Finite Group Databasebeta

← #509556

Subgroups — #509556

Order 512.

583 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×#2240C211yesmaximal, characteristic
2C2×#2226C211yesmaximal, characteristic, complemented
3#28813C211yesmaximal, characteristic, complemented
4#26254C211yesmaximal, characteristic
5#13527C211yesmaximal, characteristic, complemented
6#13480C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13452C211yesmaximal, characteristic, complemented
9#13407C211yesmaximal, characteristic
10#13363C211yesmaximal, characteristic, complemented
11#13166C211yesmaximal, characteristic, complemented
12#13131C211yesmaximal, characteristic, complemented
13#13131C211yesmaximal, characteristic, complemented
14#13104C211yesmaximal, characteristic, complemented
15#12937C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2551—21no—
21#2445—21no—
22#2240—24no—
23#2226—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C4×C16C2211yescharacteristic
38#2057—21no—
39#2057—21no—
40#2054—21no—
41#2041—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1947—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×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—41no—
67C2×#127—21no—
68C2×#123—41no—
69C2×C4×D4—21no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#500—22no—
75#500—22no—
76#500—41no—
77#500—41no—
78#493—22no—
79#493—41no—
80C4×D8—22no—
81C4×D8—41no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#395—41no—
100#395—41no—
101#394—41no—
102#394—41no—
103#390—21no—
104#390—21no—
105#390—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—41no—
110#386—41no—
111#384—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D411yescharacteristic
114#367—22no—
115#367—22no—
116#367—22no—
117#367—41no—
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—
127C4×C16—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—41no—
135C22×Q8—41no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—41no—
139C22×D4—41no—
140C23×C4—41no—
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—21no—
151C2×D8—21no—
152C2×D8—42no—
153C2×D8—42no—
154C2×D8—81no—
155C2×D8C2×D412yes—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2411yescharacteristic
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—81no—
163C4⋊Q8—81no—
164C4⋊Q8—84no—
165#127—42no—
166#127—81no—
167#123—84no—
168C4×D4—42no—
169C4×D4—81no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
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⋊C4—41no—
183C2×C4⋊C4D811yescharacteristic
184C2×C4⋊C4D811yescharacteristic
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×C22⋊C4—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
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×C16—21no—
207C2×C16C2×D412yes—
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—42no—
220#107—81no—
221#107C2×D412yes—
222#107#4112yes—
223C4⋊C8—22no—
224#102—42no—
225#102—81no—
226#102—81no—
227#102—81no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95D811yescharacteristic
239#95D811yescharacteristic
240C24—41no—
241C24—41no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8—161no—
247C2×Q8—161no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—161no—
265C2×D4—161no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—161no—
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C22×D411yescharacteristic
282D8—22no—
283D8—22no—
284D8—41no—
285D8—41no—
286D8—41no—
287D8—41no—
288D8—41no—
289D8—41no—
290D8—84no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8#13712yes—
300C2×C8C22×D411yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C2×C8H(5,𝔽2)11yescharacteristic
303C2×C82−1+411yescharacteristic
304C2×C82−1+411yescharacteristic
305C2×C82−1+411yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—161no—
326C4⋊C4SD3214yes—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—161no—
332C22⋊C4—161no—
333C22⋊C4—161no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—82no—
341C16—22no—
342C16—41no—
343C16—41no—
344C16—41no—
345C23—42no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23C22×D811yescharacteristic
360Q8—164no—
361Q8—164no—
362D4—42no—
363D4—42no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—164no—
371D4—164no—
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—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—42no—
387C2×C4—42no—
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—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—164no—
405C2×C4C2×SD3212yes—
406C2×C4C2×SD3212yes—
407C2×C4C2×SD3212yes—
408C2×C4#50912yes—
409C2×C4#53611yescharacteristic
410C2×C4D4211yescharacteristic
411C2×C4#54811yescharacteristic
412C2×C4C2×#13711yescharacteristic
413C2×C4#57611yescharacteristic
414C2×C4#57711yescharacteristic
415C8—22no—
416C8—41no—
417C8—41no—
418C8#53611yescharacteristic
419C8#54211yescharacteristic
420C8#54711yescharacteristic
421C8#56311yescharacteristic
422C22—82no—
423C22—82no—
424C22—82no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22#322611yescharacteristic
437C22D4×D811yescharacteristic
438C22#335811yescharacteristic
439C22C22×SD3211yescharacteristic
440C22C2×#50911yescharacteristic
441C22#341511yescharacteristic
442C22#341611yescharacteristic
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—41no—
448C4—41no—
449C4—82no—
450C4—82no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#267411yescharacteristic
456C4#324111yescharacteristic
457C4#326611yescharacteristic
458C4#337511yescharacteristic
459C2—162no—
460C2—321no—
461C2#3080511yescharacteristic
462C2#3379311yescharacteristic
463C2#3380711yescharacteristic
464C2#3383011yescharacteristic
465C2D4×SD3211yescharacteristic
466C2#3392911yescharacteristic
467C2#3394011yescharacteristic
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×#2226C2semidirect
3. #28813C2semidirect
5. #13527C2semidirect
6. #13480C2semidirect
7. #13476C2semidirect
8. #13452C2semidirect
10. #13363C2semidirect
11. #13166C2semidirect
12. #13131C2semidirect
13. #13131C2semidirect
14. #13104C2semidirect
15. #12937C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22402
C2×#22262
#288132
#262542
#135272
#134802
#134762
#134522
#134072
#133632
#131662
#131312
#131042
#129372