Finite Group Databasebeta

← #498313

Subgroups — #498313

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33579C211yesmaximal, characteristic
2C2×#2179C212yesmaximal, complemented
3#23179C211yesmaximal, characteristic, complemented
4#22688C211yesmaximal, characteristic, complemented
5#13208C211yesmaximal, characteristic, complemented
6#13179C211yesmaximal, characteristic, complemented
7#13152C212yesmaximal, complemented
8#13011C212yesmaximal, complemented
9#12939C212yesmaximal, complemented
10#12662C211yesmaximal, characteristic, complemented
11C4×#359C211yesmaximal, characteristic, complemented
12C2×#508C2212yes—
13#3147C2211yescharacteristic
14C2×#443C2212yes—
15C2×#443C2212yes—
16C4×#135C2211yescharacteristic
17#2935—21no—
18#2906—21no—
19#2211C412yescomplemented
20#2210C412yescomplemented
21#2209C411yescharacteristic, complemented
22#2208C411yescharacteristic, complemented
23#2207C411yescharacteristic, complemented
24#2206C411yescharacteristic, complemented
25#2179—28no—
26C2×#368C2212yes—
27C2×C4⋊C16C2211yescharacteristic
28C2×#359C2211yescharacteristic
29C2×#359C2212yes—
30C2×#359C2212yes—
31C2×#359C2211yescharacteristic
32#2138C2212yes—
33C2×#346C2212yes—
34C2×C4×C16C2211yescharacteristic
35#1980—21no—
36#1974C2212yes—
37#1945C2212yes—
38#1876—22no—
39#1866—22no—
40#1848C2211yescharacteristic
41#1806—21no—
42C4×#108C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1382C2211yescharacteristic
45#1382C2211yescharacteristic
46#1381C2212yes—
47#1374C2212yes—
48C2×#135—21no—
49C2×#135C2312yes—
50C2×#135C2311yescharacteristic
51C2×#131—22no—
52C2×#131—22no—
53#532—21no—
54C4×#41—21no—
55#508—24no—
56#508—24no—
57C2×Q32C2×C412yes—
58C2×SD32C2×C412yes—
59C2×SD32C2×C412yes—
60C2×D16C2×C412yes—
61C22×C16C2311yescharacteristic
62C22×C16C2312yes—
63C22×C16C2311yescharacteristic
64#495—21no—
65#495C2×C412yes—
66#494C2×C411yescharacteristic
67#493C2×C411yescharacteristic
68#492—21no—
69#443—24no—
70#443—24no—
71#443—42no—
72#443—42no—
73C4×Q16C2×C411yescharacteristic
74C4×Q16C2×C411yescharacteristic
75C4×SD16—21no—
76C4×SD16—21no—
77C4×D8C2×C411yescharacteristic
78C4×D8C2×C411yescharacteristic
79#436—41no—
80#435—42no—
81C8×D4—41no—
82C2×#108—21no—
83C2×#108C2312yes—
84C2×#108C2311yescharacteristic
85C2×#108C2311yescharacteristic
86#427C2311yescharacteristic
87#424—22no—
88#424—42no—
89#423—21no—
90#423—41no—
91C2×C4⋊C8—41no—
92C2×C4⋊C8D411yescharacteristic
93C2×#104—22no—
94C2×#104—22no—
95C2×#104—22no—
96#416—21no—
97#408—22no—
98C4×M16—21no—
99C2×C4×C8C2312yes—
100C2×C4×C8C2311yescharacteristic
101C4×C4⋊C4D411yescharacteristic
102#368—24no—
103C4⋊C16C2×C411yescharacteristic
104C4⋊C16C2×C411yescharacteristic
105C4⋊C16C2×C412yes—
106#359—22no—
107#359—24no—
108#359—42no—
109#359D414yescomplemented
110#358C2×C412yes—
111#358C2×C412yes—
112#357C2×C412yes—
113#357C2×C412yes—
114#346—24no—
115C4×C16—21no—
116C4×C16—21no—
117#341—21no—
118#339—22no—
119#336D412yes—
120C2×#41—21no—
121C2×#41—22no—
122#135—22no—
123#135—22no—
124#135—21no—
125#135—22no—
126#135—21no—
127#135—22no—
128#135—81no—
129#135—81no—
130#135C2×D412yes—
131#135#4112yes—
132C2×Q16—21no—
133C2×Q16C22×C412yes—
134C2×Q16C22×C411yescharacteristic
135C2×SD16—22no—
136C2×SD16—21no—
137C2×SD16—22no—
138C2×D8—21no—
139C2×D8C22×C412yes—
140C2×D8C22×C411yescharacteristic
141#131—44no—
142#131—44no—
143#131—82no—
144#131—82no—
145C2×M16—21no—
146C2×M16—22no—
147C2×M16—21no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D412yes—
154C22×C8C2411yescharacteristic
155C4⋊Q8—21no—
156#127—21no—
157#124—22no—
158#121—41no—
159#121—41no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×D4—21no—
163C4×D4—21no—
164C4×D4—41no—
165C4×D4—41no—
166C4×D4—41no—
167C4×D4—41no—
168#117—21no—
169#117—41no—
170C2×C4⋊C4—21no—
171C2×C42—41no—
172C2×C42C2×D412yes—
173C2×C42C2×D411yescharacteristic
174Q32—24no—
175SD32—24no—
176SD32—24no—
177D16—24no—
178C2×C16—22no—
179C2×C16—21no—
180C2×C16—22no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16C22×C412yes—
186C2×C16C22×C412yes—
