Finite Group Databasebeta

← #501586

Subgroups — #501586

Order 512.

561 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×#2164C211yesmaximal, characteristic, complemented
3#33216C211yesmaximal, characteristic, complemented
4#21524C211yesmaximal, characteristic
5#13494C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic
8#13381C211yesmaximal, characteristic
9#13277C211yesmaximal, characteristic, complemented
10#13177C211yesmaximal, characteristic, complemented
11#13167C211yesmaximal, characteristic, complemented
12#13166C211yesmaximal, characteristic, complemented
13#13149C211yesmaximal, characteristic, complemented
14#13147C211yesmaximal, characteristic, complemented
15#12811C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2629—21no—
21#2324—21no—
22#2228—24no—
23#2164—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×#346C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38#1992—21no—
39#1970C2211yescharacteristic
40#1943—21no—
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1936—21no—
45#1895—21no—
46#1845C2211yescharacteristic
47#1809—21no—
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1762—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×M16D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#125—21no—
65C2×#121—21no—
66C2×C4×D4—21no—
67C2×#117D411yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#499—22no—
73#499—22no—
74#466—22no—
75#466—22no—
76C4×D8—22no—
77C4×D8—22no—
78C2×#107—21no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90#405—22no—
91#405—22no—
92C2×#97D411yescharacteristic
93C2×#97D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#398—41no—
97#398—41no—
98#396—41no—
99#396—41no—
100#390—21no—
101#390—41no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—41no—
106#386—41no—
107#385—41no—
108#384—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113#367—22no—
114#367—22no—
115#367—22no—
116#366—22no—
117C4⋊C16—22no—
118C4⋊C16D414yescomplemented
119#357—22no—
120#357—22no—
121#357—22no—
122#357—22no—
123C22⋊C16—22no—
124C22⋊C16—22no—
125#346—22no—
126C4×C16—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×D4—21no—
135C22×D4—41no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8#4112yes—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—41no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2411yescharacteristic
157#125—42no—
158#125—42no—
159#121—24no—
160#121—42no—
161C4×D4—42no—
162C4×D4—42no—
163#117—22no—
164#117—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4D811yescharacteristic
180C2×C4⋊C4D811yescharacteristic
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4D811yescharacteristic
189C2×C22⋊C4D811yescharacteristic
190C2×C42—41no—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16C2×D412yes—
204C2×C16C2×D412yes—
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—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—42no—
219#107#4112yes—
220#106—42no—
221#102—24no—
222#102—42no—
223#102—42no—
224#97—22no—
225#97—22no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8C2×D412yes—
229C4×C8#4112yes—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237C24—41no—
238C24C2×D811yescharacteristic
239C2×D4—22no—
240C2×D4—44no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—161no—
247C2×D4—161no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—82no—
262C22×C4—161no—
263C22×C4—161no—
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C22×D411yescharacteristic
270D8—22no—
271D8—22no—
272D8—41no—
273D8—41no—
274M16—44no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8#12112yes—
287C2×C8#13712yes—
288C2×C8C22×D411yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8C2×#4111yescharacteristic
292C2×C82−1+411yescharacteristic
293C2×C82−1+411yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
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—41no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—161no—
313C4⋊C4—161no—
314C4⋊C4—161no—
315C4⋊C4—161no—
316C4⋊C4—161no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—44no—
320C22⋊C4—82no—
321C22⋊C4—161no—
322C22⋊C4—161no—
323C22⋊C4—161no—
324C22⋊C4—161no—
325C22⋊C4—161no—
326C42—21no—
327C42—21no—
328C42—161no—
329C42#12112yes—
330C42#13512yes—
331C16—22no—
332C16—22no—
333C16—41no—
334C16—41no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23C22×D811yescharacteristic
350D4—42no—
351D4—42no—
352D4—81no—
353D4—81no—
354D4—164no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
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—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
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#46612yes—
395C2×C4#46612yes—
396C2×C4C2×#12111yescharacteristic
397C2×C4#52511yescharacteristic
398C2×C4#53611yescharacteristic
399C2×C4C2×#13511yescharacteristic
400C2×C4C2×#13711yescharacteristic
401C2×C4#57511yescharacteristic
402C8—22no—
403C8—22no—
404C8—41no—
405C8#54711yescharacteristic
406C8#54811yescharacteristic
407C8#55211yescharacteristic
408C8#55611yescharacteristic
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22C2×#46611yescharacteristic
424C22#310011yescharacteristic
425C22#322611yescharacteristic
426C22#341511yescharacteristic
427C22#341611yescharacteristic
428C22#341611yescharacteristic
429C22#341611yescharacteristic
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—82no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4#241511yescharacteristic
443C4#305611yescharacteristic
444C4#310711yescharacteristic
445C4#323311yescharacteristic
446C2—81no—
447C2—162no—
448C2#2470011yescharacteristic
449C2#3361111yescharacteristic
450C2#3361211yescharacteristic
451C2#3367911yescharacteristic
452C2#3368311yescharacteristic
453C2#3380711yescharacteristic
454C2#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
1. C2×#2228C2semidirect
2. C2×#2164C2semidirect
3. #33216C2semidirect
5. #13494C2semidirect
6. #13492C2semidirect
9. #13277C2semidirect
10. #13177C2semidirect
11. #13167C2semidirect
12. #13166C2semidirect
13. #13149C2semidirect
14. #13147C2semidirect
15. #12811C2semidirect
118. C4⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#21642
#332162
#215242
#134942
#134922
#133812
#132772
#131772
#131672
#131662
#131492
#131472
#128112