Finite Group Databasebeta

← #413649

Subgroups — #413649

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2088C212yesmaximal
2C2×#1944C211yesmaximal, characteristic
3C2×#1944C211yesmaximal, characteristic
4C2×#1846C211yesmaximal, characteristic
5C2×#1771C211yesmaximal, characteristic
6#15234C212yesmaximal
7#11378C212yesmaximal
8#11372C212yesmaximal
9#10863C211yesmaximal, characteristic
10#10863C211yesmaximal, characteristic
11#10861C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#382C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#2088—28no—
26#1944—24no—
27#1944—24no—
28#1846—24no—
29#1771—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1442—24no—
38#1441—24no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—24no—
69C2×#106Q814yes—
70C2×#106Q814yes—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8D414yes—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97—22no—
77C2×#97—22no—
78C2×#97—22no—
79C2×#97D414yes—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8D414yes—
83#389—24no—
84#389—24no—
85#389—24no—
86#389—24no—
87#382—24no—
88#382—24no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C2×#95—24no—
92C2×#95—22no—
93C2×#95—24no—
94C2×#95—22no—
95C2×#95—24no—
96C2×#95—24no—
97C2×#95D412yes—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—22no—
104#336—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2411yescharacteristic
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8C2×D412yes—
131C22×C8C2×D412yes—
132C22×C8C2×D412yes—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8C2×D412yes—
136C22×C8C2×Q812yes—
137C22×C8C2×Q812yes—
138C22×C8#4112yes—
139C22×C8#4112yes—
140C22×C8#4112yes—
141C22×C8#4112yes—
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—24no—
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—44no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—84no—
170C2×C4⋊C4C2×Q814yes—
171C2×C4⋊C4C2×Q814yes—
172C2×C42—21no—
173C2×C42—21no—
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—21no—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42#4112yes—
190#107—44no—
191#107—44no—
192#106—28no—
193#106—28no—
194#106—48no—
195C4⋊C8—28no—
196#97—28no—
197#97—44no—
198C4×C8—28no—
199#95—24no—
200#95—48no—
201#95—48no—
202#95—44no—
203#95—44no—
204#95—84no—
205#95—84no—
206C24C22×D411yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4#12114yes—
243C22×C4#12712yes—
244C22×C4#12712yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C22×Q811yescharacteristic
249C22×C4C22×Q811yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C42−1+411yescharacteristic
256C22×C42−1+411yescharacteristic
257C22×C42−1+411yescharacteristic
258C22×C42−1+411yescharacteristic
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8#12114yes—
280C2×C8#12114yes—
281C2×C8C22⋊Q814yes—
282C2×C8C22⋊Q814yes—
283C2×C8#12314yes—
284C2×C8#12514yes—
285C2×C8#12514yes—
286C2×C8C4⋊Q814yes—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—48no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—88no—
305C4⋊C4—88no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C23C2×#12111yescharacteristic
319C23C2×#12711yescharacteristic
320C23#53211yescharacteristic
321C23#54711yescharacteristic
322C23#54711yescharacteristic
323C23D4×Q811yescharacteristic
324C23D4×Q811yescharacteristic
325C23#57514yes—
326C23#57714yes—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—84no—
350C2×C4—84no—
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—84no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4C2×#12111yescharacteristic
366C2×C4C2×#12111yescharacteristic
367C2×C4C2×C22⋊Q811yescharacteristic
368C2×C4C2×C22⋊Q811yescharacteristic
369C2×C4C2×#12311yescharacteristic
370C2×C4C2×#12511yescharacteristic
371C2×C4C2×#12511yescharacteristic
372C2×C4C2×C4⋊Q811yescharacteristic
373C2×C4#53211yescharacteristic
374C2×C4#53211yescharacteristic
375C2×C4#53611yescharacteristic
376C2×C4#53611yescharacteristic
377C2×C4#53611yescharacteristic
378C2×C4#53611yescharacteristic
379C2×C4#54111yescharacteristic
380C2×C4#54111yescharacteristic
381C2×C4#54711yescharacteristic
382C2×C4#54711yescharacteristic
383C2×C4D4×Q811yescharacteristic
384C2×C4D4×Q811yescharacteristic
385C2×C4#55211yescharacteristic
386C2×C4#55211yescharacteristic
387C2×C4#55211yescharacteristic
388C2×C4#55211yescharacteristic
389C2×C4#55711yescharacteristic
390C2×C4#55711yescharacteristic
391C2×C4#55711yescharacteristic
392C2×C4#55711yescharacteristic
393C8—24no—
394C8—42no—
395C8—42no—
396C8#39818yes—
397C22#248611yescharacteristic
398C22#250611yescharacteristic
399C22#250611yescharacteristic
400C22#254211yescharacteristic
401C22#254211yescharacteristic
402C22#254611yescharacteristic
403C22#285611yescharacteristic
404C22#305412yes—
405C22#306012yes—
406C22#315312yes—
407C22#315412yes—
408C22#317312yes—
409C22#318312yes—
410C22#334814yes—
411C22#335014yes—
412C22#338814yes—
413C22#338814yes—
414C4—44no—
415C4—42no—
416C4—42no—
417C4—82no—
418C4—84no—
419C4—84no—
420C4—162no—
421C4—162no—
422C4C2×#39811yescharacteristic
423C4#248611yescharacteristic
424C4#250611yescharacteristic
425C4#250611yescharacteristic
426C4#254611yescharacteristic
427C4#256511yescharacteristic
428C4#256511yescharacteristic
429C4#262111yescharacteristic
430C2#1515111yescharacteristic
431C2#2766512yes—
432C2#2800212yes—
433C2#2800312yes—
434C2#2854712yes—
435C2#2854912yes—
436C2#2868112yes—
437C2#3317711yescharacteristic
438C2#3318111yescharacteristic
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×#20882
C2×#19442
C2×#18462
C2×#17712
#152342
#113782
#113722
#108632
#108612