Finite Group Databasebeta

← #362289

Subgroups — #362289

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1836C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic
3C2×#1768C211yesmaximal, characteristic
4#14354C211yesmaximal, characteristic
5#13779C211yesmaximal, characteristic
6#11268C212yesmaximal
7#11267C212yesmaximal
8#10850C212yesmaximal
9#10849C212yesmaximal
10#10843C211yesmaximal, characteristic
11#10841C211yesmaximal, characteristic
12C22×C4⋊C8C2212yes—
13C22×#97C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2212yes—
24C2×#376C2211yescharacteristic
25#1836—24no—
26#1835—24no—
27#1768C418yescomplemented
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1444—22no—
38#1444—22no—
39#1436—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×C4⋊C8—24no—
60C2×C4⋊C8—24no—
61C2×C4⋊C8C2×C418yes—
62C2×#97—22no—
63C2×#97—22no—
64C2×#97C2×C414yes—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8C2×C414yes—
68#384—22no—
69#384—22no—
70#384—24no—
71#384—22no—
72#384—42no—
73#384Q814yes—
74#382—22no—
75#382—22no—
76#382—22no—
77#382—24no—
78#382—42no—
79#382D414yes—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82#376C2×C414yes—
83#376Q814yes—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D412yes—
93C2×#95D412yes—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336C2×C418yes—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
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—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8C22×C414yes—
127C22×C8C22×C414yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×Q814yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—44no—
172C2×C42—82no—
173C2×C42—82no—
174C2×C42C22×C412yes—
175C2×C42C22×C412yes—
176C2×C42C22×C412yes—
177C2×C42C2×D412yes—
178C2×C42C2×Q812yes—
179C2×C42C2×Q812yes—
180C2×C42C2×Q812yes—
181C2×C42C2×Q812yes—
182C2×C42#4114yes—
183C2×C42#4114yes—
184C2×C42#4112yes—
185C2×C42#4112yes—
186C4⋊C8—216no—
187#97—28no—
188C4×C8—28no—
189#95—28no—
190#95—42no—
191#95—42no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—82no—
196#95—82no—
197#95C4⋊C418yes—
198#95C4⋊C418yes—
199C24C22×D411yescharacteristic
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C2×C4⋊C412yes—
251C22×C4C2×C4⋊C412yes—
252C22×C4C2×C4⋊C414yes—
253C22×C4C2×C4⋊C414yes—
254C22×C4#12114yes—
255C22×C4C22⋊Q814yes—
256C22×C4#12414yes—
257C22×C4#12414yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×Q811yescharacteristic
261C22×C4C22×Q811yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4112yes—
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4H(5,𝔽2)12yes—
267C22×C42−1+412yes—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—88no—
299C4⋊C4—88no—
300C4⋊C4—88no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C42—24no—
304C42—24no—
305C42—88no—
306C42#11718yes—
307C42C4×D414yes—
308C42C4×Q814yes—
309C42#12518yes—
310C42#12514yes—
311C42C4⋊Q814yes—
312C23C22×C4⋊C411yescharacteristic
313C23C2×#12111yescharacteristic
314C23C2×C22⋊Q811yescharacteristic
315C23C2×#12411yescharacteristic
316C23C2×#12411yescharacteristic
317C23#53511yescharacteristic
318C23#53611yescharacteristic
319C23#57612yes—
320C23#57712yes—
321C23#57712yes—
322C23#57812yes—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4#38218yes—
380C2×C4#38918yes—
381C2×C4C2×#11712yes—
382C2×C4C2×C4×D411yescharacteristic
383C2×C4C2×C4×Q811yescharacteristic
384C2×C4#52012yes—
385C2×C4C2×#12511yescharacteristic
386C2×C4C2×#12512yes—
387C2×C4C2×C4⋊Q811yescharacteristic
388C2×C4#54412yes—
389C2×C4#54712yes—
390C2×C4#55012yes—
391C2×C4#55412yes—
392C2×C4#55512yes—
393C2×C4#55612yes—
394C2×C4#55612yes—
395C2×C4#55712yes—
396C2×C4#55912yes—
397C8—44no—
398C8—44no—
399C22C2×#38211yescharacteristic
400C22C2×#38911yescharacteristic
401C22#231711yescharacteristic
402C22#254711yescharacteristic
403C22#254811yescharacteristic
404C22#256311yescharacteristic
405C22#256811yescharacteristic
406C22#291212yes—
407C22#291312yes—
408C22#304512yes—
409C22#304612yes—
410C22#308112yes—
411C22#308212yes—
412C22#313612yes—
413C22#313612yes—
414C22#313712yes—
415C22#313712yes—
416C22#322011yescharacteristic
417C22#322111yescharacteristic
418C22#322211yescharacteristic
419C22#322311yescharacteristic
420C22#324312yes—
421C22#324412yes—
422C4—28no—
423C4—28no—
424C4—42no—
425C4—42no—
426C4—84no—
427C4—84no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4#235412yes—
433C4#235512yes—
434C4#255212yes—
435C4#257112yes—
436C2#1391111yescharacteristic
437C2#2076312yes—
438C2#2117512yes—
439C2#2199011yescharacteristic
440C2#2199111yescharacteristic
441C2#2873012yes—
442C2#2876011yescharacteristic
443C2#2876111yescharacteristic
444C2#2915211yescharacteristic
445C2#2915311yescharacteristic
446C2#2922912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1768C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18362
C2×#18352
C2×#17682
#143542
#137792
#112682
#112672
#108502
#108492
#108432
#108412