Finite Group Databasebeta

← #106944

Subgroups — #106944

Order 512.

861 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#20735C211yesmaximal, characteristic
3C4×#419C211yesmaximal, characteristic
4#8074C211yesmaximal, characteristic
5#8060C211yesmaximal, characteristic
6#8052C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—22no—
10#2984—22no—
11#2983—22no—
12#2983—21no—
13#2982—21no—
14Q8×M16—22no—
15Q8×M16C414yescomplemented
16#2920—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#419—21no—
21C2×C4×M16C2211yescharacteristic
22#2283C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1823—21no—
25#1805C412yescomplemented
26#1805C412yescomplemented
27#1787—21no—
28#1763C414yescomplemented
29C4×#103—22no—
30C4×#103—21no—
31C4×#103—21no—
32#1736—21no—
33#1300—22no—
34#1299—22no—
35#1297—22no—
36#1276—22no—
37C23×Q8D411yescharacteristic
38C22×M16—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8C2×C411yescharacteristic
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C412yes—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×C4×Q8D412yes—
52C2×C4×Q8Q812yes—
53C2×#117—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58#446—28no—
59#446—24no—
60#446—24no—
61#446—28no—
62C8×Q8—24no—
63C8×Q8—24no—
64#432—24no—
65#432—24no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432D412yes—
70#431—24no—
71#431—24no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431D412yes—
80#423—24no—
81#423C2×C412yes—
82#423D412yes—
83C2×C4⋊C8C2×C412yes—
84#419—28no—
85C2×#103—24no—
86C2×#103—22no—
87C2×#103—22no—
88#407—21no—
89#407D412yes—
90C2×C22⋊C8—21no—
91C4×M16—24no—
92C4×M16—21no—
93C4×M16—21no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16C2×C412yes—
98C2×#97C2×C412yes—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101#383—21no—
102#382C2×C412yes—
103#381—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2×C412yes—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111#376—21no—
112#376—21no—
113C43—21no—
114#340—22no—
115#340—22no—
116#340—24no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22⋊C412yes—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C4⋊C412yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4#4111yescharacteristic
131C2×M16—24no—
132C2×M16—22no—
133C2×M16—42no—
134C2×M16—41no—
135C2×M16—41no—
136C2×M16C22⋊C414yes—
137C22×C8—22no—
138C22×C8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—22no—
145C4×Q8—21no—
146C4×Q8—21no—
147C4×Q8—21no—
148C4×Q8—22no—
149C4×Q8—21no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—22no—
155C4×Q8C4212yes—
156C4×Q8C4212yes—
157C4×Q8C22⋊C412yes—
158C4×Q8C22⋊C412yes—
159C4×Q8C4⋊C412yes—
160C4×Q8C4⋊C412yes—
161C4×Q8C4⋊C412yes—
162C4×Q8C4⋊C412yes—
163#117—24no—
164#117—42no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C4⋊C412yes—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×Q811yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—28no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8C22⋊C414yes—
216#103—216no—
217#103—44no—
218#103—44no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224C22⋊C8—41no—
225#97—28no—
226#97—22no—
227#97—22no—
228#97—22no—
229#97—22no—
230C4×C8—28no—
231C4×C8—22no—
232C4×C8—22no—
233C4×C8—41no—
234C4×C8—41no—
235C4×C8—41no—
236C4×C8—41no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—41no—
241#95—41no—
242#95—41no—
243C24#12111yescharacteristic
244C2×Q8—22no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—21no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—44no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8#9512yes—
259C2×Q8#9512yes—
260C2×Q8#10414yes—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—82no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C4⋊C411yescharacteristic
281C22×C4#11711yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4C22⋊Q811yescharacteristic
292C22×C4C22⋊Q811yescharacteristic
293C22×C4#12311yescharacteristic
294C22×C4#12311yescharacteristic
295M16—44no—
296M16—44no—
297M16—44no—
298M16—44no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4#9512yes—
324C4⋊C4#9512yes—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—44no—
336C42—44no—
337C42—82no—
338C42—82no—
339C42C2×C4211yescharacteristic
340C42C2×C22⋊C411yescharacteristic
341C42C2×C22⋊C411yescharacteristic
342C42C2×C4⋊C411yescharacteristic
343C42C2×C4⋊C411yescharacteristic
344C42#11711yescharacteristic
345C42C4×D411yescharacteristic
346C42C4×D411yescharacteristic
347C42C4×D411yescharacteristic
348C42C4×D411yescharacteristic
349C42#12011yescharacteristic
350C42#12011yescharacteristic
351C42C22⋊Q811yescharacteristic
352C42C22⋊Q811yescharacteristic
353C42#12311yescharacteristic
354C42#12311yescharacteristic
355C23—22no—
356C23—22no—
357C23—41no—
358C23#38011yescharacteristic
359C23#38611yescharacteristic
360C23#38611yescharacteristic
361Q8—24no—
362Q8—22no—
363Q8—22no—
364Q8—22no—
365Q8—44no—
366Q8—44no—
367Q8—42no—
368Q8#33914yes—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4C2×#9511yescharacteristic
407C2×C4C4×C22⋊C411yescharacteristic
408C2×C4C4×C22⋊C411yescharacteristic
409C2×C4#37911yescharacteristic
410C2×C4#38011yescharacteristic
411C2×C4#38111yescharacteristic
412C2×C4#38511yescharacteristic
413C2×C4#38511yescharacteristic
414C2×C4#38511yescharacteristic
415C2×C4#38511yescharacteristic
416C2×C4#38611yescharacteristic
417C2×C4#38611yescharacteristic
418C2×C4C2×#10412yes—
419C2×C4#42112yes—
420C8—44no—
421C8—44no—
422C8—44no—
423C8—82no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—44no—
432C22C22⋊#9511yescharacteristic
433C22#187112yes—
434C22#188112yes—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—24no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—84no—
449C4—162no—
450C4C2×#33911yescharacteristic
451C4#173311yescharacteristic
452C4#179612yes—
453C4#187112yes—
454C2—22no—
455C2—22no—
456C2—41no—
457C2#1073211yescharacteristic
458C2#1077711yescharacteristic
459C2#1078211yescharacteristic
subgroupG/H
centerC2×C4#379
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. Q8×M16C4semidirect
25. #1805C4semidirect
26. #1805C4semidirect
28. #1763C4semidirect

Maximal subgroups

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

subgroupindex
#602482
#207352
C4×#4192
#80742
#80602
#80522
#78692