Finite Group Databasebeta

← #509029

Subgroups — #509029

Order 512.

519 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33208C211yesmaximal, characteristic, complemented
2#33129C211yesmaximal, characteristic
3#13539C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic, complemented
6#13493C211yesmaximal, characteristic, complemented
7#13453C211yesmaximal, characteristic, complemented
8#13381C211yesmaximal, characteristic
9#13380C211yesmaximal, characteristic
10#13365C211yesmaximal, characteristic, complemented
11#13166C211yesmaximal, characteristic, complemented
12#13165C211yesmaximal, characteristic, complemented
13#13147C211yesmaximal, characteristic, complemented
14#13147C211yesmaximal, characteristic, complemented
15C4×C22⋊C16C211yesmaximal, characteristic, complemented
16C2×#499C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2854—21no—
20#2851—21no—
21C2×#367C2211yescharacteristic
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×#357C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30C2×C22⋊C16C2211yescharacteristic
31C2×C4×C16C2211yescharacteristic
32C2×C4×C16C2211yescharacteristic
33#1991—21no—
34#1970C2211yescharacteristic
35#1970C2211yescharacteristic
36#1942C2211yescharacteristic
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1942C2211yescharacteristic
40#1941—21no—
41#1934—21no—
42#1934—21no—
43#1893—21no—
44#1846C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1807—21no—
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#125—21no—
63C2×#123—41no—
64C2×#121—21no—
65C22×C42D411yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70#499—41no—
71#499—41no—
72#466—41no—
73#466—41no—
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#106—21no—
82C2×#106—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×C4×C8—41no—
87C2×C4×C8—41no—
88C2×C4×C8D411yescharacteristic
89C2×C4×C8D411yescharacteristic
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#390—21no—
93#390—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#388—41no—
99#388—41no—
100#387—41no—
101#387—41no—
102#386—41no—
103#385—41no—
104#385—41no—
105#382—21no—
106#380—21no—
107#380—21no—
108#380—21no—
109#367—22no—
110#367—41no—
111#366—22no—
112#366—41no—
113#357—22no—
114#357—22no—
115#357—41no—
116#357—41no—
117C22⋊C16—22no—
118C22⋊C16—22no—
119C4×C16—41no—
120C4×C16—41no—
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128C22×D4—21no—
129C22×D4—41no—
130C23×C4D811yescharacteristic
131C23×C4D811yescharacteristic
132C23×C4C2×D411yescharacteristic
133C2×D8—21no—
134C2×D8—21no—
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—21no—
138C2×D8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—41no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2411yescharacteristic
152#125—42no—
153#125—42no—
154#123—84no—
155#121—42no—
156#121—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42—41no—
180C2×C42D811yescharacteristic
181C2×C42D811yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16#4112yes—
195C2×C16#4112yes—
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—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#106—42no—
211#106—42no—
212#102—42no—
213#102—42no—
214#102—42no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—41no—
220C4×C8—41no—
221#95—41no—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230C24—41no—
231C24C2×D811yescharacteristic
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—82no—
239C2×D4—82no—
240C2×D4—82no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
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—41no—
260C22×C4C2×D811yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C22×D411yescharacteristic
266D8—41no—
267D8—41no—
268D8—41no—
269D8—41no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—44no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8#13512yes—
283C2×C8#13712yes—
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8H(5,𝔽2)11yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C82−1+411yescharacteristic
289C2×C82−1+411yescharacteristic
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—41no—
323C42—41no—
324C16—22no—
325C16—22no—
326C16—41no—
327C16—41no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23#50812yes—
342C23C22×D811yescharacteristic
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347D4—164no—
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—44no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
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—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4#50812yes—
389C2×C4#50912yes—
390C2×C4#53211yescharacteristic
391C2×C4#53411yescharacteristic
392C2×C4#53611yescharacteristic
393C2×C4C2×#13511yescharacteristic
394C2×C4C2×#13711yescharacteristic
395C2×C4#57511yescharacteristic
396C8—22no—
397C8—22no—
398C8—41no—
399C8#53911yescharacteristic
400C8#53911yescharacteristic
401C8#54211yescharacteristic
402C8#54211yescharacteristic
403C22—22no—
404C22—22no—
405C22—22no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22#316911yescharacteristic
418C22#318611yescharacteristic
419C22#322611yescharacteristic
420C22C2×#50811yescharacteristic
421C22C2×#50811yescharacteristic
422C22C2×#50911yescharacteristic
423C22#341411yescharacteristic
424C4—22no—
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4#266311yescharacteristic
437C4#317211yescharacteristic
438C4#319011yescharacteristic
439C4#323311yescharacteristic
440C2—42no—
441C2—321no—
442C2#3035611yescharacteristic
443C2#3373511yescharacteristic
444C2#3373711yescharacteristic
445C2#3375411yescharacteristic
446C2#3375711yescharacteristic
447C2#3379411yescharacteristic
448C2#3379811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33208C2semidirect
3. #13539C2semidirect
4. #13528C2semidirect
5. #13494C2semidirect
6. #13493C2semidirect
7. #13453C2semidirect
10. #13365C2semidirect
11. #13166C2semidirect
12. #13165C2semidirect
13. #13147C2semidirect
14. #13147C2semidirect
15. C4×C22⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
#332082
#331292
#135392
#135282
#134942
#134932
#134532
#133812
#133802
#133652
#131662
#131652
#131472
C4×C22⋊C162