Finite Group Databasebeta

← #501446

Subgroups — #501446

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2234C211yesmaximal, characteristic
2C2×#2163C211yesmaximal, characteristic, complemented
3#33129C211yesmaximal, characteristic
4#21151C211yesmaximal, characteristic, complemented
5#13516C211yesmaximal, characteristic, complemented
6#13514C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13168C211yesmaximal, characteristic, complemented
10#13167C211yesmaximal, characteristic, complemented
11#13149C211yesmaximal, characteristic, complemented
12#13149C211yesmaximal, characteristic, complemented
13#12992C211yesmaximal, characteristic
14#12921C211yesmaximal, characteristic, complemented
15#12868C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2854—21no—
21#2296—21no—
22#2234—24no—
23#2163—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
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×#346C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1941—21no—
47#1941—21no—
48#1934—21no—
49#1934—21no—
50#1893—21no—
51#1808C2211yescharacteristic
52#1735—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8D411yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#123—41no—
65C2×#121—21no—
66C2×#117—21no—
67C2×#117—21no—
68C22×C42D411yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#466—22no—
74#466—22no—
75#427—22no—
76#427—22no—
77#427—22no—
78#427—22no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#106—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#390—21no—
98#390—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—41no—
106#388—41no—
107#386—41no—
108#385—41no—
109#385—41no—
110#380—21no—
111C4×C4⋊C4—41no—
112C4×C4⋊C4—41no—
113#367—22no—
114#367—22no—
115#366—22no—
116#366—22no—
117C4⋊C16—22no—
118C4⋊C16—22no—
119#357—22no—
120#357—22no—
121#357—22no—
122#357—22no—
123C22⋊C16—22no—
124C22⋊C16D414yes—
125#346—22no—
126#346—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×C4D811yescharacteristic
137C23×C4D811yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—41no—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D412yes—
155C22×C8C2×D411yescharacteristic
156C22×C8#4112yes—
157C22×C8C2411yescharacteristic
158#123—84no—
159#121—42no—
160#121—42no—
161#117—24no—
162#117—42no—
163#117—42no—
164#117—42no—
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×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42D811yescharacteristic
191C2×C42D811yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
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×C16—21no—
204C2×C16C2×D412yes—
205C2×C16C2×D412yes—
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#4112yes—
219#107#4112yes—
220#106—24no—
221#106—42no—
222#102—42no—
223#102—42no—
224#102—42no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238C24—41no—
239C24C2×D811yescharacteristic
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—82no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4#12112yes—
267C22×C4C2×D811yescharacteristic
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4#13512yes—
273C22×C4C22×D411yescharacteristic
274D8—41no—
275D8—41no—
276D8—41no—
277D8—41no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—44no—
290C2×C8#12112yes—
291C2×C8#13712yes—
292C2×C8C22×D411yescharacteristic
293C2×C8C2×#4111yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C82−1+411yescharacteristic
297C2×C82−1+411yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
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—41no—
311C4⋊C4—41no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C22⋊C4—44no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—22no—
331C42—22no—
332C42—44no—
333C42—82no—
334C16—22no—
335C16—22no—
336C16—41no—
337C16—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23#46612yes—
352C23C22×D811yescharacteristic
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—164no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
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—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#46612yes—
399C2×C4#50912yes—
400C2×C4C2×#12111yescharacteristic
401C2×C4#52511yescharacteristic
402C2×C4#53611yescharacteristic
403C2×C4C2×#13511yescharacteristic
404C2×C4C2×#13711yescharacteristic
405C2×C4#57511yescharacteristic
406C8—22no—
407C8—22no—
408C8—41no—
409C8#54711yescharacteristic
410C8#54711yescharacteristic
411C8#55211yescharacteristic
412C8#55211yescharacteristic
413C22—22no—
414C22—22no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22C2×#46611yescharacteristic
428C22#310011yescharacteristic
429C22#322611yescharacteristic
430C22C2×#50911yescharacteristic
431C22#341411yescharacteristic
432C22#341411yescharacteristic
433C22#341411yescharacteristic
434C4—22no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—82no—
440C4—82no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4—161no—
445C4—161no—
446C4#241311yescharacteristic
447C4#305611yescharacteristic
448C4#310811yescharacteristic
449C4#323211yescharacteristic
450C2—42no—
451C2—321no—
452C2#2460011yescharacteristic
453C2#3360411yescharacteristic
454C2#3360511yescharacteristic
455C2#3367111yescharacteristic
456C2#3367511yescharacteristic
457C2#3380111yescharacteristic
458C2#3380611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2163C2semidirect
4. #21151C2semidirect
5. #13516C2semidirect
6. #13514C2semidirect
7. #13476C2semidirect
8. #13474C2semidirect
9. #13168C2semidirect
10. #13167C2semidirect
11. #13149C2semidirect
12. #13149C2semidirect
14. #12921C2semidirect
15. #12868C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22342
C2×#21632
#331292
#211512
#135162
#135142
#134762
#134742
#131682
#131672
#131492
#129922
#129212
#128682