Finite Group Databasebeta

← #509371

Subgroups — #509371

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2245C211yesmaximal, characteristic, complemented
2C2×#2237C211yesmaximal, characteristic
3#26066C211yesmaximal, characteristic
4#25415C211yesmaximal, characteristic
5#13531C211yesmaximal, characteristic, complemented
6#13475C211yesmaximal, characteristic, complemented
7#13453C211yesmaximal, characteristic, complemented
8#13409C211yesmaximal, characteristic, complemented
9#13383C211yesmaximal, characteristic
10#13382C211yesmaximal, characteristic
11#13168C211yesmaximal, characteristic, complemented
12#13121C211yesmaximal, characteristic
13#12994C211yesmaximal, characteristic
14#12936C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2441—21no—
23#2429—21no—
24#2245—24no—
25#2237—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42#2084—21no—
43#2059—21no—
44#2043—21no—
45#2001—21no—
46#1969C2211yescharacteristic
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1942C2211yescharacteristic
50#1878—21no—
51#1854—21no—
52#1845C2211yescharacteristic
53#1808C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16—21no—
62C22×Q16C2311yescharacteristic
63C22×Q16C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—21no—
66C2×#125—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C4×Q8—21no—
70C2×#117—21no—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#499—22no—
76#499—41no—
77#494—22no—
78#494—41no—
79#467—22no—
80#467—22no—
81#467—41no—
82#467—41no—
83C4×Q16—22no—
84C4×Q16—41no—
85#427—22no—
86#427—41no—
87C2×#107—21no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#106—21no—
95C2×#106—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#399—41no—
104#398—41no—
105#396—41no—
106#396—41no—
107#391—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—41no—
112#389—41no—
113#388—41no—
114#387—41no—
115#384—21no—
116#380—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—41no—
120#366—22no—
121#366—22no—
122C4⋊C16—41no—
123#358—22no—
124#358—22no—
125#358—22no—
126#358—22no—
127#358—41no—
128#358—41no—
129#358—41no—
130#358D414yes—
131C22⋊C16—22no—
132C22⋊C16—41no—
133#346—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C23×C4C2×D411yescharacteristic
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—42no—
155C2×Q16—42no—
156C2×Q16—81no—
157C2×Q16C2×D412yes—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2411yescharacteristic
168C4⋊Q8—42no—
169C4⋊Q8—81no—
170#125—42no—
171#125—81no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C22⋊Q8—81no—
175C22⋊Q8—81no—
176C4×Q8—42no—
177C4×Q8—81no—
178#117—42no—
179#117—81no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4D811yescharacteristic
203C2×C22⋊C4D811yescharacteristic
204C2×C42—41no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16C2×D412yes—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—81no—
231#107C2×D412yes—
232#107#4112yes—
233#106—42no—
234#106—42no—
235#103—42no—
236#103—42no—
237#103—42no—
238#103—81no—
239C22⋊C8—22no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95D811yescharacteristic
251#95D811yescharacteristic
252C24C2×D811yescharacteristic
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C22×D411yescharacteristic
285Q16—22no—
286Q16—22no—
287Q16—41no—
288Q16—41no—
289Q16—41no—
290Q16—41no—
291Q16—41no—
292Q16—41no—
293Q16—84no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8#13712yes—
307C2×C8C22×D411yescharacteristic
308C2×C8C2×#4111yescharacteristic
309C2×C8H(5,𝔽2)11yescharacteristic
310C2×C8H(5,𝔽2)11yescharacteristic
311C2×C82−1+411yescharacteristic
312C2×C82−1+411yescharacteristic
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—161no—
336C4⋊C4—161no—
337C22⋊C4—22no—
338C22⋊C4—41no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—161no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—82no—
348C42—82no—
349C16—22no—
350C16—41no—
351C16—41no—
352C16—41no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23C22×D811yescharacteristic
361Q8—42no—
362Q8—42no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368Q8—81no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4#51012yes—
412C2×C4#53611yescharacteristic
413C2×C4D4211yescharacteristic
414C2×C4#54611yescharacteristic
415C2×C4C2×#13711yescharacteristic
416C2×C4#57611yescharacteristic
417C2×C4#57711yescharacteristic
418C8—22no—
419C8—41no—
420C8—41no—
421C8#53711yescharacteristic
422C8#54211yescharacteristic
423C8#54711yescharacteristic
424C8#56211yescharacteristic
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22#322611yescharacteristic
433C22D4×D811yescharacteristic
434C22#329411yescharacteristic
435C22C2×#51011yescharacteristic
436C22#341411yescharacteristic
437C22#341611yescharacteristic
438C22#341711yescharacteristic
439C4—22no—
440C4—41no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4—161no—
448C4—161no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4#267211yescharacteristic
453C4#324111yescharacteristic
454C4#326611yescharacteristic
455C4#333311yescharacteristic
456C2—81no—
457C2#3062911yescharacteristic
458C2#3380311yescharacteristic
459C2#3380811yescharacteristic
460C2#3383511yescharacteristic
461C2#3384411yescharacteristic
462C2#3387411yescharacteristic
463C2#3387911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2245C2semidirect
5. #13531C2semidirect
6. #13475C2semidirect
7. #13453C2semidirect
8. #13409C2semidirect
11. #13168C2semidirect
14. #12936C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22452
C2×#22372
#260662
#254152
#135312
#134752
#134532
#134092
#133832
#133822
#131682
#131212
#129942
#129362
#128122