Finite Group Databasebeta

← #503759

Subgroups — #503759

Order 512.

530 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2255C211yesmaximal, characteristic, complemented
2C2×#2158C211yesmaximal, characteristic
3#26066C211yesmaximal, characteristic
4#26064C211yesmaximal, characteristic
5#13507C211yesmaximal, characteristic, complemented
6#13506C211yesmaximal, characteristic, complemented
7#13453C211yesmaximal, characteristic, complemented
8#13452C211yesmaximal, characteristic, complemented
9#13168C211yesmaximal, characteristic, complemented
10#13167C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic
12#12921C211yesmaximal, characteristic
13#12908C211yesmaximal, characteristic, complemented
14#12835C211yesmaximal, characteristic
15#12611C211yesmaximal, characteristic
16C23×C16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#427C2211yescharacteristic
21#2441—21no—
22#2439—21no—
23#2255—24no—
24#2158—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×#366C2211yescharacteristic
33C2×#366C2211yescharacteristic
34C2×#366C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×#346C2211yescharacteristic
37C2×#346C2211yescharacteristic
38#2085—21no—
39#2084—21no—
40#2059—21no—
41#2058—21no—
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1854—21no—
47#1854—21no—
48#1846C2211yescharacteristic
49#1845C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#125—21no—
64C2×#117—21no—
65C2×#117—21no—
66C22×C16—22no—
67C22×C16—22no—
68C22×C16Q814yescomplemented
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#500—22no—
74#500—41no—
75#499—22no—
76#499—41no—
77#427—22no—
78#427—22no—
79#427—41no—
80#427—41no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×C22⋊C8D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#399—41no—
99#396—41no—
100#396—41no—
101#393—41no—
102#391—41no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—41no—
108#387—41no—
109#387—41no—
110#384—21no—
111#382—21no—
112#380—21no—
113#380—21no—
114C4×C22⋊C4D411yescharacteristic
115#367—22no—
116#367—22no—
117#367—22no—
118#367—22no—
119#367—22no—
120#367—41no—
121#366—22no—
122#366—22no—
123#366—41no—
124#366D414yes—
125C22⋊C16Q814yescomplemented
126#346—22no—
127#346—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#336D411yescharacteristic
135#336D411yescharacteristic
136C22×Q8—41no—
137C23×C4C2×D411yescharacteristic
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×Q812yes—
146C22×C8C2×Q812yes—
147C22×C8C2411yescharacteristic
148C4⋊Q8—42no—
149C4⋊Q8—81no—
150#125—42no—
151#125—81no—
152#117—42no—
153#117—42no—
154#117—81no—
155#117—81no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4D811yescharacteristic
178C2×C22⋊C4D811yescharacteristic
179C2×C42—41no—
180C2×C42—41no—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—28no—
190C2×C16C2×D412yes—
191C2×C16C2×Q812yes—
192C2×C16C2×Q812yes—
193C2×C16C2×Q812yes—
194C2×C16C2×Q812yes—
195C2×C16#4112yes—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—42no—
211#107—81no—
212#107C2×D412yes—
213#107C2×D412yes—
214#106—42no—
215#106—42no—
216#106—42no—
217#106—81no—
218C22⋊C8Q1614yes—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95D811yescharacteristic
232#95D811yescharacteristic
233C24C2×D811yescharacteristic
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—161no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C4⋊Q812yes—
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×Q1612yes—
256C22×C4C22×D411yescharacteristic
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8C4⋊Q812yes—
268C2×C8C2×Q1612yes—
269C2×C8C2×Q1612yes—
270C2×C8C22×D411yescharacteristic
271C2×C8C22×Q811yescharacteristic
272C2×C8C22×Q811yescharacteristic
273C2×C8C2×#4111yescharacteristic
274C2×C8H(5,𝔽2)11yescharacteristic
275C2×C82−1+411yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—161no—
299C4⋊C4—161no—
300C4⋊C4—161no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—161no—
306C22⋊C4—161no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—82no—
312C42—82no—
313C16—22no—
314C16—22no—
315C16—22no—
316C16C22⋊Q814yes—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23#50012yes—
324C23C22×D811yescharacteristic
325Q8—164no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4#50012yes—
369C2×C4C2×C4⋊Q811yescharacteristic
370C2×C4D4211yescharacteristic
371C2×C4#54711yescharacteristic
372C2×C4C22×Q1611yescharacteristic
373C2×C4#57611yescharacteristic
374C2×C4#57811yescharacteristic
375C8—22no—
376C8—22no—
377C8—22no—
378C8C2×C22⋊Q811yescharacteristic
379C8#54411yescharacteristic
380C8D4×Q811yescharacteristic
381C8#55411yescharacteristic
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22C2×#50011yescharacteristic
390C22D4×D811yescharacteristic
391C22#333411yescharacteristic
392C22#341411yescharacteristic
393C22#341411yescharacteristic
394C22#341611yescharacteristic
395C22#341611yescharacteristic
396C4—22no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—82no—
402C4—82no—
403C4—161no—
404C4—161no—
405C4—161no—
406C4—161no—
407C4—161no—
408C4—161no—
409C4#244011yescharacteristic
410C4#316411yescharacteristic
411C4D4×Q1611yescharacteristic
412C4#333811yescharacteristic
413C2—42no—
414C2#2606511yescharacteristic
415C2#3372811yescharacteristic
416C2#3373011yescharacteristic
417C2#3383311yescharacteristic
418C2#3383711yescharacteristic
419C2#3391511yescharacteristic
420C2#3391711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2255C2semidirect
5. #13507C2semidirect
6. #13506C2semidirect
7. #13453C2semidirect
8. #13452C2semidirect
9. #13168C2semidirect
10. #13167C2semidirect
13. #12908C2semidirect
68. C22×C16Q8semidirect
125. C22⋊C16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22552
C2×#21582
#260662
#260642
#135072
#135062
#134532
#134522
#131682
#131672
#129212
#129082
#128352
#126112