Finite Group Databasebeta

← #508508

Subgroups — #508508

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33138C211yesmaximal, characteristic
2#20803C211yesmaximal, characteristic, complemented
3#13539C211yesmaximal, characteristic, complemented
4#13537C211yesmaximal, characteristic, complemented
5#13528C211yesmaximal, characteristic, complemented
6#13527C211yesmaximal, characteristic, complemented
7#13507C211yesmaximal, characteristic, complemented
8#13506C211yesmaximal, characteristic, complemented
9#13482C211yesmaximal, characteristic, complemented
10#13480C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic
12#12991C211yesmaximal, characteristic
13#12921C211yesmaximal, characteristic, complemented
14#12920C211yesmaximal, characteristic, complemented
15#12845C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×#428—21no—
18C2×#427C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C2×#405C2211yescharacteristic
21#2624—21no—
22#2317—21no—
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33#1970C2211yescharacteristic
34#1970C2211yescharacteristic
35#1942C2211yescharacteristic
36#1942C2211yescharacteristic
37#1936—21no—
38#1936—21no—
39#1895—21no—
40#1847—21no—
41#1847—21no—
42#1846C2211yescharacteristic
43#1846C2211yescharacteristic
44#1845C2211yescharacteristic
45#1845C2211yescharacteristic
46#1808C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—21no—
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C22×C4⋊C4—21no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#466—22no—
71#466—22no—
72#428—44no—
73#427—41no—
74#427—41no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—41no—
79C2×#107—41no—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93#405—22no—
94#405—22no—
95C2×#97D411yescharacteristic
96C2×#97D411yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#399—41no—
100#397—41no—
101#394—41no—
102#394—41no—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—41no—
108#389—41no—
109#385—41no—
110#385—41no—
111#384—21no—
112#384—21no—
113#382—21no—
114#382—21no—
115#380—21no—
116#367—41no—
117#367—41no—
118#367—41no—
119#366—41no—
120#357—41no—
121#357—41no—
122#357—41no—
123#357—41no—
124C22⋊C16—22no—
125C22⋊C16—22no—
126#340—21no—
127#340—21no—
128#340—21no—
129C22×D4—21no—
130C22×D4—41no—
131C23×C4—41no—
132C23×C4C2×D411yescharacteristic
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8#4112yes—
150C22×C8#4112yes—
151C22×C8C2411yescharacteristic
152#121—42no—
153#121—42no—
154#117—22no—
155#117—22no—
156#117—42no—
157#117—42no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4D811yescharacteristic
175C2×C4⋊C4D811yescharacteristic
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4D811yescharacteristic
184C2×C22⋊C4D811yescharacteristic
185C2×C42—41no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—42no—
215#107—42no—
216#107—42no—
217#106—42no—
218#106—42no—
219#106—42no—
220#102—42no—
221#102—42no—
222#102—42no—
223#97—22no—
224#97—22no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236C24—41no—
237C24C2×D811yescharacteristic
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—82no—
245C2×D4—161no—
246C2×D4—161no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—82no—
259C22×C4—161no—
260C22×C4—161no—
261C22×C4#12312yes—
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4#13712yes—
268C22×C4C22×D411yescharacteristic
269D8—41no—
270D8—41no—
271D8—41no—
272D8—41no—
273M16—28no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8#12312yes—
285C2×C8AGL(1,ℤ/8)12yes—
286C2×C8#13712yes—
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8H(5,𝔽2)11yescharacteristic
290C2×C8H(5,𝔽2)11yescharacteristic
291C2×C82−1+411yescharacteristic
292C2×C82−1+411yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—82no—
310C4⋊C4—161no—
311C4⋊C4—161no—
312C4⋊C4—161no—
313C4⋊C4—161no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—161no—
321C22⋊C4—161no—
322C22⋊C4—161no—
323C22⋊C4—161no—
324C22⋊C4—161no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—82no—
330C16—41no—
331C16—41no—
332C16—41no—
333C16—41no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23#48012yes—
348C23C22×D811yescharacteristic
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—164no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—164no—
394C2×C4#48012yes—
395C2×C4C2×#12311yescharacteristic
396C2×C4#53511yescharacteristic
397C2×C4#53611yescharacteristic
398C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
399C2×C4C2×#13711yescharacteristic
400C2×C4C2×#13711yescharacteristic
401C8—22no—
402C8—22no—
403C8—22no—
404C8#53911yescharacteristic
405C8#53911yescharacteristic
406C8#53911yescharacteristic
407C8#53911yescharacteristic
408C22—22no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22C2×#48011yescharacteristic
423C22#320511yescharacteristic
424C22#322611yescharacteristic
425C22#341511yescharacteristic
426C22#341611yescharacteristic
427C22#341611yescharacteristic
428C22#341611yescharacteristic
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4#262311yescharacteristic
442C4#309211yescharacteristic
443C4#321811yescharacteristic
444C4#323911yescharacteristic
445C2—42no—
446C2—321no—
447C2#2990611yescharacteristic
448C2#3365011yescharacteristic
449C2#3365311yescharacteristic
450C2#3378711yescharacteristic
451C2#3379011yescharacteristic
452C2#3380711yescharacteristic
453C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20803C2semidirect
3. #13539C2semidirect
4. #13537C2semidirect
5. #13528C2semidirect
6. #13527C2semidirect
7. #13507C2semidirect
8. #13506C2semidirect
9. #13482C2semidirect
10. #13480C2semidirect
13. #12921C2semidirect
14. #12920C2semidirect
15. #12845C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#331382
#208032
#135392
#135372
#135282
#135272
#135072
#135062
#134822
#134802
#129922
#129912
#129212
#129202
#128452