Finite Group Databasebeta

← #504198

Subgroups — #504198

Order 512.

551 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26065C211yesmaximal, characteristic
2#23026C211yesmaximal, characteristic
3#13538C211yesmaximal, characteristic, complemented
4#13516C211yesmaximal, characteristic, complemented
5#13477C211yesmaximal, characteristic, complemented
6#13364C211yesmaximal, characteristic, complemented
7#13275C211yesmaximal, characteristic, complemented
8#13109C211yesmaximal, characteristic, complemented
9#13062C211yesmaximal, characteristic
10#12993C211yesmaximal, characteristic
11#12936C211yesmaximal, characteristic, complemented
12#12921C211yesmaximal, characteristic
13#12913C211yesmaximal, characteristic
14#12812C211yesmaximal, characteristic, complemented
15#12557C211yesmaximal, characteristic
16C22×M32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2362—21no—
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33#2083—21no—
34#2057—21no—
35#1969C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1927—21no—
40#1873—21no—
41#1865—21no—
42#1852—21no—
43#1845C2211yescharacteristic
44#1808C2211yescharacteristic
45C4×#107C2211yescharacteristic
46C8×C22⋊C4C2211yescharacteristic
47#1380C2211yescharacteristic
48#1380C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—41no—
64C2×C4×Q8—21no—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C2×M32—22no—
68C2×M32—41no—
69C2×M32—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#500—22no—
75#500—41no—
76#467—22no—
77#467Q814yescomplemented
78C4×Q16—22no—
79C4×Q16—41no—
80#427—22no—
81#427—41no—
82C2×#107—22no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—41no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#106—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×C22⋊C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#398—41no—
102#395—41no—
103#391—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#388—41no—
108#388—41no—
109#387—41no—
110#385—41no—
111#384—21no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—41no—
115C4×C22⋊C4D411yescharacteristic
116#367—41no—
117#366—22no—
118C4⋊C16—41no—
119C4⋊C16—41no—
120#358—22no—
121#358—41no—
122#358—41no—
123#358—41no—
124C22⋊C16—22no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#336D411yescharacteristic
131#336D411yescharacteristic
132C22×Q8—21no—
133C22×Q8—41no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16C2×Q812yes—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×Q812yes—
150C22×C8#4112yes—
151C22×C8C2411yescharacteristic
152C4⋊Q8—42no—
153C4⋊Q8—81no—
154C22⋊Q8—24no—
155C22⋊Q8—42no—
156C22⋊Q8—84no—
157C4×Q8—42no—
158C4×Q8—81no—
159#117—42no—
160#117—81no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—81no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4D811yescharacteristic
183C2×C22⋊C4D811yescharacteristic
184C2×C42—41no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189M32—44no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—42no—
214#107—42no—
215#107—81no—
216#107C2×Q812yes—
217#107#4112yes—
218#107#4112yes—
219#106—42no—
220#103—24no—
221#103—42no—
222#103—81no—
223C22⋊C8—22no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95—41no—
234#95—41no—
235#95D811yescharacteristic
236#95D811yescharacteristic
237C24C2×D811yescharacteristic
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8—161no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—161no—
262C22×C4C22⋊Q812yes—
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C2×D811yescharacteristic
268C22×C4#13712yes—
269C22×C4C22×D411yescharacteristic
270Q16—22no—
271Q16—22no—
272Q16—41no—
273Q16—41no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8C22⋊Q812yes—
287C2×C8C22×Q811yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8C2×#4111yescharacteristic
291C2×C8H(5,𝔽2)11yescharacteristic
292C2×C8H(5,𝔽2)11yescharacteristic
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—44no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—161no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—44no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—82no—
326C42—82no—
327C42—82no—
328C16—41no—
329C16—41no—
330C16—41no—
331C16—41no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23#47412yes—
339C23C22×D811yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—81no—
343Q8—81no—
344Q8—164no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—164no—
386C2×C4#47412yes—
387C2×C4#50912yes—
388C2×C4#51012yes—
389C2×C4C2×C22⋊Q811yescharacteristic
390C2×C4#54611yescharacteristic
391C2×C4#54611yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C2×C4#57711yescharacteristic
394C2×C4#57711yescharacteristic
395C8—22no—
396C8—41no—
397C8—41no—
398C8#52911yescharacteristic
399C8#53811yescharacteristic
400C8#55411yescharacteristic
401C8#55411yescharacteristic
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22C2×#47411yescharacteristic
410C22#329411yescharacteristic
411C22#329411yescharacteristic
412C22C2×#50911yescharacteristic
413C22C2×#51011yescharacteristic
414C22#341611yescharacteristic
415C22#341711yescharacteristic
416C4—22no—
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—82no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4—161no—
429C4#245011yescharacteristic
430C4#308411yescharacteristic
431C4#331611yescharacteristic
432C4#332011yescharacteristic
433C2—42no—
434C2#2640811yescharacteristic
435C2#3362911yescharacteristic
436C2#3363211yescharacteristic
437C2#3387711yescharacteristic
438C2#3388311yescharacteristic
439C2#3389511yescharacteristic
440C2#3389711yescharacteristic
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
3. #13538C2semidirect
4. #13516C2semidirect
5. #13477C2semidirect
6. #13364C2semidirect
7. #13275C2semidirect
8. #13109C2semidirect
11. #12936C2semidirect
14. #12812C2semidirect
77. #467Q8semidirect

Maximal subgroups

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

subgroupindex
#260652
#230262
#135382
#135162
#134772
#133642
#132752
#131092
#130622
#129932
#129362
#129212
#129132
#128122
#125572