Finite Group Databasebeta

← #502037

Subgroups — #502037

Order 512.

499 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28267C211yesmaximal, characteristic
2Q8×#107C211yesmaximal, characteristic, complemented
3#13527C211yesmaximal, characteristic
4#13526C211yesmaximal, characteristic
5#13506C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13264C211yesmaximal, characteristic, complemented
8#13178C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic
10#13130C211yesmaximal, characteristic
11#13112C211yesmaximal, characteristic, complemented
12#13111C211yesmaximal, characteristic, complemented
13#12908C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic
15#12637C211yesmaximal, characteristic
16C2×C8×Q8C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18#2580—21no—
19Q8×C4⋊C4—21no—
20C2×#367C2211yescharacteristic
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#346C2211yescharacteristic
32#2065—21no—
33#2065—21no—
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1947—21no—
37#1947—21no—
38#1947—21no—
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1846C2211yescharacteristic
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1804—21no—
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C2×C4×Q8—41no—
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—21no—
63C22×C16C2311yescharacteristic
64C22×C16C2311yescharacteristic
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C8×Q8—22no—
68C8×Q8—22no—
69C4×D8—22no—
70C4×D8—22no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8D411yescharacteristic
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#399—41no—
92#393—41no—
93#390—21no—
94#390—21no—
95#389—21no—
96#389—21no—
97#389—41no—
98#389—41no—
99#389—41no—
100#385—41no—
101#385—41no—
102#384—21no—
103#384—21no—
104#384—21no—
105#382—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109#367—22no—
110#367—41no—
111#367—41no—
112#366—22no—
113#366—41no—
114#366—41no—
115C4⋊C16—22no—
116#357—22no—
117#357—41no—
118#357—41no—
119#357Q814yescomplemented
120#346—22no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126C22×Q8D811yescharacteristic
127C22×D4—21no—
128C23×C4—41no—
129C2×D8—21no—
130C2×D8—21no—
131C2×D8—21no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8C2×Q812yes—
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2411yescharacteristic
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—84no—
142C4×D4—42no—
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×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4D811yescharacteristic
160C2×C4⋊C4D811yescharacteristic
161C2×C4⋊C4D811yescharacteristic
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16C2×Q812yes—
184C2×C16#4112yes—
185#107—21no—
186#107—21no—
187#107—21no—
188#107—21no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—81no—
199#107—81no—
200#107—81no—
201#107C2×Q812yes—
202#106—81no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206#102—42no—
207#102—81no—
208#102—81no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8#4112yes—
214C4×C8#4112yes—
215#95—41no—
216#95—41no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222C24—41no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×D4—22no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—82no—
240C22×C4—161no—
241C22×C4—161no—
242C22×C4C2×D811yescharacteristic
243C22×C4C2×D811yescharacteristic
244C22×C4C2×D811yescharacteristic
245C22×C4C2×D811yescharacteristic
246C22×C4C2×D811yescharacteristic
247C22×C4C2×D811yescharacteristic
248C22×C4C22×D411yescharacteristic
249D8—22no—
250D8—22no—
251D8—41no—
252D8—41no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8#12312yes—
262C2×C8C22×Q811yescharacteristic
263C2×C8C2×#4111yescharacteristic
264C2×C8C2×#4111yescharacteristic
265C2×C8C2×#4111yescharacteristic
266C2×C8H(5,𝔽2)11yescharacteristic
267C2×C82−1+411yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
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—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—161no—
289C4⋊C4—161no—
290C4⋊C4—161no—
291C4⋊C4—161no—
292C4⋊C4—161no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—161no—
297C22⋊C4—161no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—82no—
303C42—82no—
304C42—82no—
305C42#12312yes—
306C42#13712yes—
307C16—22no—
308C16—22no—
309C16—41no—
310C16—41no—
311C23—42no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23C22×D811yescharacteristic
319Q8—28no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—81no—
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—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
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—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—164no—
360C2×C4#48012yes—
361C2×C4#48012yes—
362C2×C4#50912yes—
363C2×C4#51012yes—
364C2×C4C2×#12311yescharacteristic
365C2×C4D4×Q811yescharacteristic
366C2×C4#55011yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4#57611yescharacteristic
369C2×C4#57811yescharacteristic
370C8—22no—
371C8—41no—
372C8—41no—
373C8#53311yescharacteristic
374C8#54211yescharacteristic
375C8#55211yescharacteristic
376C8#55411yescharacteristic
377C22—82no—
378C22—82no—
379C22—82no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22C2×#48011yescharacteristic
385C22Q8×D811yescharacteristic
386C22#339111yescharacteristic
387C22C2×#50911yescharacteristic
388C22C2×#51011yescharacteristic
389C22#341511yescharacteristic
390C22#341611yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—82no—
398C4—161no—
399C4—161no—
400C4—161no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4#242311yescharacteristic
405C4#309811yescharacteristic
406C4#338511yescharacteristic
407C4#339811yescharacteristic
408C2—162no—
409C2#2505611yescharacteristic
410C2#3364611yescharacteristic
411C2#3365011yescharacteristic
412C2#3395111yescharacteristic
413C2#3395211yescharacteristic
414C2#3396611yescharacteristic
415C2#3396711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#107C2semidirect
5. #13506C2semidirect
6. #13505C2semidirect
7. #13264C2semidirect
8. #13178C2semidirect
11. #13112C2semidirect
12. #13111C2semidirect
13. #12908C2semidirect
119. #357Q8semidirect

Maximal subgroups

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

subgroupindex
#282672
Q8×#1072
#135272
#135262
#135062
#135052
#132642
#131782
#131312
#131302
#131122
#131112
#129082
#128112
#126372