Finite Group Databasebeta

← #202518

Subgroups — #202518

Order 512.

689 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425C211yesmaximal, characteristic
2#17796C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic
5#17323C211yesmaximal, characteristic
6#9551C214yesmaximal, complemented
7#9549C214yesmaximal, complemented
8#7934C211yesmaximal, characteristic
9C2×#1278C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#1997C411yescharacteristic
18#1997C411yescharacteristic
19#1966C411yescharacteristic
20#1966C411yescharacteristic
21#1901C414yes—
22C4×#101C2214yes—
23#1569—22no—
24#1567—21no—
25#1566—21no—
26#1565C2214yes—
27#1497—22no—
28#1493—21no—
29#1493—21no—
30#1493—21no—
31#1493—21no—
32#1493—21no—
33#1493—21no—
34#1479C2214yes—
35C2×Q8⋊C8C2212yes—
36C2×Q8⋊C8C2212yes—
37#1452—21no—
38#1451—21no—
39#1451C2212yes—
40#1450—21no—
41C8×M16—21no—
42C8×M16C2212yes—
43#1283C2214yes—
44#1278C2212yes—
45#1278C2212yes—
46C22×M16—21no—
47C22×M16C2×C412yes—
48C23×C8—21no—
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⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2×C412yes—
62C2×C4×Q8—21no—
63C2×C4×Q8C2312yes—
64C2×#117—21no—
65C22×C42C2311yescharacteristic
66#432—21no—
67#432—21no—
68#432—21no—
69#432D412yes—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431C2312yes—
77#431C2312yes—
78#423—21no—
79#423—21no—
80#423—21no—
81#423—21no—
82#423C2×C411yescharacteristic
83#423C2×C411yescharacteristic
84#423D411yescharacteristic
85#423D411yescharacteristic
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2312yes—
89C2×#101C2×C414yes—
90C2×#101C2×C414yes—
91#407—21no—
92#407C2×C412yes—
93C2×C22⋊C8—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16C2×C411yescharacteristic
98C4×M16C2×C411yescharacteristic
99C4×M16C2312yes—
100C4×M16C2312yes—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8D411yescharacteristic
108C2×C4×C8D411yescharacteristic
109C2×C4×C8D411yescharacteristic
110C2×C4×C8D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#343C2×C414yes—
114#343C2×C414yes—
115#335—24no—
116#334—24no—
117Q8⋊C8—24no—
118Q8⋊C8—24no—
119Q8⋊C8—22no—
120Q8⋊C8—22no—
121Q8⋊C8—22no—
122Q8⋊C8—22no—
123#322—22no—
124#322—22no—
125#322—22no—
126C82—22no—
127C22×Q8—21no—
128C22×Q8C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4#4111yescharacteristic
131C23×C4#4111yescharacteristic
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—24no—
136C2×M16—42no—
137C2×M16—41no—
138C2×M16—41no—
139C2×M16—82no—
140C2×M16C22×C412yes—
141C2×M16C22×C412yes—
142C2×M16C22×C412yes—
143C2×M16C22×C412yes—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—44no—
153C22×C8—41no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162#125—41no—
163#125—41no—
164#125—41no—
165#125—41no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—41no—
171C22⋊Q8—41no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182#117—41no—
183#117—41no—
184#117—41no—
185#117—41no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—22no—
193C2×C42—22no—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42C2411yescharacteristic
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8C2×D412yes—
219C4⋊C8C2×D412yes—
220#101—28no—
221#101—44no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227C22⋊C8—41no—
228#97—21no—
229#97—21no—
230#97—22no—
231#97—22no—
232#97—22no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—41no—
246C4×C8C2×D411yescharacteristic
247C4×C8C2×D411yescharacteristic
248C4×C8C2×D411yescharacteristic
249C4×C8C2×D411yescharacteristic
250C24#11711yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—82no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4#11711yescharacteristic
266C22×C4#11711yescharacteristic
267C22×C4#11711yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4C4×D411yescharacteristic
271C22×C4C4×D411yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#12111yescharacteristic
274C22×C4#12311yescharacteristic
275C22×C4#12311yescharacteristic
276C22×C4#12411yescharacteristic
277C22×C4#12411yescharacteristic
278C22×C4#12611yescharacteristic
279C22×C4#12611yescharacteristic
280C22×C4C23×C411yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283M16—22no—
284M16—24no—
285M16—22no—
286M16—22no—
287M16—24no—
288M16—44no—
289M16—42no—
290M16—42no—
291M16—82no—
292M16—82no—
293M16#13114yes—
294C2×C8—24no—
295C2×C8—22no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—44no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—82no—
314C2×C8—82no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C42—42no—
330C42—42no—
331C42C4×D411yescharacteristic
332C42C4×D411yescharacteristic
333C42C4×D411yescharacteristic
334C42C4×D411yescharacteristic
335C42#12111yescharacteristic
336C42#12111yescharacteristic
337C42#12311yescharacteristic
338C42#12311yescharacteristic
339C42#12411yescharacteristic
340C42#12411yescharacteristic
341C42#12611yescharacteristic
342C42#12611yescharacteristic
343C42C22×D411yescharacteristic
344C42C22×D411yescharacteristic
345C42C2×#4111yescharacteristic
346C42C2×#4111yescharacteristic
347C23—22no—
348C23—22no—
349C23—41no—
350C23#38811yescharacteristic
351C23#38811yescharacteristic
352C23C2×#11711yescharacteristic
353Q8—42no—
354Q8—42no—
355Q8—44no—
356Q8—82no—
357Q8—82no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4#38811yescharacteristic
383C2×C4#38811yescharacteristic
384C2×C4#38811yescharacteristic
385C2×C4#38811yescharacteristic
386C2×C4#38811yescharacteristic
387C2×C4#38811yescharacteristic
388C2×C4#47112yes—
389C2×C4#47312yes—
390C2×C4C2×C4×D411yescharacteristic
391C2×C4C2×C4×D411yescharacteristic
392C2×C4C2×#12111yescharacteristic
393C2×C4C2×#12311yescharacteristic
394C2×C4C2×#12411yescharacteristic
395C2×C4C2×#12611yescharacteristic
396C2×C4C2×#13112yes—
397C2×C4#56812yes—
398C8—22no—
399C8—22no—
400C8—24no—
401C8—41no—
402C8—41no—
403C8—42no—
404C8—42no—
405C8—42no—
406C8—42no—
407C8—42no—
408C8—82no—
409C8—82no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—44no—
418C22#193012yes—
419C22C2×#38811yescharacteristic
420C22#292512yes—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—82no—
429C4—82no—
430C4#193012yes—
431C4#193212yes—
432C4#293111yescharacteristic
433C4#293211yescharacteristic
434C4#293312yes—
435C4C2×#47111yescharacteristic
436C4C2×#47311yescharacteristic
437C2—22no—
438C2—22no—
439C2—41no—
440C2#2105111yescharacteristic
441C2C2×#193011yescharacteristic
442C2C2×#193211yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC24#117
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9551C2semidirect
7. #9549C2semidirect

Maximal subgroups

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

subgroupindex
#214252
#177962
#174792
C2×#14932
#173232
#95512
#95492
#79342
C2×#12782