Finite Group Databasebeta

← #506176

Subgroups — #506176

Order 512.

512 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#27839C211yesmaximal, characteristic, complemented
2#27791C211yesmaximal, characteristic
3#13526C211yesmaximal, characteristic, complemented
4#13493C211yesmaximal, characteristic, complemented
5#13452C211yesmaximal, characteristic, complemented
6#13381C211yesmaximal, characteristic
7#13282C211yesmaximal, characteristic, complemented
8#13149C211yesmaximal, characteristic, complemented
9#13147C211yesmaximal, characteristic, complemented
10#13131C211yesmaximal, characteristic, complemented
11#13112C211yesmaximal, characteristic, complemented
12#13061C211yesmaximal, characteristic
13#12836C211yesmaximal, characteristic, complemented
14#12674C211yesmaximal, characteristic, complemented
15#12612C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#466C2211yescharacteristic
19#2496—21no—
20#2491—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×#346C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32#2099—21no—
33#2075—21no—
34#2033—21no—
35#2023—21no—
36#1970C2211yescharacteristic
37#1970C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1925—21no—
41#1918—21no—
42#1845C2211yescharacteristic
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4⋊Q8—21no—
62C2×#126—41no—
63C2×#124—41no—
64C2×#121—21no—
65C2×M32—22no—
66C2×M32—22no—
67C2×M32—41no—
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72#500—22no—
73#500—41no—
74#466—22no—
75#466—41no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C22⋊C8D411yescharacteristic
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#400—41no—
92#397—41no—
93#394—41no—
94#393—41no—
95#390—21no—
96#390—21no—
97#389—21no—
98#389—21no—
99#388—41no—
100#387—41no—
101#384—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4D411yescharacteristic
108#367—22no—
109#367—41no—
110#366—22no—
111#366—22no—
112#357—22no—
113#357—22no—
114#357—22no—
115#357—41no—
116C22⋊C16—22no—
117#346—22no—
118C4×C16—41no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#336D411yescharacteristic
127#336D411yescharacteristic
128C22×Q8—41no—
129C22×D4—21no—
130C22×D4—41no—
131C23×C4C2×D411yescharacteristic
132C2×D8—21no—
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148C4⋊Q8—42no—
149C4⋊Q8—81no—
150#126—84no—
151#124—84no—
152#121—42no—
153#121—81no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊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×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4D811yescharacteristic
175C2×C22⋊C4D811yescharacteristic
176C2×C42—41no—
177C2×C42—41no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180M32—44no—
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—
190C2×C16#4112yes—
191C2×C16#4112yes—
192C2×C16#4112yes—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—42no—
207#107—42no—
208#107#4112yes—
209#102—42no—
210#102—42no—
211#102—81no—
212C22⋊C8—22no—
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#95—41no—
224#95D811yescharacteristic
225#95D811yescharacteristic
226C24—41no—
227C24C2×D811yescharacteristic
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—161no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4—161no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C22×D411yescharacteristic
259D8—41no—
260D8—41no—
261D8—41no—
262D8—41no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8#13512yes—
276C2×C8C2×#4111yescharacteristic
277C2×C8C2×#4111yescharacteristic
278C2×C8C2×#4111yescharacteristic
279C2×C8C2×#4111yescharacteristic
280C2×C8H(5,𝔽2)11yescharacteristic
281C2×C82−1+411yescharacteristic
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
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—41no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—161no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—82no—
316C42—82no—
317C16—22no—
318C16—22no—
319C16—22no—
320C16—41no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23C22×D811yescharacteristic
336Q8—164no—
337D4—81no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—164no—
342C2×C4—21no—
343C2×C4—21no—
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—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
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#50912yes—
381C2×C4#50912yes—
382C2×C4#53211yescharacteristic
383C2×C4#54611yescharacteristic
384C2×C4#54711yescharacteristic
385C2×C4C2×#13511yescharacteristic
386C2×C4#57711yescharacteristic
387C2×C4#57811yescharacteristic
388C8—22no—
389C8—41no—
390C8—41no—
391C8#52911yescharacteristic
392C8#54211yescharacteristic
393C8#55111yescharacteristic
394C8#55611yescharacteristic
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—161no—
403C22—161no—
404C22—161no—
405C22—161no—
406C22—161no—
407C22—161no—
408C22—161no—
409C22#316911yescharacteristic
410C22#329411yescharacteristic
411C22#333411yescharacteristic
412C22C2×#50911yescharacteristic
413C22C2×#50911yescharacteristic
414C22#341611yescharacteristic
415C22#341711yescharacteristic
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—82no—
422C4—161no—
423C4—161no—
424C4—161no—
425C4—161no—
426C4—161no—
427C4—161no—
428C4#252111yescharacteristic
429C4#317811yescharacteristic
430C4#330011yescharacteristic
431C4#334011yescharacteristic
432C2—81no—
433C2—321no—
434C2#2806511yescharacteristic
435C2#3374511yescharacteristic
436C2#3374811yescharacteristic
437C2#3387511yescharacteristic
438C2#3387711yescharacteristic
439C2#3392011yescharacteristic
440C2#3392111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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. #27839C2semidirect
3. #13526C2semidirect
4. #13493C2semidirect
5. #13452C2semidirect
7. #13282C2semidirect
8. #13149C2semidirect
9. #13147C2semidirect
10. #13131C2semidirect
11. #13112C2semidirect
13. #12836C2semidirect
14. #12674C2semidirect
15. #12612C2semidirect

Maximal subgroups

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

subgroupindex
#278392
#277912
#135262
#134932
#134522
#133812
#132822
#131492
#131472
#131312
#131122
#130612
#128362
#126742
#126122