Finite Group Databasebeta

← #205950

Subgroups — #205950

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#21337C211yesmaximal, characteristic
3C2×#1966C211yesmaximal, characteristic
4#17813C211yesmaximal, characteristic
5#17708C211yesmaximal, characteristic
6C2×#1546C211yesmaximal, characteristic
7#17314C211yesmaximal, characteristic
8#9797C214yesmaximal, complemented
9#9794C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3229—22no—
13#3229—22no—
14#3229—22no—
15#3229—22no—
16C2×#498C2212yes—
17C2×#487C2212yes—
18C2×#483C414yes—
19C2×#475C414yes—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C22×#101—22no—
25C2×C4×M16C2211yescharacteristic
26#1998—22no—
27#1997C2212yes—
28#1997C2212yes—
29#1966—22no—
30#1966C2214yes—
31#1742—22no—
32#1570—22no—
33#1570—22no—
34#1566C2214yes—
35#1550—22no—
36#1549C2212yes—
37#1548C2212yes—
38#1546—22no—
39#1546C2214yes—
40#1542—22no—
41C2×#332—22no—
42#1450—22no—
43C8×M16—22no—
44C8×M16C2214yes—
45C23×Q8—21no—
46C22×M16—21no—
47C22×M16—21no—
48C22×M16D412yes—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57C2×C4⋊Q8—21no—
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⋊Q8C2×C412yes—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8C2×C412yes—
74C2×C4×Q8—21no—
75C2×C4×Q8C2×C412yes—
76C2×#117—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4C2×C412yes—
79C22×C42C2311yescharacteristic
80#498—24no—
81#498—24no—
82#487—24no—
83#487—24no—
84#483—28no—
85#483—28no—
86#475—28no—
87#475—28no—
88#431—21no—
89#431—21no—
90#431D412yes—
91#431D412yes—
92#431D412yes—
93#431D412yes—
94#431D412yes—
95#431C2312yes—
96C2×#106C2×C418yes—
97#423—21no—
98#423—21no—
99#423—21no—
100#423—21no—
101#423—21no—
102#423—21no—
103#423C2312yes—
104#423C2312yes—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2×C412yes—
110C2×C4⋊C8D412yes—
111C2×#103C2×C418yes—
112C2×#101—28no—
113C2×#101—48no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8C2×C412yes—
116C4×M16—21no—
117C4×M16—21no—
118C4×M16D412yes—
119C4×M16C2312yes—
120C2×#97—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2312yes—
124#343—24no—
125#343—24no—
126#334—28no—
127#333—24no—
128#332—28no—
129#332—24no—
130#322—24no—
131C82—24no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—44no—
140C22×Q8—44no—
141C22×Q8C22×C412yes—
142C23×C4—22no—
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4#4111yescharacteristic
146C2×M16—22no—
147C2×M16—24no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—41no—
151C2×M16—41no—
152C2×M16—44no—
153C2×M16—44no—
154C2×M16SD1618yes—
155C2×M16C2×D414yes—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—41no—
159C22×C8—41no—
160C22×C8C22×C414yes—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—22no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185#125—41no—
186#125—41no—
187#125—41no—
188#125—41no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208#117—41no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C22×C412yes—
227C2×C22⋊C4—22no—
228C2×C42—22no—
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42C2411yescharacteristic
241#106—216no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8C22⋊C414yes—
252C4⋊C8C22⋊C414yes—
253C4⋊C8C2×D414yes—
254#103—216no—
255#101—416no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8C22⋊C414yes—
259C22⋊C8C22⋊C414yes—
260#97—22no—
261#97—22no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—22no—
266C4×C8C2×D414yes—
267C4×C8C2×D414yes—
268C24C4×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—44no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—48no—
281C2×Q8—82no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—82no—
285C2×Q8—82no—
286C2×Q8—82no—
287C2×Q8—82no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—48no—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12411yescharacteristic
308C22×C4#12411yescharacteristic
309C22×C4#12711yescharacteristic
310C22×C4#12711yescharacteristic
311C22×C4C2×SD1614yes—
312C22×C4#13514yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316M16—24no—
317M16—28no—
318M16—44no—
319M16—44no—
320M16—82no—
321M16—82no—
322M16C2×SD1618yes—
323C2×C8—24no—
324C2×C8—28no—
325C2×C8—24no—
326C2×C8—44no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—82no—
330C2×C8—82no—
331C2×C8—82no—
332C2×C8C2×C22⋊C418yes—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C42—42no—
356C42—42no—
357C42C2×C22⋊C411yescharacteristic
358C42C2×C22⋊C411yescharacteristic
359C42C4×D411yescharacteristic
360C42C4×D411yescharacteristic
361C42#12111yescharacteristic
362C42#12111yescharacteristic
363C42#12111yescharacteristic
364C42#12111yescharacteristic
365C42#12411yescharacteristic
366C42#12411yescharacteristic
367C42#12711yescharacteristic
368C42#12711yescharacteristic
369C42C22×D411yescharacteristic
370C42C22×D411yescharacteristic
371C42C22×D411yescharacteristic
372C42C2×#4111yescharacteristic
373C23—41no—
374C23—41no—
375C23#39011yescharacteristic
376C23#39011yescharacteristic
377C23C4×SD1614yes—
378C23C2×C4×D411yescharacteristic
379Q8—44no—
380Q8—82no—
381Q8—82no—
382Q8—82no—
383Q8—82no—
384Q8—84no—
385Q8—84no—
386Q8—84no—
387Q8—82no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4#39011yescharacteristic
409C2×C4#39011yescharacteristic
410C2×C4#39011yescharacteristic
411C2×C4#39011yescharacteristic
412C2×C4#39011yescharacteristic
413C2×C4#39011yescharacteristic
414C2×C4C4×SD1614yes—
415C2×C4#46514yes—
416C2×C4#46514yes—
417C2×C4#49212yes—
418C2×C4#49212yes—
419C2×C4#49512yes—
420C2×C4#49512yes—
421C2×C4C22×C22⋊C411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C2×#12111yescharacteristic
424C2×C4C2×#12111yescharacteristic
425C2×C4C2×#12411yescharacteristic
426C2×C4C2×#12711yescharacteristic
427C2×C4C22×SD1612yes—
428C2×C4C2×#13512yes—
429C8—28no—
430C8—28no—
431C8—44no—
432C8—82no—
433C8—82no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—44no—
441C22#196812yes—
442C22#196812yes—
443C22C2×#39011yescharacteristic
444C22C2×C4×SD1612yes—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—82no—
452C4—82no—
453C4—84no—
454C4#196812yes—
455C4#196812yes—
456C4#288912yes—
457C4C2×#46512yes—
458C4#306912yes—
459C4C2×#49211yescharacteristic
460C4C2×#49511yescharacteristic
461C2—22no—
462C2—22no—
463C2—22no—
464C2#2122011yescharacteristic
465C2C2×#196811yescharacteristic
466C2#2125011yescharacteristic
subgroupG/H
center, socleC22C2×#390
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
8. #9797C2semidirect
9. #9794C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#213372
C2×#19662
#178132
#177082
C2×#15462
#173142
#97972
#97942