Finite Group Databasebeta

← #507932

Subgroups — #507932

Order 512.

523 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#21282C211yesmaximal, characteristic, complemented
2#21282C211yesmaximal, characteristic, complemented
3#13516C211yesmaximal, characteristic, complemented
4#13476C211yesmaximal, characteristic, complemented
5#13474C211yesmaximal, characteristic, complemented
6#13448C211yesmaximal, characteristic
7#13131C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13111C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic
12#13012C211yesmaximal, characteristic
13#12937C211yesmaximal, characteristic, complemented
14#12935C211yesmaximal, characteristic, complemented
15#12880C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19#2319—21no—
20#2319—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32#1970C2211yescharacteristic
33#1970C2211yescharacteristic
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1936—21no—
41#1936—21no—
42#1936—21no—
43#1936—21no—
44C4×#107C2211yescharacteristic
45C4×#107C2211yescharacteristic
46#1762—21no—
47#1762—21no—
48#1726C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1374C2211yescharacteristic
56#1374C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×D16—22no—
64C2×D16—22no—
65C2×D16—41no—
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C4×D8—22no—
71C4×D8—22no—
72C4×D8—22no—
73C4×D8—22no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#102—21no—
81C2×#102—21no—
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×#97D411yescharacteristic
87C2×#97D411yescharacteristic
88C2×#97D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#388—41no—
101#388—41no—
102#388—41no—
103#388—41no—
104#381—41no—
105#381—41no—
106#380—41no—
107#380—41no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110#376D411yescharacteristic
111#367—22no—
112#367—41no—
113#366—22no—
114C4⋊C16—41no—
115C4⋊C16—41no—
116#357—22no—
117#357—22no—
118#357—22no—
119#357—22no—
120#357—41no—
121#357—41no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×D4—21no—
129C22×D4—21no—
130C23×C4—41no—
131C23×C4—41no—
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8#4112yes—
143C2×D8#4112yes—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—42no—
151C4×D4—42no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C42—41no—
173C2×C42—41no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177D16—44no—
178C2×C16—21no—
179C2×C16—21no—
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107#4112yes—
201#107#4112yes—
202#102—24no—
203#102—24no—
204#102—42no—
205#102—42no—
206#102—42no—
207#102—42no—
208#97—41no—
209#97—41no—
210#97—41no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217#95—41no—
218#95—41no—
219#95—41no—
220#95—41no—
221#95—41no—
222#95—41no—
223#95D811yescharacteristic
224#95D811yescharacteristic
225#95D811yescharacteristic
226#95D811yescharacteristic
227C24—41no—
228C24—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—161no—
250C22×C4—161no—
251C22×C4—161no—
252C22×C4—161no—
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C2×D811yescharacteristic
259C22×C4C22×D411yescharacteristic
260D8—22no—
261D8—22no—
262D8—22no—
263D8—22no—
264D8—41no—
265D8—41no—
266D8—41no—
267D8—41no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8C2×#4111yescharacteristic
278C2×C8C2×#4111yescharacteristic
279C2×C8C2×#4111yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C8H(5,𝔽2)11yescharacteristic
282C2×C82−1+411yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—161no—
296C4⋊C4—161no—
297C4⋊C4—161no—
298C4⋊C4—161no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—161no—
302C22⋊C4—161no—
303C22⋊C4—161no—
304C22⋊C4—161no—
305C22⋊C4—161no—
306C22⋊C4—161no—
307C22⋊C4—161no—
308C22⋊C4—161no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—161no—
316C42—161no—
317C16—41no—
318C16—41no—
319C16—41no—
320C16—41no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23C22×D811yescharacteristic
336D4—42no—
337D4—42no—
338D4—42no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—81no—
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—164no—
381C2×C4—164no—
382C2×C4#52511yescharacteristic
383C2×C4#52511yescharacteristic
384C2×C4#53711yescharacteristic
385C2×C4#57511yescharacteristic
386C2×C4#57511yescharacteristic
387C2×C4#57511yescharacteristic
388C8—41no—
389C8—41no—
390C8—41no—
391C8#55111yescharacteristic
392C8#55211yescharacteristic
393C8#55211yescharacteristic
394C8#55511yescharacteristic
395C22—82no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—161no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22#310011yescharacteristic
410C22#310011yescharacteristic
411C22#324511yescharacteristic
412C22#341511yescharacteristic
413C22#341511yescharacteristic
414C22#341611yescharacteristic
415C22#341611yescharacteristic
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—82no—
423C4—82no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4#260911yescharacteristic
429C4#311111yescharacteristic
430C4#311111yescharacteristic
431C4#325611yescharacteristic
432C2—162no—
433C2—162no—
434C2#2953211yescharacteristic
435C2#3367811yescharacteristic
436C2#3367911yescharacteristic
437C2#3368211yescharacteristic
438C2#3368511yescharacteristic
439C2#3382311yescharacteristic
440C2#3382311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21282C2semidirect
2. #21282C2semidirect
3. #13516C2semidirect
4. #13476C2semidirect
5. #13474C2semidirect
7. #13131C2semidirect
8. #13131C2semidirect
9. #13111C2semidirect
10. #13109C2semidirect
13. #12937C2semidirect
14. #12935C2semidirect
15. #12880C2semidirect

Maximal subgroups

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

subgroupindex
#212822
#135162
#134762
#134742
#134482
#131312
#131112
#131092
#130122
#129372
#129352
#128802