2×#462—22no—
18C2×#451C414yes—
19Q8×M16—22no—
20Q8×M16—22no—
21#2962—22no—
22#2962C2214yes—
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#1550C2214yes—
37#1549—21no—
38#1548—22no—
39#1548—22no—
40#1548—21no—
41#1542—22no—
42C2×#333C2214yes—
43#1301C2214yes—
44#1283C2214yes—
45C23×Q8C2×C412yes—
46C22×Q16—24no—
47C22×M16—21no—
48C22×M16—21no—
49C22×M16D411yescharacteristic
50C22×M16D411yescharacteristic
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533C2×C412yes—
56#533D412yes—
57#533D412yes—
58#533C2312yes—
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⋊Q8—21no—
69C2×C4⋊Q8C2×C412yes—
70C2×C4⋊Q8D411yescharacteristic
71C2×C4⋊Q8D411yescharacteristic
72C2×C4⋊Q8C2312yes—
73C2×#125—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8C2×C412yes—
76C2×C4×Q8—21no—
77C2×C4×Q8D412yes—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C42C2311yescharacteristic
82#494—24no—
83#494—24no—
84#487—24no—
85#487—24no—
86#462—28no—
87#462—28no—
88#451—28no—
89#451C2×C4116yes—
90#446—24no—
91#446—28no—
92#446—28no—
93#446—24no—
94C8×Q8—24no—
95C8×Q8—24no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431C2×C412yes—
103#431C2312yes—
104#424—24no—
105#424—24no—
106#423—24no—
107#423—21no—
108#423—21no—
109#423—21no—
110#423—21no—
111#423—21no—
112#423D412yes—
113#423D412yes—
114#423C2312yes—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×#103—24no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8C2×C412yes—
124#405—24no—
125#405—24no—
126C4×M16—24no—
127C4×M16—21no—
128C4×M16—21no—
129C4×M16D412yes—
130C4×M16C2312yes—
131C2×#97—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135#343—24no—
136#343—24no—
137#333—24no—
138#333—28no—
139#332—24no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—84no—
147C22×Q8C22⋊C412yes—
148C22×Q8C22×C414yes—
149C23×C4—22no—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4111yescharacteristic
153C2×Q16—216no—
154C2×Q16—88no—
155C2×M16—21no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—21no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—42no—
164C2×M16—42no—
165C2×M16—41no—
166C2×M16—42no—
167C2×M16—42no—
168C2×M16Q1614yescomplemented
169C2×M16C2×D412yes—
170C2×M16C2×D412yes—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—41no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—41no—
190C4⋊Q8—41no—
191C4⋊Q8—41no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197C4⋊Q8C22⋊C412yes—
198C4⋊Q8C22⋊C412yes—
199C4⋊Q8C2×D412yes—
200C4⋊Q8C2×D412yes—
201#125—22no—
202#125—22no—
203#125—22no—
204#125—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8C22⋊C414yes—
212C22⋊Q8C22×C414yes—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8C22⋊C414yes—
220C4×Q8C2×D414yes—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
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⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4C22⋊C412yes—
243C2×C22⋊C4—22no—
244C2×C42—22no—
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4111yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42C2411yescharacteristic
257C4⋊C8—28no—
258C4⋊C8—28no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—41no—
269C4⋊C8—41no—
270C4⋊C8C2×D414yes—
271#103—216no—
272C22⋊C8—22no—
273C22⋊C8—41no—
274C22⋊C8—41no—
275C22⋊C8C22×C414yes—
276#97—28no—
277#97—22no—
278#97—22no—
279C4×C8—28no—
280C4×C8—22no—
281C4×C8—22no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—41no—
285C4×C8—41no—
286C24C4×D411yescharacteristic
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—44no—
293C2×Q8—48no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—82no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8C2×C22⋊C414yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—84no—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12311yescharacteristic
325C22×C4#12311yescharacteristic
326C22×C4C2×Q1612yes—
327C22×C4#13512yes—
328C22×C4#13712yes—
329C22×C4#13712yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333Q16—816no—
334M16—22no—
335M16—24no—
336M16—22no—
337M16—22no—
338M16—42no—
339M16—44no—
340M16—44no—
341M16—42no—
342M16—42no—
343M16—82no—
344M16C2×Q1614yescomplemented
345C2×C8—22no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—44no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—82no—
357C2×C8—82no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4C2×C22⋊C414yes—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C42—24no—
381C42—24no—
382C42C2×C22⋊C411yescharacteristic
383C42C2×C22⋊C411yescharacteristic
384C42C4×D411yescharacteristic
385C42C4×D411yescharacteristic
386C42#12011yescharacteristic
387C42#12011yescharacteristic
388C42#12111yescharacteristic
389C42#12111yescharacteristic
390C42#12111yescharacteristic
391C42#12111yescharacteristic
392C42#12311yescharacteristic
393C42#12311yescharacteristic
394C42C22×D411yescharacteristic
395C42C22×D411yescharacteristic
396C42C22×D411yescharacteristic
397C42C2×#4111yescharacteristic
398C23—41no—
399C23—41no—
400C23#38611yescharacteristic
401C23#38611yescharacteristic
402C23C4×Q1612yes—
403C23#44112yes—
404C23C2×C4×D411yescharacteristic
405Q8—44no—
406Q8—48no—
407Q8—44no—
408Q8—44no—
409Q8—82no—
410Q8—84no—
411Q8—84no—
412Q8—84no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#38611yescharacteristic
437C2×C4#38611yescharacteristic
438C2×C4#38611yescharacteristic
439C2×C4C4×Q1612yes—
440C2×C4#44112yes—
441C2×C4#45112yes—
442C2×C4#45112yes—
443C2×C4#45212yes—
444C2×C4#45212yes—
445C2×C4#46212yes—
446C2×C4#46212yes—
447C2×C4#46412yes—
448C2×C4#46412yes—
449C2×C4C22×C22⋊C411yescharacteristic
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×#12011yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×#12111yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C2×C4C22×Q1611yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×#13711yescharacteristic
458C2×C4C2×#13711yescharacteristic
459C8—24no—
460C8—44no—
461C8—42no—
462C8—44no—
463C8—82no—
464C8—82no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—42no—
473C22#187812yes—
474C22#187812yes—
475C22C2×#38611yescharacteristic
476C22C2×C4×Q1611yescharacteristic
477C22C2×#44111yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—44no—
485C4—44no—
486C4—84no—
487C4#187812yes—
488C4#187812yes—
489C4#191412yes—
490C4C2×#45111yescharacteristic
491C4C2×#45211yescharacteristic
492C4C2×#46211yescharacteristic
493C4C2×#46411yescharacteristic
494C4#306912yes—
495C2—22no—
496C2—22no—
497C2—21no—
498C2—21no—
499C2C2×#187811yescharacteristic
500C2#2100611yescharacteristic
501C2C2×#191411yescharacteristic
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. #9268C2semidirect
168. C2×M16Q16semidirect
344. M16C2×Q16semidirect

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
#177082
C2×#15482
#92682
#80792
#79322