Finite Group Databasebeta

← #509539

Subgroups — #509539

Order 512.

517 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2255C211yesmaximal, characteristic, complemented
2C2×#2234C211yesmaximal, characteristic
3#27790C211yesmaximal, characteristic
4#26064C211yesmaximal, characteristic, complemented
5#13528C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13453C211yesmaximal, characteristic, complemented
9#13381C211yesmaximal, characteristic
10#13363C211yesmaximal, characteristic, complemented
11#13178C211yesmaximal, characteristic, complemented
12#13149C211yesmaximal, characteristic, complemented
13#13121C211yesmaximal, characteristic, complemented
14#13111C211yesmaximal, characteristic, complemented
15#12921C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#427C2211yescharacteristic
20#2490—21no—
21#2439—21no—
22#2255—24no—
23#2234—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×#346C2211yescharacteristic
38#2085—21no—
39#2058—21no—
40#2034—21no—
41#2025—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1926—21no—
47#1854—21no—
48#1846C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C22⋊C4C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64C2×#125—21no—
65C2×#121—21no—
66C2×#117—21no—
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71#500—22no—
72#500—41no—
73#499—22no—
74#499—41no—
75#466—22no—
76#466—41no—
77#427—22no—
78#427—41no—
79C2×#107—21no—
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C22⋊C8D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#397—41no—
97#396—41no—
98#394—41no—
99#393—41no—
100#391—41no—
101#390—21no—
102#390—21no—
103#390—41no—
104#389—21no—
105#389—21no—
106#388—41no—
107#387—41no—
108#382—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C22⋊C4D411yescharacteristic
113#367—22no—
114#367—41no—
115#367D414yes—
116#366—22no—
117#366—22no—
118#366—22no—
119C4⋊C16—22no—
120#357—22no—
121#357—22no—
122#357—22no—
123#357—41no—
124C22⋊C16—22no—
125C22⋊C16—22no—
126#346—22no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—41no—
135C22×D4—21no—
136C22×D4—41no—
137C23×C4C2×D411yescharacteristic
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—21no—
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2411yescharacteristic
154C4⋊Q8—42no—
155C4⋊Q8—81no—
156#125—42no—
157#125—81no—
158#121—42no—
159#121—81no—
160#117—42no—
161#117—81no—
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—21no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
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×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C16—21no—
189C2×C16—21no—
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×C16C2×D412yes—
199C2×C16#4112yes—
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#107—21no—
211#107—21no—
212#107—42no—
213#107C2×D412yes—
214#107C2×D412yes—
215#106—42no—
216#106—42no—
217#106—81no—
218#102—42no—
219#102—42no—
220#102—81no—
221C22⋊C8—22no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—21no—
225C4×C8—21no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232#95—41no—
233#95D811yescharacteristic
234#95D811yescharacteristic
235C24—41no—
236C24C2×D811yescharacteristic
237C2×Q8—82no—
238C2×Q8—82no—
239C2×Q8—161no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—161no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4C2×D811yescharacteristic
266C22×C4C2×D811yescharacteristic
267C22×C4C22×D411yescharacteristic
268D8—41no—
269D8—41no—
270D8—41no—
271D8—41no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8#13712yes—
285C2×C8C22×D411yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C2×C8H(5,𝔽2)11yescharacteristic
288C2×C82−1+411yescharacteristic
289C2×C82−1+411yescharacteristic
290C2×C82−1+411yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
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—41no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—161no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—161no—
316C22⋊C4—161no—
317C22⋊C4—161no—
318C22⋊C4—161no—
319C22⋊C4—161no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—82no—
325C42—161no—
326C16—22no—
327C16—22no—
328C16—41no—
329C16—41no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23C22×D811yescharacteristic
345Q8—164no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
350D4—164no—
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—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
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—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4#53611yescharacteristic
392C2×C4D4211yescharacteristic
393C2×C4#54711yescharacteristic
394C2×C4C2×#13711yescharacteristic
395C2×C4#57611yescharacteristic
396C2×C4#57811yescharacteristic
397C8—22no—
398C8—41no—
399C8—41no—
400C8#53611yescharacteristic
401C8#54211yescharacteristic
402C8#54811yescharacteristic
403C8#56311yescharacteristic
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—161no—
412C22—161no—
413C22—161no—
414C22—161no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22#322611yescharacteristic
419C22D4×D811yescharacteristic
420C22#333411yescharacteristic
421C22#341411yescharacteristic
422C22#341411yescharacteristic
423C22#341511yescharacteristic
424C22#341611yescharacteristic
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—82no—
431C4—82no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4#267411yescharacteristic
438C4#324311yescharacteristic
439C4#327011yescharacteristic
440C4#335711yescharacteristic
441C2—81no—
442C2—321no—
443C2#3077311yescharacteristic
444C2#3380611yescharacteristic
445C2#3380711yescharacteristic
446C2#3383411yescharacteristic
447C2#3384111yescharacteristic
448C2#3391411yescharacteristic
449C2#3391711yescharacteristic
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. C2×#2255C2semidirect
4. #26064C2semidirect
5. #13528C2semidirect
6. #13492C2semidirect
7. #13474C2semidirect
8. #13453C2semidirect
10. #13363C2semidirect
11. #13178C2semidirect
12. #13149C2semidirect
13. #13121C2semidirect
14. #13111C2semidirect
15. #12921C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22552
C2×#22342
#277902
#260642
#135282
#134922
#134742
#134532
#133812
#133632
#131782
#131492
#131212
#131112
#129212