Finite Group Databasebeta

← #465228

Subgroups — #465228

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2086C211yesmaximal, characteristic
2C2×#1942C211yesmaximal, characteristic
3C2×#1916C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#15133C212yesmaximal
6#11379C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11292C212yesmaximal
9#11256C211yesmaximal, characteristic
10#11232C211yesmaximal, characteristic
11#11207C212yesmaximal
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#398—22no—
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#384C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21#2086—24no—
22#1942—24no—
23#1916—24no—
24#1834—24no—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1441—22no—
38#1441—22no—
39#1441—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—22no—
61C2×#107—24no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8D414yes—
71#398—48no—
72#389—24no—
73#389—24no—
74#389—24no—
75#389—42no—
76#384—24no—
77#384—42no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95D411yescharacteristic
89C2×#95D411yescharacteristic
90#340—22no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340D414yes—
98#336—22no—
99#336—22no—
100#336—22no—
101#336—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2411yescharacteristic
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8C2×D412yes—
130C22×C8C2×D412yes—
131C22×C8C2×D412yes—
132C22×C8#4112yes—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
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—24no—
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—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4D814yes—
172C2×C4⋊C4C2×D414yes—
173C2×C4⋊C4#4114yes—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42C2×D412yes—
189C2×C42#4112yes—
190#107—28no—
191#107—44no—
192C4⋊C8—44no—
193C4×C8—28no—
194#95—22no—
195#95—22no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—82no—
202#95—82no—
203#95—82no—
204C24C22×D411yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
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×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#12112yes—
261C22×C4#12112yes—
262C22×C4C2×D812yes—
263C22×C4C2×D812yes—
264C22×C4C2×D812yes—
265C22×C4#13512yes—
266C22×C4#13714yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4111yescharacteristic
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C42−1+412yes—
274C22×C42−1+412yes—
275C22×C42−1+412yes—
276C22×C42−1+411yescharacteristic
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8#12114yes—
297C4⋊C4—28no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—88no—
316C4⋊C4—88no—
317C4⋊C4—88no—
318C4⋊C4—88no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—41no—
329C42—41no—
330C23C2×#12111yescharacteristic
331C23#53612yes—
332C23D4211yescharacteristic
333C23#54711yescharacteristic
334C23#54812yes—
335C23C22×D811yescharacteristic
336C23C2×#13511yescharacteristic
337C23C2×#13712yes—
338C23#57611yescharacteristic
339C23#57712yes—
340C23#57811yescharacteristic
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4#46614yes—
402C2×C4#47014yes—
403C2×C4C2×#12111yescharacteristic
404C2×C4#52511yescharacteristic
405C2×C4#53611yescharacteristic
406C2×C4#53611yescharacteristic
407C2×C4#53611yescharacteristic
408C2×C4#53612yes—
409C2×C4#53711yescharacteristic
410C2×C4#54112yes—
411C2×C4#54112yes—
412C2×C4#54112yes—
413C2×C4#54212yes—
414C2×C4#54712yes—
415C2×C4#54811yescharacteristic
416C2×C4#55212yes—
417C2×C4#55212yes—
418C2×C4#55311yescharacteristic
419C2×C4#56312yes—
420C2×C4#56412yes—
421C8—24no—
422C8—42no—
423C8—81no—
424C8—81no—
425C22#241611yescharacteristic
426C22#253611yescharacteristic
427C22#262111yescharacteristic
428C22#267412yes—
429C22#271912yes—
430C22C2×#46611yescharacteristic
431C22C2×#47011yescharacteristic
432C22#306011yescharacteristic
433C22#306111yescharacteristic
434C22#322612yes—
435C22#323112yes—
436C22#324112yes—
437C22#324212yes—
438C22#326612yes—
439C22D4×D811yescharacteristic
440C22#328511yescharacteristic
441C22#333411yescharacteristic
442C22#333911yescharacteristic
443C22#334412yes—
444C22#335812yes—
445C22#336012yes—
446C22#337412yes—
447C22#337512yes—
448C4—24no—
449C4—24no—
450C4—42no—
451C4—42no—
452C4—84no—
453C4—84no—
454C4—81no—
455C4—81no—
456C4—162no—
457C4—162no—
458C4#241311yescharacteristic
459C4#262111yescharacteristic
460C4#263711yescharacteristic
461C4#271912yes—
462C4#276612yes—
463C4#282211yescharacteristic
464C2#1651711yescharacteristic
465C2#2478011yescharacteristic
466C2#2478311yescharacteristic
467C2#2818511yescharacteristic
468C2#2819611yescharacteristic
469C2#2987611yescharacteristic
470C2#2988211yescharacteristic
471C2#3080512yes—
472C2#3081412yes—
473C2#3231012yes—
474C2#3231512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20862
C2×#19422
C2×#19162
C2×#18342
#151332
#113792
#113152
#112922
#112562
#112322
#112072