Finite Group Databasebeta

← #496925

Subgroups — #496925

Order 512.

457 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33626C211yesmaximal, characteristic
2#21733C211yesmaximal, characteristic, complemented
3#21294C211yesmaximal, characteristic
4#13274C211yesmaximal, characteristic, complemented
5#13179C211yesmaximal, characteristic, complemented
6#13170C211yesmaximal, characteristic, complemented
7#13169C211yesmaximal, characteristic, complemented
8#13152C211yesmaximal, characteristic
9#13151C211yesmaximal, characteristic
10#13134C211yesmaximal, characteristic
11#13134C211yesmaximal, characteristic
12#13113C211yesmaximal, characteristic, complemented
13#13113C211yesmaximal, characteristic, complemented
14#12815C211yesmaximal, characteristic
15#12654C211yesmaximal, characteristic, complemented
16#3160C2211yescharacteristic
17#2961—21no—
18C4×#135C2211yescharacteristic
19#2923—21no—
20#2229C411yescharacteristic, complemented
21#2229C411yescharacteristic, complemented
22#2228C411yescharacteristic, complemented
23#2228C411yescharacteristic, complemented
24#2227C411yescharacteristic, complemented
25#2226C411yescharacteristic, complemented
26#2225C411yescharacteristic, complemented
27#2224C411yescharacteristic, complemented
28C2×#368C2211yescharacteristic
29C2×#368C2211yescharacteristic
30#2158C2211yescharacteristic
31#2158C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#359C2211yescharacteristic
35C2×#359C2211yescharacteristic
36#2138C2211yescharacteristic
37#2138C2211yescharacteristic
38C2×#346C2211yescharacteristic
39C2×C4×C16C2211yescharacteristic
40#1995—21no—
41#1974C2211yescharacteristic
42#1974C2211yescharacteristic
43#1946—21no—
44#1945C2211yescharacteristic
45#1945C2211yescharacteristic
46#1945C2211yescharacteristic
47#1945C2211yescharacteristic
48#1938—21no—
49#1938—21no—
50#1848C2211yescharacteristic
51C4×#108C2211yescharacteristic
52#1773—21no—
53#1772C2211yescharacteristic
54#1772C2211yescharacteristic
55#1759—21no—
56#1726C2211yescharacteristic
57#1382C2211yescharacteristic
58#1382C2211yescharacteristic
59#1382C2211yescharacteristic
60#1382C2211yescharacteristic
61#1382C2211yescharacteristic
62#1382C2211yescharacteristic
63#1381C2211yescharacteristic
64#1381C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#135C2311yescharacteristic
68#533—21no—
69C4×#41—21no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#500C2×C411yescharacteristic
75#500C2×C411yescharacteristic
76#499—21no—
77#499C2×C411yescharacteristic
78#499C2×C411yescharacteristic
79#498—21no—
80#446—41no—
81#446—41no—
82#446—41no—
83#446—41no—
84C4×Q16C2×C411yescharacteristic
85C4×Q16C2×C411yescharacteristic
86C4×SD16—21no—
87C4×SD16—21no—
88C4×D8C2×C411yescharacteristic
89C4×D8C2×C411yescharacteristic
90#433—41no—
91#433—41no—
92#432—41no—
93#432—41no—
94C2×#108—21no—
95C2×#108C2311yescharacteristic
96C2×#108C2311yescharacteristic
97C2×#108C2311yescharacteristic
98C2×#108C2311yescharacteristic
99#427—21no—
100#427C2311yescharacteristic
101#427C2311yescharacteristic
102#427C2311yescharacteristic
103#424—21no—
104#424—21no—
105#424—21no—
106#424—21no—
107#423—21no—
108C2×#104—21no—
109C2×#104—21no—
110#416—21no—
111#408—21no—
112#408—21no—
113#405—21no—
114#405—21no—
115C4×M16—21no—
116C2×#97D411yescharacteristic
117C2×#97D411yescharacteristic
118C2×#97D411yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#376D411yescharacteristic
123#368—41no—
124#368—41no—
125#367C2×C411yescharacteristic
126#367C2×C411yescharacteristic
127#367C2×C411yescharacteristic
128#367C2×C411yescharacteristic
129#366C2×C411yescharacteristic
130#366C2×C411yescharacteristic
131#366C2×C411yescharacteristic
132#366C2×C411yescharacteristic
133C4⋊C16—21no—
134C4⋊C16—21no—
135C4⋊C16C2×C411yescharacteristic
136C4⋊C16C2×C411yescharacteristic
137C4⋊C16C2×C411yescharacteristic
138C4⋊C16C2×C411yescharacteristic
139#359—41no—
140#359—41no—
141#358C2×C411yescharacteristic
142#358C2×C411yescharacteristic
143#358C2×C411yescharacteristic
144#358C2×C411yescharacteristic
145#357C2×C411yescharacteristic
146#357C2×C411yescharacteristic
147#357C2×C411yescharacteristic
148#357C2×C411yescharacteristic
149#346—41no—
150C4×C16—41no—
151#341—21no—
152#339—21no—
153#339—21no—
154#339—21no—
155#339—21no—
156#339—21no—
157#339—21no—
158C2×#41—21no—
159#135—21no—
160#135—21no—
161#135—21no—
162#135—21no—
163C2×Q16C22×C411yescharacteristic
164C2×SD16—21no—
165C2×D8C22×C411yescharacteristic
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2411yescharacteristic
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176#125—21no—
177#125—21no—
