Finite Group Databasebeta

← #504252

Subgroups — #504252

Order 512.

587 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
1C2×#2225C211yesmaximal, characteristic, complemented
2C2×#2226C211yesmaximal, characteristic
3#28813C211yesmaximal, characteristic, complemented
4#23186C211yesmaximal, characteristic
5#13475C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic
7#13364C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic
9#13254C211yesmaximal, characteristic
10#13205C211yesmaximal, characteristic
11#13177C211yesmaximal, characteristic, complemented
12#13160C211yesmaximal, characteristic, complemented
13#12938C211yesmaximal, characteristic, complemented
14#12937C211yesmaximal, characteristic
15#12674C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2551—21no—
22#2367—21no—
23#2226—24no—
24#2225—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#346C2211yescharacteristic
41#2057—21no—
42#2057—21no—
43#1983—21no—
44#1970C2211yescharacteristic
45#1969C2211yescharacteristic
46#1947—21no—
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1880—21no—
50#1877—21no—
51#1845C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#107C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—41no—
67C2×#124—21no—
68C2×C4×Q8—21no—
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#495—22no—
79#495Q814yescomplemented
80C4×Q16—22no—
81C4×Q16—41no—
82C4×D8—22no—
83C4×D8—41no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×C4⋊C8D411yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#395—41no—
102#395—41no—
103#391—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—41no—
108#388—41no—
109#388—41no—
110#385—41no—
111#384—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D411yescharacteristic
115C4×C22⋊C4—41no—
116#367—22no—
117#367—22no—
118#367D414yes—
119#366—22no—
120C4⋊C16—22no—
121C4⋊C16—22no—
122C4⋊C16—22no—
123#358—22no—
124#358—22no—
125#358—22no—
126#358—41no—
127#357—22no—
128#357—22no—
129#357—22no—
130#357—41no—
131#346—22no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—41no—
140C22×Q8—41no—
141C22×D4—21no—
142C23×C4—41no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16C2×Q812yes—
149C2×SD16—24no—
150C2×SD16—81no—
151C2×SD16—81no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8C2×Q812yes—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2411yescharacteristic
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—81no—
165C4⋊Q8—81no—
166C4⋊Q8—84no—
167#124—24no—
168#124—42no—
169C4×Q8—42no—
170C4×Q8—81no—
171C4×D4—42no—
172C4×D4—81no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4D811yescharacteristic
187C2×C4⋊C4D811yescharacteristic
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C42—41no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
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×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16C2×D412yes—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—42no—
221#107—42no—
222#107—81no—
223#107C2×D412yes—
224#107C2×D412yes—
225#107#4112yes—
226C4⋊C8—22no—
227#103—42no—
228#103—81no—
229#102—42no—
230#102—81no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8C2×Q812yes—
236C4×C8#4112yes—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95D811yescharacteristic
244#95D811yescharacteristic
245C24—41no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—82no—
253C2×Q8—82no—
254C2×Q8—82no—
255C2×Q8—82no—
256C2×Q8—161no—
257C2×Q8—161no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—161no—
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×D811yescharacteristic
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C22×D411yescharacteristic
281Q16—22no—
282Q16—22no—
283Q16—41no—
284Q16—41no—
285SD16—84no—
286D8—22no—
287D8—22no—
288D8—41no—
289D8—41no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8C22⋊Q812yes—
299C2×C8C22×D411yescharacteristic
300C2×C8C22×Q811yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C2×C8C2×#4111yescharacteristic
303C2×C8H(5,𝔽2)11yescharacteristic
304C2×C82−1+411yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
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—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—161no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—82no—
336C42—82no—
337C42—161no—
338C42C22⋊Q812yes—
339C42#13712yes—
340C16—22no—
341C16—41no—
342C16—41no—
343C16—41no—
344C23—42no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23C22×D811yescharacteristic
352Q8—42no—
353Q8—42no—
354Q8—81no—
355Q8—81no—
356Q8—164no—
357Q8—164no—
358D4—42no—
359D4—42no—
360D4—81no—
361D4—81no—
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—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
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—164no—
398C2×C4#47412yes—
399C2×C4#47412yes—
400C2×C4#50912yes—
401C2×C4#51012yes—
402C2×C4C2×C22⋊Q811yescharacteristic
403C2×C4D4211yescharacteristic
404C2×C4#54811yescharacteristic
405C2×C4C2×#13711yescharacteristic
406C2×C4#57611yescharacteristic
407C2×C4#57711yescharacteristic
408C8—22no—
409C8—41no—
410C8—41no—
411C8#53211yescharacteristic
412C8#54211yescharacteristic
413C8D4×Q811yescharacteristic
414C8#55411yescharacteristic
415C22—82no—
416C22—82no—
417C22—82no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22C2×#47411yescharacteristic
423C22D4×D811yescharacteristic
424C22#335811yescharacteristic
425C22C2×#50911yescharacteristic
426C22C2×#51011yescharacteristic
427C22#341611yescharacteristic
428C22#341611yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—82no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4—161no—
440C4—161no—
441C4—161no—
442C4#245211yescharacteristic
443C4#308111yescharacteristic
444C4#327811yescharacteristic
445C4#337511yescharacteristic
446C2—162no—
447C2#2646011yescharacteristic
448C2#3362911yescharacteristic
449C2#3363411yescharacteristic
450C2#3383111yescharacteristic
451C2#3383211yescharacteristic
452C2#3393611yescharacteristic
453C2#3394011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
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
1. C2×#2225C2semidirect
3. #28813C2semidirect
5. #13475C2semidirect
7. #13364C2semidirect
11. #13177C2semidirect
12. #13160C2semidirect
13. #12938C2semidirect
15. #12674C2semidirect
79. #495Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22252
C2×#22262
#288132
#231862
#134752
#134742
#133642
#133632
#132542
#132052
#131772
#131602
#129382
#129372
#126742