Finite Group Databasebeta

← #388097

Subgroups — #388097

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1850C211yesmaximal, characteristic
2#13876C212yesmaximal
3#11267C212yesmaximal, complemented
4#11232C212yesmaximal, complemented
5#11096C211yesmaximal, characteristic
6#11096C211yesmaximal, characteristic
7#10907C212yesmaximal, complemented
8#10849C212yesmaximal, complemented
9#10721C212yesmaximal
10C24×C8C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C2×#388—22no—
16C2×#384C2212yes—
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1850C418yescomplemented
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31#1442—22no—
32#1439—22no—
33#1436—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C24×C4C2311yescharacteristic
37C23×C8—41no—
38C23×C8—41no—
39C23×C8—42no—
40C23×C8—42no—
41C23×C8C2×C414yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2312yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4D411yescharacteristic
55C22×C42—22no—
56C22×C42C2312yes—
57C2×#107—24no—
58C2×#107—41no—
59C2×#107—41no—
60C2×#107—42no—
61C2×C4⋊C8—42no—
62C2×C4⋊C8—42no—
63C2×C4⋊C8C2×C418yes—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8C2×C414yes—
67#388—48no—
68#384—24no—
69#384—42no—
70#380—24no—
71#380—24no—
72#380—42no—
73#380—42no—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—42no—
76C4×C22⋊C4C2×C414yes—
77C4×C22⋊C4Q814yes—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95D412yes—
86#340—24no—
87#340—41no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—41no—
93#340—42no—
94#336C2×C418yes—
95C25C2×D411yescharacteristic
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C22×C412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×Q812yes—
113C23×C4C2411yescharacteristic
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—48no—
129C22×C8—44no—
130C22×C8—48no—
131C22×C8—44no—
132C22×C8—44no—
133C22×C8C22×C414yes—
134C22×C8C22×C412yes—
135C22×C8C22×C412yes—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4#4114yes—
165C2×C4⋊C4#4114yes—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4C4⋊C414yes—
175C2×C22⋊C4C4⋊C414yes—
176C2×C22⋊C4D814yes—
177C2×C22⋊C4Q1614yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—44no—
183C2×C42—44no—
184C2×C42—82no—
185C2×C42C22×C414yes—
186C2×C42C2×Q814yes—
187#107—44no—
188#107—44no—
189C4⋊C8—48no—
190C22⋊C8—44no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—82no—
198#95C4⋊C418yes—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—22no—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—22no—
207C24C2×C4⋊C412yes—
208C24C2×D812yes—
209C24C2×Q1612yes—
210C24C22×D411yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C2×C4⋊C412yes—
272C22×C4C2×C4⋊C414yes—
273C22×C4C2×C4⋊C414yes—
274C22×C4C2×D814yes—
275C22×C4C2×Q1614yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×Q811yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4112yes—
280C22×C4H(5,𝔽2)12yes—
281C22×C4H(5,𝔽2)12yes—
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C42−1+411yescharacteristic
285C22×C42−1+411yescharacteristic
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—416no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—42no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—88no—
321C4⋊C4—88no—
322C22⋊C4—44no—
323C22⋊C4—88no—
324C22⋊C4—88no—
325C22⋊C4#107116yescomplemented
326C42—24no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—88no—
331C23—22no—
332C23—24no—
333C23—22no—
334C23—24no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—24no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—42no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23C2×#10714yes—
354C23C22×C4⋊C411yescharacteristic
355C23#53711yescharacteristic
356C23#53711yescharacteristic
357C23#54612yes—
358C23#54612yes—
359C23C22×D811yescharacteristic
360C23C22×Q1611yescharacteristic
361C23#57511yescharacteristic
362C23#57511yescharacteristic
363C23#57712yes—
364C23#57712yes—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4C2×#10718yes—
418C2×C4C4×#4112yes—
419C2×C4#51812yes—
420C2×C4#51811yescharacteristic
421C2×C4#51911yescharacteristic
422C2×C4#53912yes—
423C2×C4#53912yes—
424C2×C4#54212yes—
425C2×C4#54212yes—
426C2×C4#54411yescharacteristic
427C2×C4#54411yescharacteristic
428C2×C4#55111yescharacteristic
429C2×C4#55312yes—
430C2×C4#55412yes—
431C2×C4#55412yes—
432C2×C4#55911yescharacteristic
433C2×C4#56012yes—
434C2×C4#56011yescharacteristic
435C2×C4H(3,𝔽4)11yescharacteristic
436C8—42no—
437C8—42no—
438C8—44no—
439C22—28no—
440C22—24no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—44no—
448C22—42no—
449C22—44no—
450C22#231611yescharacteristic
451C22#234912yes—
452C22#269112yes—
453C22#269311yescharacteristic
454C22#271211yescharacteristic
455C22C22×#10711yescharacteristic
456C22#291111yescharacteristic
457C22#291212yes—
458C22#324511yescharacteristic
459C22#324511yescharacteristic
460C22#324911yescharacteristic
461C22#324911yescharacteristic
462C22#326012yes—
463C22#326311yescharacteristic
464C22#326311yescharacteristic
465C22#329412yes—
466C22#329412yes—
467C22#329912yes—
468C22#329912yes—
469C22#330712yes—
470C22#331112yes—
471C22#331712yes—
472C22#332112yes—
473C4—28no—
474C4—42no—
475C4—44no—
476C4—84no—
477C4—84no—
478C4—162no—
479C4—162no—
480C4#233212yes—
481C4#237411yescharacteristic
482C4#237511yescharacteristic
483C4#276311yescharacteristic
484C4#279012yes—
485C4#280711yescharacteristic
486C2—44no—
487C2#1451411yescharacteristic
488C2#2196111yescharacteristic
489C2#2196711yescharacteristic
490C2#2270512yes—
491C2#2271012yes—
492C2#3148612yes—
493C2#3149212yes—
494C2#3155011yescharacteristic
495C2#3155411yescharacteristic
496C2#3211011yescharacteristic
497C2#3211411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11267C2semidirect
4. #11232C2semidirect
7. #10907C2semidirect
8. #10849C2semidirect
21. #1850C4semidirect
325. C22⋊C4#107semidirect

Maximal subgroups

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

subgroupindex
C2×#18502
#138762
#112672
#112322
#110962
#109072
#108492
#107212