Finite Group Databasebeta

← #196013

Subgroups — #196013

Order 512.

1167 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic, complemented
2C2×#2637C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic, complemented
4C2×#1473C212yesmaximal
5#9086C212yesmaximal, complemented
6#9084C212yesmaximal
7#9082C212yesmaximal, complemented
8#9080C212yesmaximal
9#7914C212yesmaximal, complemented
10C22×#126C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2983—28no—
13C2×#446—24no—
14C2×#432C2212yes—
15C2×#432C2212yes—
16C2×#431C2212yes—
17C2×#431C2212yes—
18#2637—24no—
19#2637—24no—
20C2×#398C411yescharacteristic
21C2×#398C411yescharacteristic
22C2×#398C411yescharacteristic
23C2×#397C411yescharacteristic
24C2×#396C411yescharacteristic
25C2×#396C411yescharacteristic
26C2×#395C411yescharacteristic
27C2×#393C411yescharacteristic
28C2×#388C2211yescharacteristic
29C2×#388C2211yescharacteristic
30C2×#387C2211yescharacteristic
31C2×#387C2211yescharacteristic
32C2×#382C2211yescharacteristic
33#1984—24no—
34#1817C2212yes—
35#1817C2212yes—
36#1473—28no—
37C2×#324C2214yes—
38C2×#323C2214yes—
39#1280C2214yes—
40#1280C2214yes—
41C23×Q8C2×C411yescharacteristic
42C2×#126—22no—
43C2×#126—22no—
44C2×#126—24no—
45C2×#126—24no—
46C2×C4×Q8—24no—
47C2×C4×Q8—24no—
48C2×C4×Q8D414yes—
49C2×C4×Q8D414yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#446—232no—
73#432—24no—
74#432—28no—
75#432D418yes—
76#431—28no—
77#431—24no—
78#431—24no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8C2314yes—
84C2×C22⋊C8C2314yes—
85C2×#97—22no—
86C2×#97—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89#398—24no—
90#398—24no—
91#398—24no—
92#397—24no—
93#396—24no—
94#396—24no—
95#395—24no—
96#393—24no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#382—22no—
106#382—22no—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C411yescharacteristic
117C2×#95C2311yescharacteristic
118C2×#95C2311yescharacteristic
119#336—24no—
120#336—24no—
121#324—28no—
122#323—28no—
123C25C22×C411yescharacteristic
124C22×Q8—24no—
125C22×Q8—24no—
126C22×Q8—42no—
127C22×Q8C22⋊C414yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—24no—
143C22×C8—24no—
144#126—44no—
145#126—44no—
146#126—44no—
147#126—44no—
148C4×Q8—28no—
149C4×Q8—28no—
150C4×Q8—28no—
151C4×Q8—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—81no—
172C2×C4⋊C4—81no—
173C2×C4⋊C4—81no—
174C2×C4⋊C4—81no—
175C2×C4⋊C4C22⋊C414yes—
176C2×C4⋊C4C22⋊C414yes—
177C2×C4⋊C4C22⋊C414yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—81no—
193C2×C22⋊C4—81no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D414yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42#4114yes—
206C4⋊C8—28no—
207C4⋊C8—28no—
208C4⋊C8—28no—
209C4⋊C8—44no—
210C22⋊C8—24no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8C2×D418yes—
214#97—28no—
215#97—28no—
216C4×C8—28no—
217C4×C8—44no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24C23×C411yescharacteristic
241C2×Q8—28no—
242C2×Q8—28no—
243C2×Q8—44no—
244C2×Q8—416no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
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—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C4×D414yes—
297C22×C4C4×D414yes—
298C22×C4#13114yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D412yes—
302C22×C4C2×#4112yes—
303C2×C8—28no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—44no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C42—22no—
333C42—22no—
334C42—24no—
335C42—24no—
336C42—22no—
337C42—22no—
338C42#12014yes—
339C42#12114yes—
340C42#12114yes—
341C42#12314yes—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—44no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—82no—
360C23C22×C22⋊C411yescharacteristic
361C23C2×C4×D412yes—
362C23C2×#13112yes—
363C23#56812yes—
364Q8—416no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—44no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4#38614yes—
421C2×C4#38614yes—
422C2×C4#40814yes—
423C2×C4C2×#12011yescharacteristic
424C2×C4C2×#12112yes—
425C2×C4C2×#12311yescharacteristic
426C8—48no—
427C8—84no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—164no—
442C22C2×#38611yescharacteristic
443C22C2×#40811yescharacteristic
444C22#287911yescharacteristic
445C22#288411yescharacteristic
446C22#288411yescharacteristic
447C22#288811yescharacteristic
448C22#288911yescharacteristic
449C22#293314yes—
450C4—24no—
451C4—24no—
452C4—24no—
453C4—21no—
454C4—22no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—81no—
466C2—24no—
467C2—161no—
468C2#1739011yescharacteristic
469C2#1739111yescharacteristic
470C2#1740911yescharacteristic
471C2#1741011yescharacteristic
472C2#2089711yescharacteristic
473C2#2098611yescharacteristic
474C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2983C2semidirect
3. #21818C2semidirect
5. #9086C2semidirect
7. #9082C2semidirect
9. #7914C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
C2×#26372
#218182
C2×#14732
#90862
#90842
#90822
#90802
#79142