Finite Group Databasebeta

← #509578

Subgroups — #509578

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2244C211yesmaximal, characteristic
2C2×#2228C211yesmaximal, characteristic, complemented
3#28814C211yesmaximal, characteristic, complemented
4#26254C211yesmaximal, characteristic
5#13527C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13407C211yesmaximal, characteristic
10#13365C211yesmaximal, characteristic, complemented
11#13168C211yesmaximal, characteristic, complemented
12#13130C211yesmaximal, characteristic, complemented
13#13130C211yesmaximal, characteristic, complemented
14#13105C211yesmaximal, characteristic, complemented
15#12937C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2552—21no—
21#2445—21no—
22#2244—24no—
23#2228—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#346C2211yescharacteristic
38#2059—21no—
39#2059—21no—
40#2054—21no—
41#2041—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1948—21no—
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1879—21no—
50#1845C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8—21no—
62C22×D8C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C2×#127—21no—
65C2×#125—21no—
66C2×#125—21no—
67C2×#123—41no—
68C2×C4×D4—21no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#499—22no—
74#499—22no—
75#499—41no—
76#499—41no—
77#493—22no—
78#493—41no—
79C4×D8—22no—
80C4×D8—41no—
81C2×#107—21no—
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#106—21no—
89C2×#106—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#398—41no—
99#398—41no—
100#394—41no—
101#394—41no—
102#390—21no—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—41no—
109#386—41no—
110#384—21no—
111#384—41no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114#367—22no—
115#367—41no—
116#366—22no—
117#366—22no—
118C4⋊C16—22no—
119#357—22no—
120#357—22no—
121#357—22no—
122#357—22no—
123#357—22no—
124#357—22no—
125#357—41no—
126#357D414yes—
127#346—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—41no—
137C22×D4—41no—
138C23×C4—41no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—42no—
151C2×D8—42no—
152C2×D8—81no—
153C2×D8C2×D412yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2411yescharacteristic
158#127—42no—
159#127—81no—
160#125—42no—
161#125—42no—
162#125—81no—
163#125—81no—
164#123—84no—
165C4×D4—42no—
166C4×D4—81no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
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⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4D811yescharacteristic
183C2×C4⋊C4D811yescharacteristic
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C42—41no—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
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×C16—21no—
205C2×C16—21no—
206C2×C16C2×D412yes—
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—81no—
218#107C2×D412yes—
219#107#4112yes—
220#106—42no—
221#106—42no—
222C4⋊C8—22no—
223#102—42no—
224#102—81no—
225#102—81no—
226#102—81no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95D811yescharacteristic
238#95D811yescharacteristic
239C24—41no—
240C24—41no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—82no—
254C2×D4—82no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—161no—
258C2×D4—161no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—161no—
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C22×D411yescharacteristic
275D8—22no—
276D8—22no—
277D8—41no—
278D8—41no—
279D8—41no—
280D8—41no—
281D8—41no—
282D8—41no—
283D8—84no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8#13712yes—
293C2×C8C22×D411yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8H(5,𝔽2)11yescharacteristic
296C2×C82−1+411yescharacteristic
297C2×C82−1+411yescharacteristic
298C2×C82−1+411yescharacteristic
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
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—41no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—161no—
320C4⋊C4—161no—
321C4⋊C4—161no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—161no—
327C22⋊C4—161no—
328C22⋊C4—161no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—82no—
336C16—22no—
337C16—41no—
338C16—41no—
339C16—41no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23C22×D811yescharacteristic
355D4—42no—
356D4—42no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—164no—
364D4—164no—
365C2×C4—21no—
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—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
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—81no—
399C2×C4—81no—
400C2×C4—164no—
401C2×C4#50912yes—
402C2×C4#53611yescharacteristic
403C2×C4D4211yescharacteristic
404C2×C4#54811yescharacteristic
405C2×C4C2×#13711yescharacteristic
406C2×C4#57611yescharacteristic
407C2×C4#57711yescharacteristic
408C8—22no—
409C8—41no—
410C8—41no—
411C8#53611yescharacteristic
412C8#54211yescharacteristic
413C8#54711yescharacteristic
414C8#56311yescharacteristic
415C22—82no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22—161no—
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22#322611yescharacteristic
430C22D4×D811yescharacteristic
431C22#335811yescharacteristic
432C22C2×#50911yescharacteristic
433C22#341411yescharacteristic
434C22#341511yescharacteristic
435C22#341611yescharacteristic
436C4—22no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—82no—
443C4—82no—
444C4—161no—
445C4—161no—
446C4—161no—
447C4—161no—
448C4#267411yescharacteristic
449C4#324111yescharacteristic
450C4#326611yescharacteristic
451C4#337511yescharacteristic
452C2—162no—
453C2—321no—
454C2#3080511yescharacteristic
455C2#3380011yescharacteristic
456C2#3380711yescharacteristic
457C2#3383411yescharacteristic
458C2#3383811yescharacteristic
459C2#3393311yescharacteristic
460C2#3393811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#493C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2228C2semidirect
3. #28814C2semidirect
5. #13527C2semidirect
6. #13480C2semidirect
7. #13474C2semidirect
8. #13453C2semidirect
10. #13365C2semidirect
11. #13168C2semidirect
12. #13130C2semidirect
13. #13130C2semidirect
14. #13105C2semidirect
15. #12937C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22442
C2×#22282
#288142
#262542
#135272
#134802
#134742
#134532
#134072
#133652
#131682
#131302
#131052
#129372