Finite Group Databasebeta

← #370468

Subgroups — #370468

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1940C211yesmaximal, characteristic
2C2×#1771C211yesmaximal, characteristic
3C4×#382C211yesmaximal, characteristic
4#11255C212yesmaximal
5#11255C212yesmaximal
6#10835C211yesmaximal, characteristic
7#10835C211yesmaximal, characteristic
8#10832C211yesmaximal, characteristic
9#10798C212yesmaximal
10#10797C212yesmaximal
11#10693C211yesmaximal, characteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#97C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#382C412yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C43C2211yescharacteristic
23#1940—24no—
24#1771—24no—
25C2×#336C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C4⋊#95C412yes—
37#1434C412yes—
38#1434C412yes—
39C4×#95C2212yes—
40C4×#95C2212yes—
41C23×C8C2312yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C42C2×C411yescharacteristic
50C22×C42C2×C412yes—
51C22×C42C2×C411yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2312yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—24no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—42no—
64C2×C4⋊C8—42no—
65C2×C4⋊C8Q814yes—
66C2×#97—22no—
67C2×#97—22no—
68C2×#97—22no—
69C2×#97—24no—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97—22no—
73C2×#97D414yes—
74C2×C4×C8—22no—
75C2×C4×C8—24no—
76#382—28no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C2×#95C2×C412yes—
82C2×#95C2×C412yes—
83C2×#95C2×C412yes—
84C2×#95C2×C412yes—
85C2×#95C2×C412yes—
86C2×#95C2×C411yescharacteristic
87C2×#95C2×C412yes—
88C2×#95C2×C411yescharacteristic
89C2×#95C2×C412yes—
90C2×#95C2×C412yes—
91C2×#95C2×C412yes—
92C43—22no—
93C43—22no—
94#336—24no—
95#336—22no—
96#336—22no—
97#336—24no—
98#336—24no—
99#336—24no—
100#336—24no—
101#336—42no—
102#336—42no—
103#336—42no—
104#336—42no—
105C23×C4C22×C412yes—
106C23×C4C22×C412yes—
107C23×C4C22×C412yes—
108C23×C4C22×C412yes—
109C23×C4C22×C412yes—
110C23×C4C22×C412yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C2411yescharacteristic
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8C2×Q812yes—
145C22×C8C2×Q812yes—
146C22×C8#4114yes—
147C22×C8#4114yes—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C42—22no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—22no—
158C2×C42—24no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42C2×D412yes—
182C2×C42C2×Q812yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42#4112yes—
187C4⋊C8—28no—
188C4⋊C8—48no—
189C4⋊C8—44no—
190#97—28no—
191#97—44no—
192#97—44no—
193C4×C8—44no—
194#95—22no—
195#95—24no—
196#95—22no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—42no—
203#95—42no—
204#95M1618yes—
205C24C23×C411yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
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—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—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4#11712yes—
249C22×C4#11712yes—
250C22×C4C4×D412yes—
251C22×C4C4×D412yes—
252C22×C4C4×Q812yes—
253C22×C4C4×Q812yes—
254C22×C4C2×M1614yes—
255C22×C4C2×M1614yes—
256C22×C4C2×M1614yes—
257C22×C4#13114yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C22×Q811yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4112yes—
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4112yes—
266C22×C4H(5,𝔽2)12yes—
267C22×C42−1+412yes—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8#12414yes—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C42—28no—
305C42—48no—
306C42—44no—
307C42—41no—
308C42—41no—
309C42—42no—
310C42—42no—
311C42—41no—
312C42—41no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42C22⋊Q814yes—
322C42#12314yes—
323C42#12514yes—
324C42#12614yes—
325C23C2×#11711yescharacteristic
326C23C2×C4×D411yescharacteristic
327C23C2×C4×Q811yescharacteristic
328C23C4×#4112yes—
329C23#52012yes—
330C23C22×M1612yes—
331C23C2×#13112yes—
332C23#56812yes—
333C23#56812yes—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4#38714yes—
385C2×C4#38714yes—
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4C2×#12311yescharacteristic
388C2×C4C2×#12411yescharacteristic
389C2×C4C2×#12511yescharacteristic
390C2×C4C2×#12611yescharacteristic
391C2×C4#52912yes—
392C2×C4#53211yescharacteristic
393C2×C4#53511yescharacteristic
394C2×C4#53611yescharacteristic
395C2×C4#54212yes—
396C2×C4#55212yes—
397C2×C4#55212yes—
398C2×C4#55212yes—
399C2×C4#55312yes—
400C2×C4#55412yes—
401C2×C4#55512yes—
402C2×C4#55512yes—
403C2×C4#55712yes—
404C8—24no—
405C8—42no—
406C8—42no—
407C8—44no—
408C8—42no—
409C8—82no—
410C22C2×#38711yescharacteristic
411C22C4×#12411yescharacteristic
412C22#232611yescharacteristic
413C22#234612yes—
414C22#237012yes—
415C22#292011yescharacteristic
416C22#292111yescharacteristic
417C22#292412yes—
418C22D4×M1612yes—
419C22#293512yes—
420C22#296011yescharacteristic
421C22#296111yescharacteristic
422C22Q8×M1612yes—
423C22#296512yes—
424C22#296712yes—
425C22#296812yes—
426C22#297012yes—
427C22#298912yes—
428C22#299012yes—
429C22#299212yes—
430C22#299512yes—
431C4—24no—
432C4—24no—
433C4—44no—
434C4—44no—
435C4—44no—
436C4—44no—
437C4—82no—
438C4—82no—
439C4#249512yes—
440C4#250011yescharacteristic
441C4#250311yescharacteristic
442C4#250812yes—
443C4#256311yescharacteristic
444C4#256511yescharacteristic
445C2#1405111yescharacteristic
446C2#2102311yescharacteristic
447C2#2102611yescharacteristic
448C2#2163212yes—
449C2#2221612yes—
450C2#2263412yes—
451C2#2263512yes—
452C2#2325512yes—
453C2#2325612yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socle, prime radicalC24C23×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×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×#19402
C2×#17712
C4×#3822
#112552
#108352
#108322
#107982
#107972
#106932