Finite Group Databasebeta

← #495987

Subgroups — #495987

Order 512.

606 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×#2228C211yesmaximal, characteristic
2#21723C211yesmaximal, characteristic, complemented
3#20508C211yesmaximal, characteristic
4#13277C211yesmaximal, characteristic, complemented
5#13176C211yesmaximal, characteristic, complemented
6#13112C211yesmaximal, characteristic, complemented
7#13111C211yesmaximal, characteristic, complemented
8#12935C211yesmaximal, characteristic
9#12935C211yesmaximal, characteristic
10#12811C211yesmaximal, characteristic
11#12674C211yesmaximal, characteristic, complemented
12C4×#366C211yesmaximal, characteristic, complemented
13#12654C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic
15C4×#357C211yesmaximal, characteristic
16C2×#499C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2330—21no—
21#2275—21no—
22#2228C418yescomplemented
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#346C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#1992—21no—
36#1944—21no—
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1845C2211yescharacteristic
40#1776—21no—
41C4×#107C2211yescharacteristic
42C4×#107C2211yescharacteristic
43C4×#107C2211yescharacteristic
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46#1762—21no—
47#1762—21no—
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×#125—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#499—22no—
70#499C2×C414yescomplemented
71C4×D8—22no—
72C4×D8—22no—
73C4×D8—22no—
74C4×D8—22no—
75C4×D8C2×C414yes—
76C4×D8Q814yes—
77C2×#107—21no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#106—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#97D411yescharacteristic
90C2×#97D411yescharacteristic
91C2×#97D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#389—21no—
96#389—21no—
97#389—41no—
98#389—41no—
99#387—41no—
100#387—41no—
101#384—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4—41no—
108C4×C22⋊C4—41no—
109C4×C22⋊C4—41no—
110#376D411yescharacteristic
111C2×#95—41no—
112#367C2×C414yescomplemented
113#366—22no—
114#366—22no—
115#366C2×C414yescomplemented
116C4⋊C16—22no—
117C4⋊C16C2×C414yescomplemented
118#357—22no—
119#357—22no—
120#357C2×C414yes—
121#357C2×C414yes—
122#346—22no—
123C4×C16—22no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131C22×D4—21no—
132C23×C4—41no—
133C23×C4—41no—
134C23×C4—41no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8C22×C412yes—
139C2×D8C2×Q812yes—
140C2×D8#4112yes—
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2411yescharacteristic
145#125—24no—
146#125—42no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—42no—
151C4×D4—42no—
152C4×D4—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C42—41no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C16—21no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16C22×C412yes—
188C2×C16C22×C412yes—
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—42no—
199#107C22×C412yes—
200#107C22×C412yes—
201#107C22×C412yes—
202#107C2×Q812yes—
203#107#4112yes—
204#106—24no—
205#102—24no—
206#102—24no—
207#102—24no—
208#97—22no—
209#97—22no—
210#97—41no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8C22×C412yes—
214C4×C8C2×Q812yes—
215C4×C8#4112yes—
216C4×C8#4112yes—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95D811yescharacteristic
225#95D811yescharacteristic
226#95D811yescharacteristic
227#95D811yescharacteristic
228C24—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—44no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—161no—
246C22×C4—161no—
247C22×C4—161no—
248C22×C4—161no—
249C22×C4—161no—
250C22×C4—161no—
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C22×D411yescharacteristic
258D8—22no—
259D8—22no—
260D8—22no—
261D8C4×Q814yes—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8C4×D412yes—
270C2×C8C22⋊Q812yes—
271C2×C8C23×C411yescharacteristic
272C2×C8C22×Q811yescharacteristic
273C2×C8C2×#4111yescharacteristic
274C2×C8C2×#4111yescharacteristic
275C2×C8C2×#4111yescharacteristic
276C2×C8C2×#4111yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—161no—
294C4⋊C4—161no—
295C4⋊C4—161no—
296C4⋊C4—161no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C42—21no—
301C42—21no—
302C42—161no—
303C42—161no—
304C42—161no—
305C42—161no—
306C42—161no—
307C42C4×D412yes—
308C42C22⋊Q812yes—
309C42#13512yes—
310C42#13712yes—
311C16—22no—
312C16—22no—
313C16—41no—
314C16—41no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23C22×D811yescharacteristic
323D4—24no—
324D4—42no—
325D4—42no—
326D4—42no—
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—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
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—164no—
363C2×C4—164no—
364C2×C4—164no—
365C2×C4C4×D812yes—
366C2×C4C4×D812yes—
367C2×C4#47412yes—
368C2×C4#47412yes—
369C2×C4C2×C4×D411yescharacteristic
370C2×C4C2×C22⋊Q811yescharacteristic
371C2×C4#52511yescharacteristic
372C2×C4C2×#13511yescharacteristic
373C2×C4C2×#13711yescharacteristic
374C2×C4#57511yescharacteristic
375C8—22no—
376C8—22no—
377C8—41no—
378C8C2×C4×Q811yescharacteristic
379C8C4×#4111yescharacteristic
380C8#52911yescharacteristic
381C8#53311yescharacteristic
382C22—44no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22C2×C4×D811yescharacteristic
390C22C2×#47411yescharacteristic
391C22#310011yescharacteristic
392C22#341511yescharacteristic
393C22#341511yescharacteristic
394C22#341611yescharacteristic
395C22#341711yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—82no—
403C4—82no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—161no—
408C4—161no—
409C4C4×C22⋊Q811yescharacteristic
410C4C4×#13711yescharacteristic
411C4#307611yescharacteristic
412C4#310711yescharacteristic
413C2—84no—
414C2C4×#47411yescharacteristic
415C2#3345411yescharacteristic
416C2#3345911yescharacteristic
417C2#3363111yescharacteristic
418C2#3363311yescharacteristic
419C2#3368011yescharacteristic
420C2#3368211yescharacteristic
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. #21723C2semidirect
4. #13277C2semidirect
5. #13176C2semidirect
6. #13112C2semidirect
7. #13111C2semidirect
11. #12674C2semidirect
12. C4×#366C2semidirect
13. #12654C2semidirect
22. #2228C4semidirect
70. #499C2×C4semidirect
112. #367C2×C4semidirect
115. #366C2×C4semidirect
117. C4⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
#217232
#205082
#132772
#131762
#131122
#131112
#129352
#128112
#126742
C4×#3662
#126542
#126372
C4×#3572