Finite Group Databasebeta

← #233975

Subgroups — #233975

Order 512.

1344 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C22×#475C211yesmaximal, characteristic
3#35179C211yesmaximal, characteristic
4#21834C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic
7#10234C218yesmaximal, complemented
8#8039C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494—21no—
15C2×#487—21no—
16C2×#475—24no—
17C2×#475—24no—
18C2×#475—22no—
19C2×#475C2214yes—
20C2×#462C2214yes—
21C2×#451—22no—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C22×#106—21no—
25C22×C4⋊C8C2211yescharacteristic
26C22×#103—21no—
27#2855—22no—
28#2627—22no—
29#2582C2214yes—
30#2581—22no—
31#2568—21no—
32#2567—21no—
33#2565—22no—
34#2317—21no—
35#2310—21no—
36#2285C2211yescharacteristic
37#1990C2214yes—
38#1987—22no—
39#1824—22no—
40#1481—24no—
41#1479—24no—
42C2×Q8⋊C8C2214yes—
43C2×Q8⋊C8C2214yes—
44#1296C2214yes—
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C22×Q16—24no—
48C23×C8—21no—
49#533—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D414yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8C2312yes—
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8D412yes—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8D412yes—
74C2×C4×Q8C2312yes—
75C2×C4×Q8C2312yes—
76C2×#117—21no—
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#494—28no—
85#487—28no—
86#475—216no—
87#475—28no—
88#475—28no—
89#462—28no—
90#462—28no—
91#451—216no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431D414yes—
96C2×#106—28no—
97C2×#106—24no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8D414yes—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2312yes—
104C2×#103—28no—
105C2×#103—24no—
106C2×#103—24no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2312yes—
111#400—24no—
112#398—28no—
113#395—24no—
114#391—22no—
115#391—22no—
116#391—24no—
117#391—24no—
118#389—24no—
119#389—24no—
120#387—28no—
121#385—24no—
122#384—22no—
123#384C2312yes—
124#382—22no—
125#382—21no—
126#381—24no—
127#380—21no—
128#380D412yes—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D412yes—
131#340—28no—
132Q8⋊C8—28no—
133Q8⋊C8—28no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—24no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—41no—
144C22×Q8—42no—
145C22×Q8—84no—
146C22×Q8Q1614yescomplemented
147C22×Q8C2×D412yes—
148C22×Q8C2×D412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×Q16—216no—
156C2×Q16—88no—
157C22×C8—24no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—42no—
162C22×C8—42no—
163C4⋊Q8—24no—
164C4⋊Q8—22no—
165C4⋊Q8—24no—
166C4⋊Q8—22no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8C2×D414yes—
177#125—42no—
178#125—42no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199#117—42no—
200#117—42no—
201#117—42no—
202#117—42no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42#4112yes—
244C2×C42C2411yescharacteristic
245#106—216no—
246C4⋊C8—24no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—42no—
250C4⋊C8C2×D414yes—
251C4⋊C8C2×D414yes—
252#103—216no—
253#103—216no—
254C22⋊C8—24no—
255C22⋊C8—42no—
256C4×C8—24no—
257C4×C8—22no—
258C4×C8—22no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—22no—
263#95—22no—
264C24#12011yescharacteristic
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—28no—
275C2×Q8—28no—
276C2×Q8—28no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—42no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—48no—
285C2×Q8—82no—
286C2×Q8—88no—
287C2×Q8—84no—
288C2×Q8—88no—
289C2×Q8C2×Q1614yes—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—82no—
304C22×C4—84no—
305C22×C4#12012yes—
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12012yes—
309C22×C4#12011yescharacteristic
310C22×C4#12112yes—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12712yes—
314C22×C4C2×Q1612yes—
315C22×C4#13512yes—
316C22×C4#13712yes—
317C22×C4#13712yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321Q16—816no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—44no—
327C2×C8—48no—
328C2×C8—44no—
329C2×C8—42no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—42no—
354C4⋊C4—88no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—44no—
362C42—44no—
363C42#12012yes—
364C42#12012yes—
365C42#12112yes—
366C42#12112yes—
367C42#12112yes—
368C42#12712yes—
369C42C22×D411yescharacteristic
370C42C22×D411yescharacteristic
371C42C22×D411yescharacteristic
372C42C2×#4111yescharacteristic
373C23—22no—
374C23—22no—
375C23#39212yes—
376C23#45112yes—
377C23#45212yes—
378C23C2×#12011yescharacteristic
379Q8—24no—
380Q8—24no—
381Q8—42no—
382Q8—48no—
383Q8—42no—
384Q8—44no—
385Q8—48no—
386Q8—44no—
387Q8—84no—
388Q8—88no—
389Q8—84no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—84no—
412C2×C4—88no—
413C2×C4—84no—
414C2×C4#39212yes—
415C2×C4#39212yes—
416C2×C4#39212yes—
417C2×C4#45114yes—
418C2×C4#45112yes—
419C2×C4#45214yes—
420C2×C4#45212yes—
421C2×C4#46714yes—
422C2×C4#47014yes—
423C2×C4C2×#12011yescharacteristic
424C2×C4C2×#12011yescharacteristic
425C2×C4C2×#12111yescharacteristic
426C2×C4C2×#12111yescharacteristic
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×#12711yescharacteristic
429C2×C4C22×Q1611yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×#13711yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4#57714yes—
434C8—44no—
435C8—44no—
436C8—82no—
437C22—22no—
438C22—22no—
439C22—24no—
440C22—24no—
441C22#200114yes—
442C22C2×#39211yescharacteristic
443C22C2×#45111yescharacteristic
444C22C2×#45211yescharacteristic
445C22#301012yes—
446C22#302212yes—
447C4—22no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—44no—
452C4—82no—
453C4—84no—
454C4—82no—
455C4—82no—
456C4—84no—
457C4#200114yes—
458C4C2×#45111yescharacteristic
459C4C2×#45211yescharacteristic
460C4#300912yes—
461C4C2×#46711yescharacteristic
462C4C2×#47011yescharacteristic
463C4#305712yes—
464C2—24no—
465C2#1890712yes—
466C2#1892512yes—
467C2C2×#200111yescharacteristic
468C2#2338811yescharacteristic
469C2#2340811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10234C2semidirect
146. C22×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C22×#4752
#351792
#218342
C2×#14812
C2×#14792
#102342
#80392