Finite Group Databasebeta

← #417050

Subgroups — #417050

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4C2×#1835C211yesmaximal, characteristic
5#14354C211yesmaximal, characteristic
6#14282C211yesmaximal, characteristic
7#11373C212yesmaximal
8#11372C212yesmaximal
9#11372C212yesmaximal
10#11371C212yesmaximal
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#384C2212yes—
17C2×#382C2211yescharacteristic
18C2×#382C2212yes—
19C2×#382C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1993—24no—
22#1990—24no—
23#1836—24no—
24#1835—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1446—22no—
38#1445—22no—
39#1444—22no—
40#1444—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×Q8—21no—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8Q814yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×#97—22no—
66C2×#97—22no—
67C2×#97D414yes—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8D414yes—
71#384—22no—
72#384—22no—
73#384—24no—
74#384—24no—
75#382—22no—
76#382—22no—
77#382—22no—
78#382—22no—
79#382—24no—
80#382—24no—
81#376—22no—
82#376—22no—
83C2×#95—24no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#336—24no—
103#336—24no—
104C22×Q8—24no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—82no—
108C22×Q8—82no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2×D414yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8C2×D414yes—
136C4⋊Q8—28no—
137C4⋊Q8—28no—
138C4⋊Q8—44no—
139C4⋊Q8—44no—
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—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
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—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4—84no—
177C2×C4⋊C4C2×Q814yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×Q812yes—
185C2×C42#4114yes—
186C2×C42#4114yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189C2×C42#4112yes—
190#97—28no—
191C4×C8—28no—
192#95—24no—
193#95—24no—
194#95—48no—
195#95—42no—
196#95—42no—
197#95—42no—
198#95—42no—
199#95—84no—
200#95—84no—
201C24C22×D411yescharacteristic
202C2×Q8—48no—
203C2×Q8—88no—
204C2×Q8—88no—
205C2×Q8—84no—
206C2×Q8—84no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4C22⋊Q812yes—
244C22×C4C22⋊Q812yes—
245C22×C4#12312yes—
246C22×C4#12312yes—
247C22×C4#12418yes—
248C22×C4#12712yes—
249C22×C4#12712yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×Q811yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258C22×C42−1+412yes—
259C22×C42−1+412yes—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—24no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—24no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—48no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C4⋊C4—816no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—24no—
305C42C22⋊Q814yes—
306C42C22⋊Q814yes—
307C42#12314yes—
308C42#12314yes—
309C23C2×C22⋊Q811yescharacteristic
310C23C2×#12311yescharacteristic
311C23C2×#12412yes—
312C23C2×#12711yescharacteristic
313C23#53612yes—
314C23#57612yes—
315C23#57712yes—
316C23#57712yes—
317C23#57812yes—
318Q8—816no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—28no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—44no—
336C2×C4—84no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4#39914yes—
361C2×C4#39914yes—
362C2×C4C2×C22⋊Q811yescharacteristic
363C2×C4C2×C22⋊Q811yescharacteristic
364C2×C4C2×#12311yescharacteristic
365C2×C4C2×#12311yescharacteristic
366C2×C4#53211yescharacteristic
367C2×C4#53211yescharacteristic
368C2×C4#53212yes—
369C2×C4#53611yescharacteristic
370C2×C4#53611yescharacteristic
371C2×C4#53612yes—
372C2×C4#55212yes—
373C2×C4#55212yes—
374C2×C4#55212yes—
375C2×C4#55212yes—
376C2×C4#55612yes—
377C2×C4#55612yes—
378C2×C4#55712yes—
379C2×C4#55712yes—
380C8—42no—
381C8—42no—
382C8—44no—
383C22C2×#39911yescharacteristic
384C22#250312yes—
385C22#256812yes—
386C22#265811yescharacteristic
387C22#285811yescharacteristic
388C22#308112yes—
389C22#308212yes—
390C22#309612yes—
391C22#309712yes—
392C22#313614yes—
393C22#313714yes—
394C22#315411yescharacteristic
395C22#315411yescharacteristic
396C22#315511yescharacteristic
397C22#315611yescharacteristic
398C22#324314yes—
399C22#324414yes—
400C4—24no—
401C4—24no—
402C4—48no—
403C4—84no—
404C4—161no—
405C4—161no—
406C4—161no—
407C4—162no—
408C4—161no—
409C4#249911yescharacteristic
410C4#249911yescharacteristic
411C4#249912yes—
412C4#256511yescharacteristic
413C4#256511yescharacteristic
414C4#256512yes—
415C2#1524011yescharacteristic
416C2#2373511yescharacteristic
417C2#2373611yescharacteristic
418C2#2797114yes—
419C2#2922914yes—
420C2#3028911yescharacteristic
421C2#3029011yescharacteristic
422C2#3322811yescharacteristic
423C2#3322911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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×#19932
C2×#19902
C2×#18362
C2×#18352
#143542
#142822
#113732
#113722
#113712
#106942