Finite Group Databasebeta

← #414228

Subgroups — #414228

Order 512.

1196 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1941C211yesmaximal, characteristic, complemented
2C2×C8×C22⋊C4C211yesmaximal, characteristic
3#15220C212yesmaximal
4#11327C211yesmaximal, characteristic, complemented
5#11257C211yesmaximal, characteristic, complemented
6#11208C212yesmaximal, complemented
7#11095C211yesmaximal, characteristic
8#11095C211yesmaximal, characteristic
9#10880C212yesmaximal, complemented
10#10842C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#382C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1941—24no—
25C8×C22⋊C4—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1443—22no—
37#1441—22no—
38#1439—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C23×C8—22no—
43C23×C8—22no—
44C23×C8—42no—
45C23×C8—42no—
46C23×C8Q814yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D412yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#106—22no—
63C2×#106—24no—
64C2×#106—22no—
65C2×#106—42no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8Q814yes—
69C2×C4×C8—22no—
70C2×C4×C8—42no—
71C2×C4×C8—42no—
72C2×C4×C8D414yes—
73#389—24no—
74#389—24no—
75#389—24no—
76#389—42no—
77#382—24no—
78#382—42no—
79#380—24no—
80#380—24no—
81#380—42no—
82#380—42no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95D412yes—
94#340—24no—
95#340—22no—
96#340—24no—
97#340—22no—
98#340—24no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—22no—
103#336—22no—
104C25C2×D411yescharacteristic
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×Q812yes—
121C23×C4C2×Q812yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—28no—
136C22×C8—48no—
137C22×C8—44no—
138C22×C8—44no—
139C22×C8—48no—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×Q812yes—
143C22×C8C2×Q812yes—
144C22×C8C2×Q812yes—
145C22×C8C2×Q812yes—
146C22×C8#4112yes—
147C22×C8#4112yes—
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—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4#4114yes—
177C2×C4⋊C4#4114yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C2×D412yes—
194C2×C42#4112yes—
195#106—28no—
196#106—44no—
197C22⋊C8—28no—
198C4×C8—44no—
199C4×C8—44no—
200#95—24no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—82no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24C4⋊Q812yes—
217C24#13514yes—
218C24C22×D411yescharacteristic
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
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×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4#12114yes—
282C22×C4C4⋊Q812yes—
283C22×C4#13514yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×Q811yescharacteristic
286C22×C4C22×Q811yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4C2×#4112yes—
291C22×C4C2×#4111yescharacteristic
292C22×C4H(5,𝔽2)12yes—
293C22×C4H(5,𝔽2)12yes—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—416no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8C22⋊Q814yes—
314C2×C8C22⋊Q814yes—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—88no—
332C4⋊C4—88no—
333C22⋊C4—216no—
334C22⋊C4—88no—
335C22⋊C4—88no—
336C42—22no—
337C42—22no—
338C42—24no—
339C42—41no—
340C42—41no—
341C42—42no—
342C23—22no—
343C23—22no—
344C23—24no—
345C23—24no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23#49914yes—
363C23C2×#12111yescharacteristic
364C23C2×C4⋊Q811yescharacteristic
365C23#53211yescharacteristic
366C23#54612yes—
367C23#54612yes—
368C23C2×#13512yes—
369C23C2×#13512yes—
370C23#57712yes—
371C23#57712yes—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
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×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4C2×C22⋊Q811yescharacteristic
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4#52511yescharacteristic
428C2×C4#52912yes—
429C2×C4#53211yescharacteristic
430C2×C4#53311yescharacteristic
431C2×C4#53311yescharacteristic
432C2×C4#53312yes—
433C2×C4#53411yescharacteristic
434C2×C4#53511yescharacteristic
435C2×C4#53812yes—
436C2×C4#54312yes—
437C2×C4#55312yes—
438C2×C4#55312yes—
439C2×C4#55312yes—
440C2×C4#55312yes—
441C2×C4#55412yes—
442C2×C4#55412yes—
443C8—24no—
444C8—24no—
445C8—42no—
446C8—42no—
447C22—28no—
448C22—24no—
449C22—42no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—42no—
455C22—44no—
456C22#253511yescharacteristic
457C22#254311yescharacteristic
458C22#254912yes—
459C22#255912yes—
460C22#285711yescharacteristic
461C22#305012yes—
462C22#305712yes—
463C22C2×#49911yescharacteristic
464C22#316011yescharacteristic
465C22#316312yes—
466C22#316812yes—
467C22#317311yescharacteristic
468C22#317311yescharacteristic
469C22#329612yes—
470C22#329612yes—
471C22#329612yes—
472C22#330112yes—
473C22#330112yes—
474C22#330112yes—
475C22#330312yes—
476C22#330412yes—
477C4—42no—
478C4—44no—
479C4—44no—
480C4—84no—
481C4—84no—
482C4—162no—
483C4—162no—
484C4#239611yescharacteristic
485C4#256611yescharacteristic
486C4#257911yescharacteristic
487C4#258712yes—
488C4#260812yes—
489C4#262611yescharacteristic
490C2—44no—
491C2#1516511yescharacteristic
492C2#2814412yes—
493C2#2855612yes—
494C2#2877012yes—
495C2#2877412yes—
496C2#2904412yes—
497C2#2904612yes—
498C2#3320811yescharacteristic
499C2#3321311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
1. C2×#1941C2semidirect
4. #11327C2semidirect
5. #11257C2semidirect
6. #11208C2semidirect
9. #10880C2semidirect
10. #10842C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19412
C2×C8×C22⋊C42
#152202
#113272
#112572
#112082
#110952
#108802
#108422
#107212