Finite Group Databasebeta

← #509905

Subgroups — #509905

Order 512.

576 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2235C211yesmaximal, characteristic
2C2×#2225C211yesmaximal, characteristic, complemented
3#26065C211yesmaximal, characteristic
4#25415C211yesmaximal, characteristic
5#13506C211yesmaximal, characteristic, complemented
6#13477C211yesmaximal, characteristic, complemented
7#13409C211yesmaximal, characteristic, complemented
8#13382C211yesmaximal, characteristic
9#13382C211yesmaximal, characteristic
10#13364C211yesmaximal, characteristic, complemented
11#13203C211yesmaximal, characteristic, complemented
12#13166C211yesmaximal, characteristic, complemented
13#13120C211yesmaximal, characteristic
14#12993C211yesmaximal, characteristic
15#12938C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2429—21no—
24#2235—24no—
25#2225—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
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×C4×C16C2211yescharacteristic
42#2083—21no—
43#2057—21no—
44#2043—21no—
45#2001—21no—
46#1969C2211yescharacteristic
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1942C2211yescharacteristic
50#1878—21no—
51#1852—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×C4⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—41no—
70C2×C4×Q8—21no—
71C22×C4⋊C4—21no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500—22no—
77#500—41no—
78#494—22no—
79#494—41no—
80#467—22no—
81#467—22no—
82#467—41no—
83#467—41no—
84C4×Q16—22no—
85C4×Q16—41no—
86C2×#107—22no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—41no—
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×C22⋊C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#398—41no—
105#398—41no—
106#396—41no—
107#395—41no—
108#391—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—41no—
113#388—41no—
114#385—41no—
115#384—21no—
116#380—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#367—22no—
120#367—41no—
121#367D414yes—
122C4⋊C16—22no—
123#358—22no—
124#358—22no—
125#358—22no—
126#358—22no—
127#358—22no—
128#358—41no—
129#358—41no—
130#358—41no—
131C22⋊C16—22no—
132C22⋊C16—41no—
133C4×C16—41no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—41no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
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—21no—
155C2×Q16—21no—
156C2×Q16—42no—
157C2×Q16—42no—
158C2×Q16—81no—
159C2×Q16#4112yes—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—81no—
173C4⋊Q8—81no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—81no—
177C22⋊Q8—81no—
178C22⋊Q8—84no—
179C4×Q8—42no—
180C4×Q8—81no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
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×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—81no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4D811yescharacteristic
205C2×C22⋊C4D811yescharacteristic
206C2×C42—41no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
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×C16—21no—
220C2×C16C2×D412yes—
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—21no—
231#107—42no—
232#107—42no—
233#107—81no—
234#107C2×D412yes—
235#107C2×D412yes—
236#103—42no—
237#103—42no—
238#103—42no—
239#103—81no—
240C22⋊C8—22no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95D811yescharacteristic
252#95D811yescharacteristic
253C24C2×D811yescharacteristic
254C2×Q8—22no—
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—
265C2×Q8—41no—
266C2×Q8—82no—
267C2×Q8—82no—
268C2×Q8—161no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—161no—
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C22×D411yescharacteristic
292Q16—22no—
293Q16—22no—
294Q16—41no—
295Q16—41no—
296Q16—41no—
297Q16—41no—
298Q16—41no—
299Q16—41no—
300Q16—84no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8#13712yes—
314C2×C8C22×D411yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8H(5,𝔽2)11yescharacteristic
317C2×C8H(5,𝔽2)11yescharacteristic
318C2×C82−1+411yescharacteristic
319C2×C82−1+411yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—161no—
344C22⋊C4—22no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4SD3214yes—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—82no—
354C16—22no—
355C16—41no—
356C16—41no—
357C16—41no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23C2×SD3212yes—
365C23C22×D811yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—81no—
369Q8—81no—
370Q8—81no—
371Q8—81no—
372Q8—81no—
373Q8—81no—
374Q8—164no—
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—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
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—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—164no—
416C2×C4C2×SD3212yes—
417C2×C4C2×SD3212yes—
418C2×C4#51012yes—
419C2×C4#53611yescharacteristic
420C2×C4D4211yescharacteristic
421C2×C4#54611yescharacteristic
422C2×C4C2×#13711yescharacteristic
423C2×C4#57611yescharacteristic
424C2×C4#57711yescharacteristic
425C8—22no—
426C8—41no—
427C8—41no—
428C8#53711yescharacteristic
429C8#53911yescharacteristic
430C8#54811yescharacteristic
431C8#56211yescharacteristic
432C22—22no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22#322611yescharacteristic
440C22D4×D811yescharacteristic
441C22#329411yescharacteristic
442C22C22×SD3211yescharacteristic
443C22C2×#51011yescharacteristic
444C22#341611yescharacteristic
445C22#341711yescharacteristic
446C4—22no—
447C4—22no—
448C4—22no—
449C4—41no—
450C4—41no—
451C4—82no—
452C4—82no—
453C4—82no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4#268111yescharacteristic
460C4#324111yescharacteristic
461C4#327811yescharacteristic
462C4#331411yescharacteristic
463C2—42no—
464C2#3111611yescharacteristic
465C2#3379511yescharacteristic
466C2#3380811yescharacteristic
467C2#3384211yescharacteristic
468C2D4×SD3211yescharacteristic
469C2#3386511yescharacteristic
470C2#3388311yescharacteristic
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
2. C2×#2225C2semidirect
5. #13506C2semidirect
6. #13477C2semidirect
7. #13409C2semidirect
10. #13364C2semidirect
11. #13203C2semidirect
12. #13166C2semidirect
15. #12938C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22352
C2×#22252
#260652
#254152
#135062
#134772
#134092
#133822
#133642
#132032
#131662
#131202
#129932
#129382