Finite Group Databasebeta

← #411133

Subgroups — #411133

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2084C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic
3#14277C211yesmaximal, characteristic
4#13779C211yesmaximal, characteristic
5#11357C212yesmaximal
6#11325C212yesmaximal
7#11208C211yesmaximal, characteristic
8#11208C211yesmaximal, characteristic
9#10908C212yesmaximal
10#10907C212yesmaximal
11#10796C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C2×#389C2211yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21C2×#376C2211yescharacteristic
22#2084—24no—
23#2083—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1444—22no—
35#1442—22no—
36#1436—22no—
37#1436—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
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⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C42—22no—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#107—24no—
59C2×#107—24no—
60C2×#107—42no—
61C2×#106—24no—
62C2×#106—24no—
63C2×#106—42no—
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#384—22no—
69#384—24no—
70#384D414yes—
71#382—24no—
72#382—22no—
73#382—22no—
74C4×C4⋊C4—42no—
75C4×C4⋊C4—42no—
76#376—24no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D412yes—
88C2×#95D412yes—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—42no—
94#340—42no—
95#336—22no—
96#336—22no—
97#336—24no—
98#336Q818yes—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2411yescharacteristic
110C22×C8—22no—
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×C8C2×Q814yes—
123C22×C8C2×Q814yes—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
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—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
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—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4#4114yes—
165C2×C4⋊C4#4112yes—
166C2×C4⋊C4#4112yes—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—44no—
176C2×C42—82no—
177C2×C42—82no—
178C2×C42C2×D412yes—
179C2×C42C2×Q814yes—
180C2×C42#4112yes—
181C2×C42#4114yes—
182#107—48no—
183#106—48no—
184#95—24no—
185#95—24no—
186#95—28no—
187#95—24no—
188#95—24no—
189#95—44no—
190#95—42no—
191#95—42no—
192#95—82no—
193#95—82no—
194#95—82no—
195#95—82no—
196C24C22×D411yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
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—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4#12112yes—
252C22×C4#12112yes—
253C22×C4#12314yes—
254C22×C4C4⋊Q814yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×Q812yes—
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4112yes—
259C22×C4C2×#4112yes—
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4H(5,𝔽2)11yescharacteristic
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C42−1+412yes—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C4⋊C4—88no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C42—24no—
305C42—24no—
306C42—24no—
307C42—24no—
308C42—24no—
309C42—24no—
310C42—88no—
311C23C2×#12111yescharacteristic
312C23C2×#12311yescharacteristic
313C23#52511yescharacteristic
314C23#52511yescharacteristic
315C23C2×C4⋊Q811yescharacteristic
316C23#53511yescharacteristic
317C23#53611yescharacteristic
318C23#57612yes—
319C23#57712yes—
320C23#57712yes—
321C23#57812yes—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
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—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
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—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4#52912yes—
381C2×C4#53212yes—
382C2×C4#53312yes—
383C2×C4#53312yes—
384C2×C4#53912yes—
385C2×C4#53912yes—
386C2×C4#54611yescharacteristic
387C2×C4#54611yescharacteristic
388C2×C4D4×Q812yes—
389C2×C4#55212yes—
390C2×C4#55212yes—
391C2×C4#55311yescharacteristic
392C2×C4#55311yescharacteristic
393C2×C4#55412yes—
394C2×C4#55412yes—
395C2×C4#55712yes—
396C8—44no—
397C8—44no—
398C22#239911yescharacteristic
399C22#248411yescharacteristic
400C22#250211yescharacteristic
401C22#253511yescharacteristic
402C22#254611yescharacteristic
403C22#263311yescharacteristic
404C22#285411yescharacteristic
405C22#304111yescharacteristic
406C22#304211yescharacteristic
407C22#304311yescharacteristic
408C22#304411yescharacteristic
409C22#309612yes—
410C22#309712yes—
411C22#311912yes—
412C22#311912yes—
413C22#312012yes—
414C22#312012yes—
415C22#316312yes—
416C22#316412yes—
417C22#322412yes—
418C22#322512yes—
419C22#324312yes—
420C22#324412yes—
421C4—44no—
422C4—42no—
423C4—42no—
424C4—44no—
425C4—82no—
426C4—84no—
427C4—82no—
428C4—84no—
429C4—161no—
430C4—161no—
431C4#255612yes—
432C4#255612yes—
433C4#259012yes—
434C4#259012yes—
435C2#1510611yescharacteristic
436C2#2384312yes—
437C2#2761912yes—
438C2#2795912yes—
439C2#2816711yescharacteristic
440C2#2816811yescharacteristic
441C2#2869611yescharacteristic
442C2#2869711yescharacteristic
443C2#3008812yes—
444C2#3311811yescharacteristic
445C2#3311911yescharacteristic
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×#20842
C2×#20832
#142772
#137792
#113572
#113252
#112082
#109082
#109072
#107962