Finite Group Databasebeta

← #496918

Subgroups — #496918

Order 512.

516 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2226C211yesmaximal, characteristic
2#21725C211yesmaximal, characteristic, complemented
3#21282C211yesmaximal, characteristic
4#13275C211yesmaximal, characteristic, complemented
5#13178C211yesmaximal, characteristic, complemented
6#13167C211yesmaximal, characteristic, complemented
7#13166C211yesmaximal, characteristic, complemented
8#13149C211yesmaximal, characteristic
9#13147C211yesmaximal, characteristic
10#13131C211yesmaximal, characteristic
11#13130C211yesmaximal, characteristic
12#13111C211yesmaximal, characteristic, complemented
13#13110C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic
15#12654C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2331—21no—
19#2319—21no—
20#2226C418yescomplemented
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#346C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1993—21no—
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1943—21no—
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1936—21no—
42#1936—21no—
43#1846C2211yescharacteristic
44#1776—21no—
45C4×#107C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48#1762—21no—
49#1726C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×C4×D4—21no—
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67#500—22no—
68#500C2×C414yes—
69C4×D8—22no—
70C4×D8C2×C414yes—
71C2×#107—21no—
72C2×#107C2311yescharacteristic
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#106—21no—
80C2×#102—21no—
81C2×#102—21no—
82C2×#102—21no—
83C2×#97D411yescharacteristic
84C2×#97D411yescharacteristic
85C2×#97D411yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#391—41no—
90#391—41no—
91#390—21no—
92#390—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#388—41no—
98#388—41no—
99#387—41no—
100#387—41no—
101#382—21no—
102#381—41no—
103#380—41no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107#376D411yescharacteristic
108#367—41no—
109#367C2×C414yes—
110#367C2×C414yes—
111#366—41no—
112C4⋊C16—22no—
113C4⋊C16C2×C414yes—
114#357—41no—
115#357—41no—
116#357C2×C414yes—
117#357C2×C414yes—
118#346—41no—
119C4×C16—41no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127C22×Q8—41no—
128C22×D4—21no—
129C23×C4—41no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8C22×C412yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2411yescharacteristic
140C4⋊Q8—24no—
141C4⋊Q8—42no—
142C4×D4—24no—
143C4×D4—42no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C16—21no—
168C2×C16—21no—
169C2×C16—21no—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16C22×C412yes—
178C2×C16C22×C412yes—
179#107—24no—
180#107—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#107—21no—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107C22×C412yes—
192#107C22×C412yes—
193#107C22×C412yes—
194#106—42no—
195#102—24no—
196#102—42no—
197#102—42no—
198#97—22no—
199#97—22no—
200#97—41no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—21no—
204C4×C8—21no—
205C4×C8C22×C412yes—
206C4×C8#4112yes—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212#95—41no—
213#95—41no—
214#95—41no—
215#95—41no—
216#95D811yescharacteristic
217#95D811yescharacteristic
218#95D811yescharacteristic
219#95D811yescharacteristic
220C24—41no—
221C2×Q8—44no—
222C2×Q8—161no—
223C2×Q8—161no—
224C2×D4—22no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—44no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—161no—
239C22×C4—161no—
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C2×D811yescharacteristic
243C22×C4C2×D811yescharacteristic
244C22×C4C2×D811yescharacteristic
245C22×C4C2×D811yescharacteristic
246C22×C4C22×D411yescharacteristic
247D8—22no—
248D8—22no—
249D8—41no—
250D8—41no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8C4×D412yes—
259C2×C8#13712yes—
260C2×C8C23×C411yescharacteristic
261C2×C8C2×#4111yescharacteristic
262C2×C8H(5,𝔽2)11yescharacteristic
263C2×C82−1+411yescharacteristic
264C2×C82−1+411yescharacteristic
265C2×C82−1+411yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—44no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—161no—
282C4⋊C4—161no—
283C4⋊C4—161no—
284C4⋊C4—161no—
285C22⋊C4—44no—
286C22⋊C4—161no—
287C22⋊C4—161no—
288C22⋊C4—161no—
289C22⋊C4—161no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—161no—
295C42—161no—
296C42—161no—
297C42C4×D412yes—
298C42#13512yes—
299C16—22no—
300C16—22no—
301C16—41no—
302C16—41no—
303C23—42no—
304C23—81no—
305C23—81no—
306C23—81no—
307C23—81no—
308C23—81no—
309C23—81no—
310C23C22×D811yescharacteristic
311Q8—164no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4—81no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—164no—
354C2×C4C4×D812yes—
355C2×C4C4×D812yes—
356C2×C4C2×C4×D411yescharacteristic
357C2×C4#53611yescharacteristic
358C2×C4#53711yescharacteristic
359C2×C4C2×#13511yescharacteristic
360C2×C4C2×#13711yescharacteristic
361C2×C4#57511yescharacteristic
362C8—22no—
363C8—22no—
364C8—41no—
365C8#51911yescharacteristic
366C8#52011yescharacteristic
367C8#54111yescharacteristic
368C8#54211yescharacteristic
369C22—82no—
370C22—82no—
371C22—82no—
372C22—161no—
373C22—161no—
374C22—161no—
375C22—161no—
376C22C2×C4×D811yescharacteristic
377C22#322611yescharacteristic
378C22#324511yescharacteristic
379C22#341511yescharacteristic
380C22#341611yescharacteristic
381C22#341611yescharacteristic
382C22#341611yescharacteristic
383C4—22no—
384C4—22no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—82no—
390C4—82no—
391C4—82no—
392C4—161no—
393C4—161no—
394C4—161no—
395C4—161no—
396C4#232311yescharacteristic
397C4C4×#13711yescharacteristic
398C4#323311yescharacteristic
399C4#324811yescharacteristic
400C2—162no—
401C2#2211011yescharacteristic
402C2#3345411yescharacteristic
403C2#3345711yescharacteristic
404C2#3380711yescharacteristic
405C2#3381011yescharacteristic
406C2#3382311yescharacteristic
407C2#3382911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21725C2semidirect
4. #13275C2semidirect
5. #13178C2semidirect
6. #13167C2semidirect
7. #13166C2semidirect
12. #13111C2semidirect
13. #13110C2semidirect
15. #12654C2semidirect
20. #2226C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22262
#217252
#212822
#132752
#131782
#131672
#131662
#131492
#131472
#131312
#131302
#131112
#131102
#128132
#126542