Finite Group Databasebeta

← #496965

Subgroups — #496965

Order 512.

585 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#33579C211yesmaximal, characteristic
2#21705C211yesmaximal, characteristic, complemented
3#21292C211yesmaximal, characteristic, complemented
4#13251C211yesmaximal, characteristic, complemented
5#13241C212yesmaximal, complemented
6#13179C211yesmaximal, characteristic, complemented
7#13151C212yesmaximal, complemented
8#13011C212yesmaximal, complemented
9#12939C212yesmaximal, complemented
10#12815C211yesmaximal, characteristic, complemented
11C4×C4⋊C16C211yesmaximal, characteristic, complemented
12C2×#508C2212yes—
13#3147C2211yescharacteristic
14C2×#443—21no—
15C2×#443C2212yes—
16C4×#135C2211yescharacteristic
17#2931—21no—
18#2922—21no—
19#2211C412yescomplemented
20#2210C412yescomplemented
21#2209C411yescharacteristic, complemented
22#2208C411yescharacteristic, complemented
23#2207C411yescharacteristic, complemented
24#2206C411yescharacteristic, complemented
25C2×#368C2212yes—
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#359C2212yes—
29C2×#359C2212yes—
30C2×#359C2212yes—
31#2138C2212yes—
32C2×C4×C16C2212yes—
33#1974—21no—
34#1974C2211yescharacteristic
35#1974C2211yescharacteristic
36#1974C2212yes—
37#1962—21no—
38#1945C2211yescharacteristic
39#1945C2211yescharacteristic
40#1937—21no—
41#1937—21no—
42#1848C2211yescharacteristic
43C4×#108C2211yescharacteristic
44C4×#104—21no—
45C42×C8C2211yescharacteristic
46#1382C2212yes—
47#1381C2212yes—
48#1374C2212yes—
49C2×#135—21no—
50C2×#135C2312yes—
51C2×#135C2311yescharacteristic
52C2×#131—22no—
53#532—21no—
54C4×#41—21no—
55#508—42no—
56#508—42no—
57C2×Q32C2×C412yes—
58C2×SD32C2×C412yes—
59C2×SD32C2×C412yes—
60C2×D16C2×C412yes—
61C22×C16C2312yes—
62C22×C16C2312yes—
63#495—21no—
64#495C2×C412yes—
65#494C2×C411yescharacteristic
66#493C2×C411yescharacteristic
67#492—21no—
68#443—42no—
69#443—42no—
70#443—44no—
71C4×Q16C2×C411yescharacteristic
72C4×Q16C2×C411yescharacteristic
73C4×SD16—21no—
74C4×SD16—21no—
75C4×D8C2×C411yescharacteristic
76C4×D8C2×C411yescharacteristic
77#436—41no—
78#436—41no—
79C8×D4—41no—
80C8×D4—41no—
81#433—41no—
82#433—41no—
83#431—41no—
84#431—41no—
85C2×#108—21no—
86C2×#108C2312yes—
87C2×#108C2311yescharacteristic
88C2×#108C2311yescharacteristic
89#427C2311yescharacteristic
90#424—21no—
91#424—21no—
92#423—21no—
93C2×#104—22no—
94C2×#104—22no—
95C2×#104—21no—
96C2×#104—21no—
97#416—21no—
98#408—21no—
99#408—21no—
100#408—22no—
101C4×M16—21no—
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C2×C4×C8D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C43D411yescharacteristic
109#368—42no—
110C4⋊C16—21no—
111C4⋊C16—21no—
112C4⋊C16C2×C411yescharacteristic
113C4⋊C16C2×C411yescharacteristic
114C4⋊C16C2×C412yes—
115#359—42no—
116#359—42no—
117#359—42no—
118#358C2×C412yes—
119#358C2×C412yes—
120#357C2×C412yes—
121#357C2×C412yes—
122C4×C16—42no—
123#341—21no—
124#339—21no—
125#339—21no—
126C2×#41—21no—
127C2×#41—22no—
128#135—21no—
129#135—22no—
130#135—22no—
131#135—22no—
132#135—21no—
133#135—21no—
134#135—22no—
135#135—21no—
136#135—42no—
137#135—42no—
138C2×Q16—21no—
139C2×Q16C22×C412yes—
140C2×Q16C22×C411yescharacteristic
141C2×SD16—22no—
142C2×SD16—21no—
143C2×SD16—22no—
144C2×D8—21no—
145C2×D8C22×C412yes—
146C2×D8C22×C411yescharacteristic
147#131—44no—
148#131—44no—
149C2×M16—22no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—42no—
153C22×C8—42no—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2411yescharacteristic
158C4⋊Q8—21no—
159#127—21no—
160#124—22no—
161#121—41no—
162#121—41no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—41no—
168C4×D4—41no—
169C4×D4—41no—
170C4×D4—41no—
171#117—21no—
172#117—41no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—41no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179Q32—42no—
180SD32—42no—
181SD32—42no—
182D16—42no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16—22no—
186C2×C16—22no—
187C2×C16—22no—
188C2×C16C22×C412yes—
189C2×C16C22×C412yes—
190#108—21no—
191#108—21no—
192#108—22no—
193#108—21no—
194#108—22no—
195#108—21no—
196#108—42no—
197#107C22×C411yescharacteristic