178C22⋊Q8—41no—
179C22⋊Q8—41no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4—41no—
188#117—21no—
189#117—21no—
190#117—21no—
191#117—41no—
192C2×C42—41no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16C22×C411yescharacteristic
207C2×C16C22×C411yescharacteristic
208C2×C16C22×C411yescharacteristic
209C2×C16C22×C411yescharacteristic
210#108—21no—
211#108—21no—
212#108—21no—
213#108—21no—
214#108—21no—
215#108—21no—
216#108—21no—
217#108—21no—
218#108—42no—
219#107—21no—
220#107—21no—
221#107C22×C411yescharacteristic
222#107C22×C411yescharacteristic
223#107C22×C411yescharacteristic
224#107C22×C411yescharacteristic
225#107C22×C411yescharacteristic
226#107C22×C411yescharacteristic
227#106—21no—
228#106—21no—
229#106—21no—
230#106—21no—
231#106—21no—
232C4⋊C8—41no—
233C4⋊C8—41no—
234C4⋊C8—41no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237C4⋊C8—41no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241C4⋊C8—41no—
242#104—42no—
243#104—42no—
244#103—21no—
245#103—21no—
246#102—21no—
247#102—21no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251C22⋊C8—41no—
252#97—22no—
253#97—41no—
254#97—41no—
255#97—41no—
256#97—41no—
257#97D811yescharacteristic
258#97D811yescharacteristic
259#97D811yescharacteristic
260#97D811yescharacteristic
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—41no—
266C4×C8—41no—
267C4×C8—41no—
268C4×C8C22×C411yescharacteristic
269C4×C8C22×C411yescharacteristic
270C4×C8#4111yescharacteristic
271C4×C8#4111yescharacteristic
272#95—21no—
273#95—21no—
274#41—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×D4—21no—
282C2×D4—41no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4#13511yescharacteristic
289C22×C4#13511yescharacteristic
290C22×C4C22×D411yescharacteristic
291Q16—21no—
292Q16—21no—
293SD16—41no—
294SD16—41no—
295D8—21no—
296D8—21no—
297M16—41no—
298M16—41no—
299M16—41no—
300M16—41no—
301M16—41no—
302M16—41no—
303M16—41no—
304M16—41no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8C4×D411yescharacteristic
316C2×C8C4×D411yescharacteristic
317C2×C8C2×D811yescharacteristic
318C2×C8C2×D811yescharacteristic
319C2×C8C2×D811yescharacteristic
320C2×C8C2×D811yescharacteristic
321C2×C8C23×C411yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8H(5,𝔽2)11yescharacteristic
324C2×C82−1+411yescharacteristic
325C2×C82−1+411yescharacteristic
326C2×C82−1+411yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42C4×D411yescharacteristic
355C42C4×D411yescharacteristic
356C42C2×D811yescharacteristic
357C42C2×D811yescharacteristic
358C42#13712yes—
359C16—21no—
360C16—21no—
361C16—21no—
362C16—21no—
363C16—41no—
364C16—41no—
365C23—41no—
366C23C2×#13511yescharacteristic
367Q8—41no—
368Q8—41no—
369Q8—161no—
370Q8—161no—
371D4—41no—
372D4—41no—
373D4—81no—
374D4—81no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—161no—
393C2×C4—161no—
394C2×C4C4×D811yescharacteristic
395C2×C4C4×D811yescharacteristic
396C2×C4C4×D811yescharacteristic
397C2×C4C4×D811yescharacteristic
398C2×C4C2×C4×D411yescharacteristic
399C2×C4#53611yescharacteristic
400C2×C4#53711yescharacteristic
401C2×C4C22×D811yescharacteristic
402C2×C4C2×#13711yescharacteristic
403C2×C4#57511yescharacteristic
404C8—21no—
405C8—21no—
406C8—21no—
407C8—21no—
408C8—21no—
409C8—21no—
410C8—81no—
411C8—81no—
412C8—81no—
413C8—81no—
414C8#51911yescharacteristic
415C8#52011yescharacteristic
416C8#54111yescharacteristic
417C8#54211yescharacteristic
418C22—21no—
419C22—21no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22C2×C4×D811yescharacteristic
424C22#323311yescharacteristic
425C22#324811yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—81no—
432C4—81no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4#232311yescharacteristic
440C4C4×#13711yescharacteristic
441C4#322611yescharacteristic
442C4#324511yescharacteristic
443C2—21no—
444C2—21no—
445C2—161no—
446C2#2211011yescharacteristic
447C2#3346411yescharacteristic
448C2#3346511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radicalC2×#135C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21733C2semidirect
4. #13274C2semidirect
5. #13179C2semidirect
6. #13170C2semidirect
7. #13169C2semidirect
12. #13113C2semidirect
13. #13113C2semidirect
15. #12654C2semidirect
20. #2229C4semidirect
21. #2229C4semidirect
22. #2228C4semidirect
23. #2228C4semidirect
24. #2227C4semidirect
25. #2226C4semidirect
26. #2225C4semidirect
27. #2224C4semidirect

Maximal subgroups

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

subgroupindex
#336262
#217332
#212942
#132742
#131792
#131702
#131692
#131522
#131512
#131342
#131132
#128152
#126542