Finite Group Databasebeta

← #484028

Subgroups — #484028

Order 512.

453 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33666C211yesmaximal, characteristic
2#18066C211yesmaximal, characteristic, complemented
3#18039C211yesmaximal, characteristic, complemented
4#12086C211yesmaximal, characteristic, complemented
5#12081C211yesmaximal, characteristic, complemented
6#12075C211yesmaximal, characteristic, complemented
7#12073C211yesmaximal, characteristic, complemented
8#12071C211yesmaximal, characteristic, complemented
9#12069C211yesmaximal, characteristic, complemented
10#12043C211yesmaximal, characteristic, complemented
11C8×SD32C211yesmaximal, characteristic, complemented
12#12035C211yesmaximal, characteristic, complemented
13#11675C211yesmaximal, characteristic, complemented
14#11592C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#2971—21no—
18C4×#135C2211yescharacteristic
19#2935—21no—
20#2234C411yescharacteristic
21#2234C411yescharacteristic
22#2227C411yescharacteristic
23#2226C411yescharacteristic
24#2207C411yescharacteristic
25#2206C411yescharacteristic
26#2186C411yescharacteristic
27#2186C411yescharacteristic
28C4×SD32C2211yescharacteristic
29C4×SD32C2211yescharacteristic
30C4×D16C2211yescharacteristic
31C4×D16C2211yescharacteristic
32#2162C2211yescharacteristic
33C4×M32C2211yescharacteristic
34#1589C2211yescharacteristic
35#1589C2211yescharacteristic
36#1589C2211yescharacteristic
37#1589C2211yescharacteristic
38#1583—21no—
39#1583—21no—
40#1581—21no—
41#1581—21no—
42C8×Q16C2211yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×D8C2211yescharacteristic
45C8×D8C2211yescharacteristic
46#1568C2211yescharacteristic
47C2×#335C2211yescharacteristic
48#1488—21no—
49#1480—21no—
50C8×M16C2211yescharacteristic
51#1372C2211yescharacteristic
52#1372C2211yescharacteristic
53#1371C2211yescharacteristic
54#1371C2211yescharacteristic
55#1332C2211yescharacteristic
56#1332C2211yescharacteristic
57#1331C2211yescharacteristic
58#1331C2211yescharacteristic
59#1331C2211yescharacteristic
60#1331C2211yescharacteristic
61#1331C2211yescharacteristic
62#1331C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68C2×#131—41no—
69#532—21no—
70C4×#41—21no—
71C2×SD32C2×C411yescharacteristic
72C2×SD32C2×C411yescharacteristic
73C2×D16C2×C411yescharacteristic
74C2×D16C2×C411yescharacteristic
75C2×M32—21no—
76#500C2×C411yescharacteristic
77#493C2×C411yescharacteristic
78#486—21no—
79#466—21no—
80#466C2×C411yescharacteristic
81#466C2×C411yescharacteristic
82C8×Q8—21no—
83C8×Q8—21no—
84C4×Q16C2×C411yescharacteristic
85C4×Q16C2311yescharacteristic
86C4×SD16—21no—
87C4×SD16—21no—
88C4×D8—21no—
89C4×D8C2×C411yescharacteristic
90C4×D8C2311yescharacteristic
91C4×D8C2311yescharacteristic
92C4×D8C2311yescharacteristic
93#436—21no—
94#436—21no—
95#436—21no—
96#436—21no—
97#436—41no—
98#436—41no—
99#435—41no—
100#435—41no—
101C8×D4—21no—
102C8×D4—21no—
103#432—21no—
104#432—41no—
105#431D411yescharacteristic
106#427C2×C411yescharacteristic
107C2×C4⋊C8—21no—
108#416—21no—
109#408—41no—
110C4×M16—41no—
111C4×M16D411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#367C2×C411yescharacteristic
114#367C2×C411yescharacteristic
115#366C2×C411yescharacteristic
116#366C2×C411yescharacteristic
117C4⋊C16C2×C411yescharacteristic
118C4⋊C16C2×C411yescharacteristic
119#358C2×C411yescharacteristic
120#358C2×C411yescharacteristic
121#357C2×C411yescharacteristic
122#357C2×C411yescharacteristic
123#357C2×C411yescharacteristic
124#357C2×C411yescharacteristic
125#357C2×C411yescharacteristic
126#357C2×C411yescharacteristic
127C22⋊C16C2×C411yescharacteristic
128C22⋊C16C2×C411yescharacteristic
129#346—21no—
130C4×C16C2311yescharacteristic
131C4×C16C2311yescharacteristic
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134#335—21no—
135#335C2311yescharacteristic
136#335C2311yescharacteristic
137#335C2311yescharacteristic
138#335C2311yescharacteristic
139#334—21no—
140Q8⋊C8—21no—
141Q8⋊C8—21no—
142D4⋊C8—21no—
143D4⋊C8—21no—
144D4⋊C8—21no—
145D4⋊C8—21no—
146D4⋊C8—21no—
147D4⋊C8—21no—
148#322D411yescharacteristic
149#322D411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C2×#41—21no—
153C2×#41—41no—
154#135—22no—
155#135—41no—
156C2×Q16C22×C411yescharacteristic
157C2×SD16—21no—
158C2×D8—21no—
159C2×D8C22×C411yescharacteristic
160C2×D8C22×C411yescharacteristic
161C2×D8C22×C411yescharacteristic
