Finite Group Databasebeta

← #415176

Subgroups — #415176

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1992C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3#14354C211yesmaximal, characteristic
4#14283C211yesmaximal, characteristic
5#11358C212yesmaximal
6#11357C212yesmaximal
7#11268C212yesmaximal
8#11267C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1992—24no—
24#1991—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1444—22no—
36#1444—22no—
37#1441—22no—
38#1441—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#125—22no—
44C2×#125—22no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125D414yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#97—22no—
65C2×#97—22no—
66C2×#97—22no—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70#384—22no—
71#384—24no—
72#384—22no—
73#384—22no—
74#384—22no—
75#384—42no—
76#382—22no—
77#382—22no—
78#382—24no—
79#382—42no—
80#376—22no—
81#376—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D412yes—
91C2×#95D412yes—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#336—24no—
101#336Q818yes—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4C2×D412yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2411yescharacteristic
113C22×C8—22no—
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×C8C2×Q814yes—
126C22×C8C2×Q814yes—
127#125—28no—
128#125—28no—
129#125—44no—
130#125—44no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
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—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C2×D414yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42C2×D412yes—
182C2×C42C2×Q814yes—
183C2×C42#4114yes—
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189#97—44no—
190C4×C8—44no—
191#95—24no—
192#95—24no—
193#95—42no—
194#95—42no—
195#95—44no—
196#95—44no—
197#95—82no—
198#95—82no—
199#95—82no—
200#95—82no—
201C24C22×D411yescharacteristic
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
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—24no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
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#12114yes—
255C22×C4#12314yes—
256C22×C4#12414yes—
257C22×C4#12414yes—
258C22×C4C4⋊Q812yes—
259C22×C4C4⋊Q812yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×Q812yes—
262C22×C4C2×#4111yescharacteristic
263C22×C4C2×#4111yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4H(5,𝔽2)12yes—
269C22×C42−1+412yes—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—416no—
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—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—88no—
302C4⋊C4—88no—
303C4⋊C4—88no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42#12314yes—
311C42#12314yes—
312C42#12614yes—
313C42#12614yes—
314C23C2×#12111yescharacteristic
315C23C2×#12311yescharacteristic
316C23C2×#12411yescharacteristic
317C23C2×#12411yescharacteristic
318C23C2×C4⋊Q811yescharacteristic
319C23#53511yescharacteristic
320C23#53611yescharacteristic
321C23#57612yes—
322C23#57712yes—
323C23#57712yes—
324C23#57812yes—
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—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
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—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4#39718yes—
383C2×C4C2×#12311yescharacteristic
384C2×C4C2×#12311yescharacteristic
385C2×C4C2×#12611yescharacteristic
386C2×C4C2×#12611yescharacteristic
387C2×C4#53312yes—
388C2×C4#53312yes—
389C2×C4#54212yes—
390C2×C4#54212yes—
391C2×C4D4×Q812yes—
392C2×C4#55012yes—
393C2×C4#55212yes—
394C2×C4#55212yes—
395C2×C4#55412yes—
396C2×C4#55412yes—
397C2×C4#55712yes—
398C2×C4#55912yes—
399C8—44no—
400C8—44no—
401C22C2×#39711yescharacteristic
402C22#250211yescharacteristic
403C22#250311yescharacteristic
404C22#254611yescharacteristic
405C22#254811yescharacteristic
406C22#264011yescharacteristic
407C22#286211yescharacteristic
408C22#304512yes—
409C22#304612yes—
410C22#309812yes—
411C22#309912yes—
412C22#313612yes—
413C22#313612yes—
414C22#313712yes—
415C22#313712yes—
416C22#316312yes—
417C22#316412yes—
418C22#322011yescharacteristic
419C22#322111yescharacteristic
420C22#322211yescharacteristic
421C22#322311yescharacteristic
422C22#324112yes—
423C22#324212yes—
424C4—28no—
425C4—42no—
426C4—42no—
427C4—44no—
428C4—84no—
429C4—162no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4#242312yes—
435C4#242312yes—
436C4#251112yes—
437C4#251112yes—
438C2#1518511yescharacteristic
439C2#2363412yes—
440C2#2795711yescharacteristic
441C2#2795811yescharacteristic
442C2#2797012yes—
443C2#2869812yes—
444C2#2876011yescharacteristic
445C2#2876111yescharacteristic
446C2#3020911yescharacteristic
447C2#3021011yescharacteristic
448C2#3329212yes—
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×#19922
C2×#19912
#143542
#142832
#113582
#113572
#112682
#112672
#108432
#108412
#106942