Finite Group Databasebeta

← #465229

Subgroups — #465229

Order 512.

912 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2086C211yesmaximal, characteristic
2C2×#1942C211yesmaximal, characteristic
3C2×#1916C211yesmaximal, characteristic
4C2×#1835C211yesmaximal, characteristic
5#15221C212yesmaximal
6#11379C211yesmaximal, characteristic
7#11316C212yesmaximal
8#11293C212yesmaximal
9#11256C211yesmaximal, characteristic
10#11232C211yesmaximal, characteristic
11#11208C212yesmaximal
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#382C2212yes—
19C2×C4×C4⋊C4C2211yescharacteristic
20#2086—24no—
21#1942—24no—
22#1916—24no—
23#1835—24no—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1445—22no—
37#1444—22no—
38#1442—22no—
39#1442—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#107—22no—
59C2×#107—24no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8D414yes—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—42no—
73#382—24no—
74#382—42no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95D411yescharacteristic
88C2×#95D411yescharacteristic
89#340—22no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—42no—
96#340D414yes—
97#336—22no—
98#336—22no—
99#336—22no—
100#336—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8C2×D412yes—
129C22×C8C2×D412yes—
130C22×C8C2×D412yes—
131C22×C8#4112yes—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
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—22no—
143C2×C4⋊C4—22no—
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—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4#4114yes—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42C2×D412yes—
182C2×C42#4112yes—
183#107—28no—
184#107—44no—
185C4⋊C8—44no—
186C4×C8—28no—
187#95—22no—
188#95—22no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—82no—
197#95—82no—
198#95—82no—
199C24C22×D411yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
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—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—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4#12112yes—
259C22×C4#12112yes—
260C22×C4#13512yes—
261C22×C4#13512yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4112yes—
266C22×C4C2×#4111yescharacteristic
267C22×C4H(5,𝔽2)11yescharacteristic
268C22×C42−1+412yes—
269C22×C42−1+412yes—
270C22×C42−1+412yes—
271C22×C42−1+411yescharacteristic
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8#12114yes—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—88no—
311C4⋊C4—88no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—42no—
321C42—41no—
322C42—41no—
323C23C2×#12111yescharacteristic
324C23#53612yes—
325C23D4211yescharacteristic
326C23#54711yescharacteristic
327C23#54812yes—
328C23C2×#13511yescharacteristic
329C23C2×#13511yescharacteristic
330C23#57512yes—
331C23#57611yescharacteristic
332C23#57611yescharacteristic
333C23#57812yes—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4C2×#12111yescharacteristic
397C2×C4#52511yescharacteristic
398C2×C4#53611yescharacteristic
399C2×C4#53611yescharacteristic
400C2×C4#53611yescharacteristic
401C2×C4#53612yes—
402C2×C4#53711yescharacteristic
403C2×C4#54112yes—
404C2×C4#54112yes—
405C2×C4#54112yes—
406C2×C4#54212yes—
407C2×C4#54712yes—
408C2×C4#54811yescharacteristic
409C2×C4#55212yes—
410C2×C4#55212yes—
411C2×C4#55311yescharacteristic
412C2×C4#56312yes—
413C2×C4#56412yes—
414C8—24no—
415C8—42no—
416C8—81no—
417C8—81no—
418C22#241611yescharacteristic
419C22#253611yescharacteristic
420C22#262111yescharacteristic
421C22#267412yes—
422C22#271912yes—
423C22#305011yescharacteristic
424C22#305411yescharacteristic
425C22#305811yescharacteristic
426C22#305911yescharacteristic
427C22#323212yes—
428C22#323312yes—
429C22#324312yes—
430C22#324312yes—
431C22#326912yes—
432C22#328011yescharacteristic
433C22#328711yescharacteristic
434C22#333911yescharacteristic
435C22#334111yescharacteristic
436C22#334912yes—
437C22#336412yes—
438C22#336412yes—
439C22#337012yes—
440C22#337012yes—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—84no—
446C4—84no—
447C4—81no—
448C4—81no—
449C4—162no—
450C4—162no—
451C4#241311yescharacteristic
452C4#262111yescharacteristic
453C4#263711yescharacteristic
454C4#271912yes—
455C4#276612yes—
456C4#282211yescharacteristic
457C2#1651711yescharacteristic
458C2#2477811yescharacteristic
459C2#2478511yescharacteristic
460C2#2819011yescharacteristic
461C2#2819111yescharacteristic
462C2#2987911yescharacteristic
463C2#2988011yescharacteristic
464C2#3080912yes—
465C2#3081812yes—
466C2#3232112yes—
467C2#3232412yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#20862
C2×#19422
C2×#19162
C2×#18352
#152212
#113792
#113162
#112932
#112562
#112322
#112082