162#131—84no—
163C2×M16—41no—
164C2×M16—41no—
165C2×M16—41no—
166C2×M16—41no—
167C2×M16D811yescharacteristic
168C2×M16D811yescharacteristic
169C22×C8—41no—
170C22×C8—41no—
171C22×C8—41no—
172C22×C8C22×C411yescharacteristic
173C22×C8#4111yescharacteristic
174C4⋊Q8—21no—
175#127—21no—
176#124—41no—
177#121—21no—
178#121—21no—
179#121—41no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189#117—21no—
190#117—41no—
191C2×C42—41no—
192C2×C42C2×D411yescharacteristic
193SD32—22no—
194SD32—22no—
195D16—41no—
196D16—41no—
197M32—42no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16C22×C411yescharacteristic
201C2×C16C22×C411yescharacteristic
202C2×C16C22×C411yescharacteristic
203C2×C16C22×C411yescharacteristic
204#107—21no—
205#107C22×C411yescharacteristic
206#107C22×C411yescharacteristic
207#107C22×C411yescharacteristic
208#107C22×C411yescharacteristic
209#106—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218#103—21no—
219#103—21no—
220#102—21no—
221#102—21no—
222#102—21no—
223#102—21no—
224#102—21no—
225#102—21no—
226C22⋊C8—21no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234#97—41no—
235#97D811yescharacteristic
236#97D811yescharacteristic
237C4×C8—41no—
238C4×C8—41no—
239C4×C8C22×C411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8C2×D411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D411yescharacteristic
244C4×C8C2×D411yescharacteristic
245C4×C8#4111yescharacteristic
246C4×C8C2411yescharacteristic
247#41—42no—
248#41—81no—
249#41—82no—
250#41—82no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C4×D411yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4#13511yescharacteristic
269Q16—21no—
270Q16#13112yescomplemented
271SD16—42no—
272D8—21no—
273D8—21no—
274D8—21no—
275D8—21no—
276D8—21no—
277D8—21no—
278D8—81no—
279M16—22no—
280M16—41no—
281M16—82no—
282M16—82no—
283M16—82no—
284M16—82no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—82no—
295C2×C8—82no—
296C2×C8—82no—
297C2×C8C4×D411yescharacteristic
298C2×C8C4×D411yescharacteristic
299C2×C8C2×D811yescharacteristic
300C2×C8C2×D811yescharacteristic
301C2×C8C2×D811yescharacteristic
302C2×C8C2×D811yescharacteristic
303C2×C8C23×C411yescharacteristic
304C2×C8C2×#4111yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42C4×D411yescharacteristic
323C42C2×D811yescharacteristic
324C42#13511yescharacteristic
325C42C22×D411yescharacteristic
326C16—22no—
327C16—21no—
328C16—21no—
329C16—21no—
330C16—21no—
331C16—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23C4×D811yescharacteristic
336Q8—22no—
337Q8—41no—
338Q8—82no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—41no—
344D4—41no—
345D4—82no—
346D4—82no—
347D4—82no—
348D4—82no—
349D4—161no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—161no—
368C2×C4C4×D811yescharacteristic
369C2×C4C4×D811yescharacteristic
370C2×C4C4×D811yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4C22×D811yescharacteristic
373C2×C4C2×#13511yescharacteristic
374C8—21no—
375C8—21no—
376C8—21no—
377C8—21no—
378C8—21no—
379C8—21no—
380C8—41no—
381C8—41no—
382C8—81no—
383C8—81no—
384C8—81no—
385C8—81no—
386C8#50912yes—
387C8C2×#13111yescharacteristic
388C8#56811yescharacteristic
389C22—21no—
390C22—21no—
391C22—21no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C2×C4×D811yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—42no—
407C4—81no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4#293311yescharacteristic
412C4#295611yescharacteristic
413C4C2×#50911yescharacteristic
414C4#341411yescharacteristic
415C2—41no—
416C2—161no—
417C2—161no—
418C2—161no—
419C2#1799411yescharacteristic
420C2C4×#50911yescharacteristic
421C2#3346511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C2×C4×D8
prime radical#33666C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18066C2semidirect
3. #18039C2semidirect
4. #12086C2semidirect
5. #12081C2semidirect
6. #12075C2semidirect
7. #12073C2semidirect
8. #12071C2semidirect
9. #12069C2semidirect
10. #12043C2semidirect
11. C8×SD32C2semidirect
12. #12035C2semidirect
13. #11675C2semidirect
14. #11592C2semidirect
15. #11409C2semidirect
270. Q16#131semidirect

Maximal subgroups

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

subgroupindex
#336662
#180662
#180392
#120862
#120812
#120752
#120732
#120712
#120692
#120432
C8×SD322
#120352
#116752
#115922
#114092