Finite Group Databasebeta

← #416769

Subgroups — #416769

Order 512.

932 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1847C211yesmaximal, characteristic
2C2×#1834C211yesmaximal, characteristic
3#14348C211yesmaximal, characteristic
4#14276C211yesmaximal, characteristic
5#11257C212yesmaximal
6#11256C212yesmaximal
7#11220C212yesmaximal
8#11219C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10842C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#376C2211yescharacteristic
22#1847—24no—
23#1834—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1442—22no—
33#1442—22no—
34#1441—22no—
35#1441—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×#107—24no—
58C2×#107—24no—
59C2×#107—42no—
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—42no—
63C2×#97—22no—
64C2×#97—22no—
65C2×#97Q814yes—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8Q814yes—
69#389—42no—
70#389—42no—
71#384—22no—
72#384—22no—
73#384—22no—
74#384D414yes—
75#382—22no—
76#382—22no—
77#382—22no—
78#382—22no—
79#376—22no—
80#376—22no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D412yes—
93C2×#95D412yes—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#336—24no—
101#336—24no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
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×C8C2×Q814yes—
124C22×C8C2×Q814yes—
125C22×C8#4114yes—
126C22×C8#4114yes—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
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—21no—
145C2×C4⋊C4—21no—
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—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
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⋊C4C2×D414yes—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42C2×D412yes—
175C2×C42C2×Q812yes—
176C2×C42C2×Q812yes—
177C2×C42#4112yes—
178C2×C42#4112yes—
179C2×C42#4112yes—
180#107—48no—
181#106—48no—
182#97—28no—
183C4×C8—28no—
184#95—24no—
185#95—24no—
186#95—44no—
187#95—42no—
188#95—42no—
189#95—42no—
190#95—42no—
191#95—82no—
192#95—82no—
193#95—82no—
194#95—82no—
195#95—82no—
196#95—82no—
197C24C22×D411yescharacteristic
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—22no—
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—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—42no—
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—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—42no—
249C22×C4—42no—
250C22×C4#12112yes—
251C22×C4#12112yes—
252C22×C4#12312yes—
253C22×C4#12312yes—
254C22×C4C4⋊Q812yes—
255C22×C4C4⋊Q812yes—
256C22×C4C22×D411yescharacteristic
257C22×C4C22×Q811yescharacteristic
258C22×C4C22×Q811yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4112yes—
264C22×C4H(5,𝔽2)12yes—
265C22×C42−1+412yes—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8#12518yes—
280C4⋊C4—24no—
281C4⋊C4—24no—
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—42no—
295C4⋊C4—42no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C4⋊C4—88no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42C22⋊Q814yes—
306C42C22⋊Q814yes—
307C42#12514yes—
308C42#12514yes—
309C23C2×#12111yescharacteristic
310C23C2×#12311yescharacteristic
311C23C2×C4⋊Q811yescharacteristic
312C23#53211yescharacteristic
313C23#53211yescharacteristic
314C23#53511yescharacteristic
315C23#53611yescharacteristic
316C23#57612yes—
317C23#57712yes—
318C23#57712yes—
319C23#57812yes—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
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—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4#39814yes—
377C2×C4#39814yes—
378C2×C4C2×C22⋊Q811yescharacteristic
379C2×C4C2×C22⋊Q811yescharacteristic
380C2×C4C2×#12511yescharacteristic
381C2×C4C2×#12511yescharacteristic
382C2×C4C2×#12512yes—
383C2×C4#53312yes—
384C2×C4#54412yes—
385C2×C4#54412yes—
386C2×C4#54612yes—
387C2×C4#54812yes—
388C2×C4#55212yes—
389C2×C4#55312yes—
390C2×C4#55512yes—
391C2×C4#55612yes—
392C2×C4#55612yes—
393C2×C4#55912yes—
394C8—28no—
395C8—44no—
396C22C2×#39811yescharacteristic
397C22#249911yescharacteristic
398C22#250011yescharacteristic
399C22#254411yescharacteristic
400C22#254511yescharacteristic
401C22#266011yescharacteristic
402C22#285911yescharacteristic
403C22#304111yescharacteristic
404C22#304211yescharacteristic
405C22#304311yescharacteristic
406C22#304411yescharacteristic
407C22#309812yes—
408C22#309912yes—
409C22#316312yes—
410C22#316412yes—
411C22#317812yes—
412C22#317812yes—
413C22#317912yes—
414C22#317912yes—
415C22#322412yes—
416C22#322512yes—
417C22#324112yes—
418C22#324212yes—
419C4—24no—
420C4—24no—
421C4—82no—
422C4—84no—
423C4—82no—
424C4—162no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4#243912yes—
430C4#244212yes—
431C4#257012yes—
432C4#257112yes—
433C2#1523111yescharacteristic
434C2#2366011yescharacteristic
435C2#2366111yescharacteristic
436C2#2788812yes—
437C2#2790412yes—
438C2#2862711yescharacteristic
439C2#2862811yescharacteristic
440C2#2866611yescharacteristic
441C2#2866711yescharacteristic
442C2#3031512yes—
443C2#3326212yes—
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×#18472
C2×#18342
#143482
#142762
#112572
#112562
#112202
#112192
#108432
#108422
#106942