Finite Group Databasebeta

← #504244

Subgroups — #504244

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2227C211yesmaximal, characteristic, complemented
2C2×#2228C211yesmaximal, characteristic
3#28812C211yesmaximal, characteristic, complemented
4#23186C211yesmaximal, characteristic
5#13475C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic
7#13366C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic
9#13254C211yesmaximal, characteristic
10#13204C211yesmaximal, characteristic
11#13178C211yesmaximal, characteristic, complemented
12#13159C211yesmaximal, characteristic, complemented
13#12938C211yesmaximal, characteristic, complemented
14#12935C211yesmaximal, characteristic
15C4×#366C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21#2550—21no—
22#2367—21no—
23#2228—24no—
24#2227—24no—
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
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×C4×C16C2211yescharacteristic
41#2059—21no—
42#2058—21no—
43#1983—21no—
44#1970C2211yescharacteristic
45#1969C2211yescharacteristic
46#1949—21no—
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1880—21no—
50#1877—21no—
51#1846C2211yescharacteristic
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×#125—21no—
66C2×#124—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#500—22no—
74#500—41no—
75#499—22no—
76#499—41no—
77#495—22no—
78#495Q814yescomplemented
79C4×Q16—22no—
80C4×Q16—41no—
81C4×D8—22no—
82C4×D8—41no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91C2×#106—21no—
92C2×C4⋊C8D411yescharacteristic
93C2×#103—21no—
94C2×#103—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#398—41no—
101#395—41no—
102#391—21no—
103#391—41no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—41no—
108#388—41no—
109#388—41no—
110#385—41no—
111#382—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D411yescharacteristic
115C4×C22⋊C4—41no—
116#367—22no—
117#366—22no—
118#366—22no—
119#366D414yes—
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—
131C4×C16—22no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C22×Q8—21no—
139C22×Q8—41no—
140C22×D4—21no—
141C23×C4—41no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16C2×Q812yes—
148C2×SD16—24no—
149C2×SD16—81no—
150C2×SD16—81no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8C2×Q812yes—
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2411yescharacteristic
161C4⋊Q8—42no—
162C4⋊Q8—81no—
163#125—42no—
164#125—81no—
165#124—24no—
166#124—42no—
167C4×Q8—42no—
168C4×Q8—81no—
169C4×D4—42no—
170C4×D4—81no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
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×C42—41no—
191C2×C42—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—42no—
218#107C2×D412yes—
219#107C2×D412yes—
220#107#4112yes—
221#106—42no—
222#106—81no—
223C4⋊C8—22no—
224#103—42no—
225#103—81no—
226#102—42no—
227#102—81no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8C2×Q812yes—
233C4×C8#4112yes—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24—41no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—82no—
252C2×Q8—82no—
253C2×Q8—161no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—161no—
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C22×D411yescharacteristic
277Q16—22no—
278Q16—22no—
279Q16—41no—
280Q16—41no—
281SD16—84no—
282D8—22no—
283D8—22no—
284D8—41no—
285D8—41no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8C22⋊Q812yes—
295C2×C8C22×D411yescharacteristic
296C2×C8C22×Q811yescharacteristic
297C2×C8C2×#4111yescharacteristic
298C2×C8C2×#4111yescharacteristic
299C2×C8H(5,𝔽2)11yescharacteristic
300C2×C82−1+411yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
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—41no—
314C4⋊C4—41no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—161no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—161no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—82no—
331C42—161no—
332C42—161no—
333C42C22⋊Q812yes—
334C42#13712yes—
335C16—22no—
336C16—41no—
337C16—41no—
338C16—41no—
339C23—42no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23C22×D811yescharacteristic
347Q8—42no—
348Q8—42no—
349Q8—81no—
350Q8—81no—
351Q8—164no—
352D4—42no—
353D4—42no—
354D4—81no—
355D4—81no—
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—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
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—164no—
393C2×C4#47412yes—
394C2×C4#47412yes—
395C2×C4#50812yes—
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4D4211yescharacteristic
398C2×C4#54811yescharacteristic
399C2×C4C2×#13711yescharacteristic
400C2×C4#57611yescharacteristic
401C2×C4#57711yescharacteristic
402C8—22no—
403C8—41no—
404C8—41no—
405C8#53211yescharacteristic
406C8#54211yescharacteristic
407C8D4×Q811yescharacteristic
408C8#55411yescharacteristic
409C22—82no—
410C22—82no—
411C22—82no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22C2×#47411yescharacteristic
417C22D4×D811yescharacteristic
418C22#335811yescharacteristic
419C22C2×#50811yescharacteristic
420C22#341411yescharacteristic
421C22#341511yescharacteristic
422C22#341711yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#245211yescharacteristic
437C4#308111yescharacteristic
438C4#327811yescharacteristic
439C4#337511yescharacteristic
440C2—162no—
441C2#2646011yescharacteristic
442C2#3362611yescharacteristic
443C2#3363311yescharacteristic
444C2#3383911yescharacteristic
445C2#3385111yescharacteristic
446C2#3392811yescharacteristic
447C2#3394411yescharacteristic
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×#2227C2semidirect
3. #28812C2semidirect
5. #13475C2semidirect
7. #13366C2semidirect
11. #13178C2semidirect
12. #13159C2semidirect
13. #12938C2semidirect
15. C4×#366C2semidirect
78. #495Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22272
C2×#22282
#288122
#231862
#134752
#134742
#133662
#133632
#132542
#132042
#131782
#131592
#129382
#129352
C4×#3662