198#107C22×C411yescharacteristic
199#106—21no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C4⋊C8—41no—
205C4⋊C8—41no—
206#104—42no—
207#104—44no—
208#104—42no—
209#104—44no—
210#103—21no—
211#103—21no—
212#102—21no—
213#102—21no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217C22⋊C8—41no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220#97—41no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—22no—
226C4×C8—41no—
227C4×C8—41no—
228C4×C8D811yescharacteristic
229C4×C8D811yescharacteristic
230C4×C8D811yescharacteristic
231C4×C8D811yescharacteristic
232C4×C8C22×C411yescharacteristic
233C4×C8C22×C411yescharacteristic
234C4×C8#4111yescharacteristic
235C4×C8#4111yescharacteristic
236#41—42no—
237#41—41no—
238#41—42no—
239#41—41no—
240#41—42no—
241#41—41no—
242#41—41no—
243#41—42no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×D4—21no—
247C2×D4—22no—
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4#13511yescharacteristic
259C22×C4#13511yescharacteristic
260C22×C4C22×D411yescharacteristic
261Q16—21no—
262Q16—22no—
263Q16—22no—
264Q16—21no—
265Q16—42no—
266SD16—44no—
267SD16—41no—
268SD16—42no—
269SD16—41no—
270SD16—42no—
271D8—21no—
272D8—22no—
273D8—22no—
274D8—21no—
275D8—42no—
276M16—42no—
277M16—41no—
278M16—41no—
279M16—42no—
280M16—41no—
281M16—41no—
282M16—84no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—84no—
293C2×C8C4×D411yescharacteristic
294C2×C8C4×D411yescharacteristic
295C2×C8C2×D811yescharacteristic
296C2×C8C2×D811yescharacteristic
297C2×C8C2×D811yescharacteristic
298C2×C8C2×D811yescharacteristic
299C2×C8C23×C411yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C2×C8H(5,𝔽2)11yescharacteristic
302C2×C8H(5,𝔽2)11yescharacteristic
303C2×C8H(5,𝔽2)11yescharacteristic
304C2×C82−1+411yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C42—24no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42C4×D411yescharacteristic
325C42C4×D411yescharacteristic
326C42C2×D811yescharacteristic
327C42C2×D811yescharacteristic
328C42AGL(1,ℤ/8)12yes—
329C16—22no—
330C16—22no—
331C16—42no—
332C23—41no—
333C23—42no—
334C23C2×#13511yescharacteristic
335Q8—41no—
336Q8—42no—
337Q8—42no—
338Q8—41no—
339D4—41no—
340D4—42no—
341D4—42no—
342D4—41no—
343D4—82no—
344D4—81no—
345D4—82no—
346D4—81no—
347D4—82no—
348D4—82no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4C4×D811yescharacteristic
368C2×C4C4×D811yescharacteristic
369C2×C4C4×D811yescharacteristic
370C2×C4C4×D811yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4#53511yescharacteristic
373C2×C4#53711yescharacteristic
374C2×C4C22×D811yescharacteristic
375C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
376C2×C4#57511yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8—21no—
381C8—21no—
382C8—21no—
383C8—82no—
384C8—81no—
385C8—81no—
386C8#51811yescharacteristic
387C8#52011yescharacteristic
388C8#54011yescharacteristic
389C8#54211yescharacteristic
390C22—21no—
391C22—21no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—82no—
396C22—81no—
397C22—82no—
398C22C2×C4×D811yescharacteristic
399C22#321211yescharacteristic
400C22#324711yescharacteristic
401C4—24no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—42no—
406C4—81no—
407C4—82no—
408C4—81no—
409C4—81no—
410C4#217812yes—
411C4#232411yescharacteristic
412C4C4×AGL(1,ℤ/8)11yescharacteristic
413C4#320511yescharacteristic
414C4#324511yescharacteristic
415C2—21no—
416C2—21no—
417C2—161no—
418C2—162no—
419C2#2214511yescharacteristic
420C2C2×#217811yescharacteristic
421C2#3346311yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radical#33579C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21705C2semidirect
3. #21292C2semidirect
4. #13251C2semidirect
5. #13241C2semidirect
6. #13179C2semidirect
7. #13151C2semidirect
8. #13011C2semidirect
9. #12939C2semidirect
10. #12815C2semidirect
11. C4×C4⋊C16C2semidirect
19. #2211C4semidirect
20. #2210C4semidirect
21. #2209C4semidirect
22. #2208C4semidirect
23. #2207C4semidirect
24. #2206C4semidirect

Maximal subgroups

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

subgroupindex
#335792
#217052
#212922
#132512
#132412
#131792
#131512
#130112
#129392
#128152
C4×C4⋊C162