187C2×C16C2×D412yes—
188#108—22no—
189#108—21no—
190#108—21no—
191#108—22no—
192#108—21no—
193#108—81no—
194#108C2×D412yes—
195#107C22×C411yescharacteristic
196#107C22×C411yescharacteristic
197#106—21no—
198C4⋊C8—42no—
199C4⋊C8—42no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8D811yescharacteristic
203C4⋊C8D811yescharacteristic
204C4⋊C8D812yes—
205#104—44no—
206#104—44no—
207#104—82no—
208#103—21no—
209#103—21no—
210#102—21no—
211#102—21no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214#97—41no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—41no—
218C4×C8C22×C411yescharacteristic
219C4×C8C22×C411yescharacteristic
220C4×C8#4111yescharacteristic
221C4×C8#4111yescharacteristic
222#95—21no—
223#41—22no—
224#41—22no—
225#41—41no—
226#41—41no—
227#41—42no—
228#41—42no—
229#41—42no—
230#41—42no—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×D4—21no—
234C2×D4—22no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4#13511yescharacteristic
245C22×C4#13511yescharacteristic
246C22×C4C22×D411yescharacteristic
247Q16—22no—
248Q16—21no—
249Q16—22no—
250Q16—81no—
251Q16C4×D412yes—
252SD16—22no—
253SD16—22no—
254SD16—42no—
255SD16—42no—
256SD16—82no—
257D8—22no—
258D8—21no—
259D8—22no—
260D8—81no—
261D8C4×D412yes—
262M16—22no—
263M16—41no—
264M16—42no—
265M16—41no—
266M16—42no—
267M16—41no—
268M16—84no—
269M16—84no—
270C2×C8—22no—
271C2×C8—21no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—84no—
278C2×C8—84no—
279C2×C8C4×D411yescharacteristic
280C2×C8C4×D411yescharacteristic
281C2×C8C2×D812yes—
282C2×C8C2×D812yes—
283C2×C8C23×C411yescharacteristic
284C2×C8C22×D411yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C82−1+411yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C42—22no—
302C42—22no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42C2×D811yescharacteristic
309C42C2×D811yescharacteristic
310C16—22no—
311C16—22no—
312C16—22no—
313C16—41no—
314C23—41no—
315C23—42no—
316C23C2×#13511yescharacteristic
317Q8—22no—
318Q8—41no—
319Q8—42no—
320Q8—42no—
321D4—22no—
322D4—42no—
323D4—42no—
324D4—42no—
325D4—41no—
326D4—42no—
327D4—82no—
328D4—82no—
329D4—82no—
330D4—161no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—161no—
346C2×C4C4×D811yescharacteristic
347C2×C4C4×D811yescharacteristic
348C2×C4C4×D811yescharacteristic
349C2×C4C4×D811yescharacteristic
350C2×C4C2×C4×D411yescharacteristic
351C2×C4D4211yescharacteristic
352C2×C4#54811yescharacteristic
353C2×C4C22×D811yescharacteristic
354C2×C4#57611yescharacteristic
355C2×C4#57711yescharacteristic
356C8—21no—
357C8—21no—
358C8—22no—
359C8—22no—
360C8—42no—
361C8—81no—
362C8—82no—
363C8C2×C4×D411yescharacteristic
364C8#52011yescharacteristic
365C8#54711yescharacteristic
366C8#55311yescharacteristic
367C22—21no—
368C22—21no—
369C22—42no—
370C22—81no—
371C22—81no—
372C22—82no—
373C22—82no—
374C22—82no—
375C22C2×C4×D811yescharacteristic
376C22#328411yescharacteristic
377C22#336111yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—82no—
383C4—81no—
384C4—81no—
385C4—82no—
386C4#217912yescomplemented
387C4#236111yescharacteristic
388C4#294611yescharacteristic
389C4D4×D811yescharacteristic
390C4#335811yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2—162no—
395C2#2299511yescharacteristic
396C2C2×#217911yescharacteristic
397C2#3346411yescharacteristic
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 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2179C2semidirect
3. #23179C2semidirect
4. #22688C2semidirect
5. #13208C2semidirect
6. #13179C2semidirect
7. #13152C2semidirect
8. #13011C2semidirect
9. #12939C2semidirect
10. #12662C2semidirect
11. C4×#359C2semidirect
19. #2211C4semidirect
20. #2210C4semidirect
21. #2209C4semidirect
22. #2208C4semidirect
23. #2207C4semidirect
24. #2206C4semidirect
109. #359D4semidirect
386. C4#2179semidirect

Maximal subgroups

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

subgroupindex
#335792
C2×#21792
#231792
#226882
#132082
#131792
#131522
#130112
#129392
#126622
C4×#3592