Finite Group Databasebeta

← #504635

Subgroups — #504635

Order 512.

625 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2233C211yesmaximal, characteristic, complemented
2C2×#2228C211yesmaximal, characteristic, complemented
3#27790C211yesmaximal, characteristic, complemented
4#23025C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13453C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13304C211yesmaximal, characteristic
9#13235C211yesmaximal, characteristic, complemented
10#13131C211yesmaximal, characteristic, complemented
11#13130C211yesmaximal, characteristic, complemented
12#13120C211yesmaximal, characteristic, complemented
13#13012C211yesmaximal, characteristic, complemented
14#12919C211yesmaximal, characteristic, complemented
15C4×#367C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2490—21no—
23#2361—21no—
24#2233—24no—
25#2228—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#2034—21no—
42#2025—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1970C2211yescharacteristic
46#1942C2211yescharacteristic
47#1926—21no—
48#1925—21no—
49#1874—21no—
50#1862—21no—
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic
53C4×#107C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#125—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×C4×D4—21no—
68C2×C4×D4—41no—
69C22×C4⋊C4—21no—
70C2×D16—22no—
71C2×D16—22no—
72C2×D16—41no—
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77#499—22no—
78#499—41no—
79#466—22no—
80#466—22no—
81#466—22no—
82#466—41no—
83C4×D8—22no—
84C4×D8—41no—
85C2×#107—22no—
86C2×#107—21no—
87C2×#107—41no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#106—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C22⋊C8D411yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#397—41no—
105#394—41no—
106#390—21no—
107#390—21no—
108#390—21no—
109#390—41no—
110#389—21no—
111#388—41no—
112#388—41no—
113#386—41no—
114#385—41no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C22⋊C4D411yescharacteristic
119#367—22no—
120#367—22no—
121#367D414yescomplemented
122#366—22no—
123#366—41no—
124C4⋊C16—22no—
125#357—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357—22no—
130#357—22no—
131C22⋊C16—22no—
132C22⋊C16—22no—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—41no—
143C22×D4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8#4112yes—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2411yescharacteristic
169#125—42no—
170#125—81no—
171#121—24no—
172#121—42no—
173#121—42no—
174#121—81no—
175C4×D4—42no—
176C4×D4—81no—
177C4×D4—84no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—81no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4D811yescharacteristic
201C2×C22⋊C4D811yescharacteristic
202C2×C42—41no—
203C2×C42—41no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206D16—44no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16C2×D412yes—
218C2×C16#4112yes—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—42no—
229#107C2×D412yes—
230#107C2×D412yes—
231#107#4112yes—
232#106—42no—
233#102—24no—
234#102—42no—
235#102—42no—
236#102—42no—
237#102—81no—
238#102—81no—
239C22⋊C8—41no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95D811yescharacteristic
249#95D811yescharacteristic
250C24—41no—
251C24—41no—
252C24C2×D811yescharacteristic
253C2×D4—22no—
254C2×D4—44no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C2×D4—161no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—161no—
288C22×C4—161no—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C22×D411yescharacteristic
295D8—22no—
296D8—22no—
297D8—41no—
298D8—41no—
299D8—41no—
300D8—41no—
301D8—41no—
302D8—41no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8C22×D411yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8H(5,𝔽2)11yescharacteristic
321C2×C82−1+411yescharacteristic
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—44no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—161no—
346C22⋊C4—161no—
347C22⋊C4—161no—
348C22⋊C4—161no—
349C22⋊C4—161no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—82no—
355C42—161no—
356C16—22no—
357C16—22no—
358C16—41no—
359C16—41no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—42no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23#50812yes—
381C23C22×D811yescharacteristic
382D4—42no—
383D4—42no—
384D4—88no—
385D4—81no—
386D4—81no—
387D4—81no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—164no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—164no—
429C2×C4—164no—
430C2×C4#50912yes—
431C2×C4#52511yescharacteristic
432C2×C4D4211yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4#57511yescharacteristic
435C2×C4#57611yescharacteristic
436C2×C4#57811yescharacteristic
437C8—41no—
438C8—41no—
439C8—41no—
440C8#53211yescharacteristic
441C8#54211yescharacteristic
442C8#54811yescharacteristic
443C8#55111yescharacteristic
444C22—22no—
445C22—22no—
446C22—22no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22#310011yescharacteristic
466C22D4×D811yescharacteristic
467C22#333411yescharacteristic
468C22C2×#50811yescharacteristic
469C22C2×#50911yescharacteristic
470C22#341511yescharacteristic
471C22#341611yescharacteristic
472C4—22no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4#246111yescharacteristic
484C4#312411yescharacteristic
485C4#327711yescharacteristic
486C4#334711yescharacteristic
487C2—42no—
488C2—162no—
489C2—321no—
490C2#2678411yescharacteristic
491C2#3366511yescharacteristic
492C2#3367911yescharacteristic
493C2#3384011yescharacteristic
494C2#3384811yescharacteristic
495C2#3390611yescharacteristic
496C2#3391211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#13304C2
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×#2233C2semidirect
2. C2×#2228C2semidirect
3. #27790C2semidirect
4. #23025C2semidirect
5. #13476C2semidirect
6. #13453C2semidirect
7. #13381C2semidirect
9. #13235C2semidirect
10. #13131C2semidirect
11. #13130C2semidirect
12. #13120C2semidirect
13. #13012C2semidirect
14. #12919C2semidirect
15. C4×#367C2semidirect
121. #367D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22332
C2×#22282
#277902
#230252
#134762
#134532
#133812
#133042
#132352
#131312
#131302
#131202
#130122
#129192
C4×#3672