Finite Group Databasebeta

← #417099

Subgroups — #417099

Order 512.

960 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1836C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic
3#14354C211yesmaximal, characteristic
4#14277C211yesmaximal, characteristic
5#11358C212yesmaximal
6#11357C212yesmaximal
7#11208C212yesmaximal
8#11207C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×#97C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#389C2212yes—
15C2×#384C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1836—24no—
25#1835—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1444—22no—
37#1444—22no—
38#1444—22no—
39#1442—22no—
40C4⋊#95C2212yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#97—22no—
59C2×#97—22no—
60C2×#97Q814yes—
61C2×C4×C8—22no—
62C2×C4×C8—22no—
63C2×C4×C8Q814yes—
64#389—24no—
65#389—24no—
66#384—22no—
67#384—24no—
68#384—22no—
69#384—22no—
70#384—22no—
71#384—42no—
72#382—22no—
73#382—22no—
74#382—22no—
75#382—24no—
76#382—42no—
77#382D414yes—
78#376—22no—
79#376—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95D412yes—
89C2×#95D412yes—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#336—24no—
99#336—24no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2411yescharacteristic
111C22×C8—22no—
112C22×C8—22no—
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×C8C2×Q814yes—
124C22×C8C2×Q814yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
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—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C2×D414yes—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42C2×D412yes—
176C2×C42C2×Q812yes—
177C2×C42C2×Q812yes—
178C2×C42#4114yes—
179C2×C42#4114yes—
180C2×C42#4112yes—
181C2×C42#4112yes—
182C2×C42#4112yes—
183#97—28no—
184C4×C8—28no—
185#95—28no—
186#95—24no—
187#95—24no—
188#95—42no—
189#95—42no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—82no—
194#95—82no—
195C24C22×D411yescharacteristic
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
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—24no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
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#12114yes—
249C22×C4#12314yes—
250C22×C4#12412yes—
251C22×C4#12414yes—
252C22×C4#12412yes—
253C22×C4C4⋊Q814yes—
254C22×C4C22×D411yescharacteristic
255C22×C4C22×Q811yescharacteristic
256C22×C4C22×Q811yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4112yes—
261C22×C4C2×#4112yes—
262C22×C4H(5,𝔽2)12yes—
263C22×C42−1+412yes—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—88no—
295C4⋊C4—88no—
296C4⋊C4—88no—
297C4⋊C4—88no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C42—24no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42C22⋊Q814yes—
305C42C22⋊Q814yes—
306C42#12514yes—
307C42#12514yes—
308C23C2×#12111yescharacteristic
309C23C2×#12311yescharacteristic
310C23C2×#12411yescharacteristic
311C23C2×#12411yescharacteristic
312C23C2×C4⋊Q811yescharacteristic
313C23#53511yescharacteristic
314C23#53611yescharacteristic
315C23#57612yes—
316C23#57712yes—
317C23#57712yes—
318C23#57812yes—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4#39818yes—
377C2×C4C2×C22⋊Q811yescharacteristic
378C2×C4C2×C22⋊Q811yescharacteristic
379C2×C4C2×#12511yescharacteristic
380C2×C4C2×#12511yescharacteristic
381C2×C4#53312yes—
382C2×C4#53312yes—
383C2×C4#54412yes—
384C2×C4#54412yes—
385C2×C4#54712yes—
386C2×C4#55012yes—
387C2×C4#55212yes—
388C2×C4#55212yes—
389C2×C4#55312yes—
390C2×C4#55312yes—
391C2×C4#55612yes—
392C2×C4#55912yes—
393C8—44no—
394C8—44no—
395C22C2×#39811yescharacteristic
396C22#250311yescharacteristic
397C22#250411yescharacteristic
398C22#254711yescharacteristic
399C22#254811yescharacteristic
400C22#266011yescharacteristic
401C22#286211yescharacteristic
402C22#304512yes—
403C22#304612yes—
404C22#309612yes—
405C22#309712yes—
406C22#313612yes—
407C22#313612yes—
408C22#313712yes—
409C22#313712yes—
410C22#316312yes—
411C22#316412yes—
412C22#322011yescharacteristic
413C22#322111yescharacteristic
414C22#322211yescharacteristic
415C22#322311yescharacteristic
416C22#324312yes—
417C22#324412yes—
418C4—28no—
419C4—44no—
420C4—42no—
421C4—42no—
422C4—84no—
423C4—162no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4#244112yes—
429C4#244112yes—
430C4#257012yes—
431C4#257012yes—
432C2#1524211yescharacteristic
433C2#2366212yes—
434C2#2797112yes—
435C2#2798511yescharacteristic
436C2#2798611yescharacteristic
437C2#2873012yes—
438C2#2876011yescharacteristic
439C2#2876111yescharacteristic
440C2#3031311yescharacteristic
441C2#3031411yescharacteristic
442C2#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×#18362
C2×#18352
#143542
#142772
#113582
#113572
#112082
#112072
#108432
#108412
#106942