Finite Group Databasebeta

← #501978

Subgroups — #501978

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2233C211yesmaximal, characteristic, complemented
2C2×#2234C211yesmaximal, characteristic, complemented
3#28227C211yesmaximal, characteristic
4#22656C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13190C211yesmaximal, characteristic, complemented
10#13177C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic, complemented
12#12920C211yesmaximal, characteristic, complemented
13#12908C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2537—21no—
23#2352—21no—
24#2234—24no—
25#2233—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42#2033—21no—
43#2033—21no—
44#1949—21no—
45#1854—21no—
46#1853—21no—
47#1846C2211yescharacteristic
48#1845C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1802—21no—
54C4×#107C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—41no—
67C2×C4×D4—21no—
68C2×C4×D4D411yescharacteristic
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#466—22no—
76#466—22no—
77#466—41no—
78#466—41no—
79C4×D8—22no—
80C4×D8—22no—
81C8×D4—22no—
82C8×D4—22no—
83#427—22no—
84#427—41no—
85C2×#107—22no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—41no—
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#106—21no—
97C2×#106—21no—
98C2×C4⋊C8D411yescharacteristic
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×C22⋊C8D411yescharacteristic
103C2×C22⋊C8D411yescharacteristic
104C2×C4×C8C2311yescharacteristic
105#394—41no—
106#394—41no—
107#388—41no—
108#387—41no—
109#386—41no—
110#384—21no—
111#382—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—41no—
117C4×C4⋊C4—21no—
118C4×C22⋊C4—41no—
119#367—22no—
120#367—22no—
121#367—22no—
122#367—41no—
123#366—22no—
124#366—41no—
125C4⋊C16—22no—
126#357—22no—
127#357—22no—
128#357—22no—
129#357D414yescomplemented
130C22⋊C16—22no—
131C22⋊C16—22no—
132C22⋊C16—22no—
133C22⋊C16—22no—
134#346—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×D4—21no—
141C22×D4—41no—
142C22×D4—41no—
143C22×D4D811yescharacteristic
144C23×C4—41no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8C2×D412yes—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170#121—42no—
171#121—42no—
172#121—81no—
173#121—81no—
174#121—84no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—42no—
178C4×D4—42no—
179#117—42no—
180#117—81no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4D811yescharacteristic
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⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4D811yescharacteristic
203C2×C22⋊C4D811yescharacteristic
204C2×C42—41no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
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—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—42no—
233#107—81no—
234#107C2×D412yes—
235#106—42no—
236#106—81no—
237C4⋊C8—22no—
238#102—42no—
239#102—42no—
240#102—42no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C4×C8#4112yes—
244C4×C8#4112yes—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24—41no—
253C24C2×D811yescharacteristic
254C24C2×D811yescharacteristic
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—161no—
269C2×D4—161no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—161no—
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C2×D811yescharacteristic
299C22×C4C22×D411yescharacteristic
300D8—22no—
301D8—22no—
302D8—41no—
303D8—41no—
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×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8#12312yes—
321C2×C8C22×D411yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8H(5,𝔽2)11yescharacteristic
326C2×C82−1+411yescharacteristic
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—161no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—161no—
355C42—82no—
356C42—161no—
357C42#12312yes—
358C42#13712yes—
359C16—22no—
360C16—22no—
361C16—41no—
362C16—41no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—42no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23#50912yes—
384C23C22×D811yescharacteristic
385D4—44no—
386D4—42no—
387D4—42no—
388D4—81no—
389D4—81no—
390D4—164no—
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—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—81no—
412C2×C4—81no—
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—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—164no—
432C2×C4—164no—
433C2×C4#48012yes—
434C2×C4#48012yes—
435C2×C4C2×#12311yescharacteristic
436C2×C4D4211yescharacteristic
437C2×C4#54711yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4#57611yescharacteristic
440C2×C4#57811yescharacteristic
441C8—22no—
442C8—41no—
443C8—41no—
444C8#53211yescharacteristic
445C8#54211yescharacteristic
446C8#54711yescharacteristic
447C8#55311yescharacteristic
448C22—22no—
449C22—22no—
450C22—22no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22C2×#48011yescharacteristic
470C22D4×D811yescharacteristic
471C22#333411yescharacteristic
472C22C2×#50911yescharacteristic
473C22#341411yescharacteristic
474C22#341611yescharacteristic
475C22#341611yescharacteristic
476C4—22no—
477C4—22no—
478C4—41no—
479C4—41no—
480C4—82no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4#242211yescharacteristic
488C4#309811yescharacteristic
489C4#327611yescharacteristic
490C4#335711yescharacteristic
491C2—42no—
492C2—81no—
493C2—162no—
494C2#2501011yescharacteristic
495C2#3364811yescharacteristic
496C2#3365311yescharacteristic
497C2#3383111yescharacteristic
498C2#3383311yescharacteristic
499C2#3391211yescharacteristic
500C2#3391511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28227C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ 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×#2234C2semidirect
4. #22656C2semidirect
5. #13494C2semidirect
6. #13492C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13190C2semidirect
10. #13177C2semidirect
11. #12921C2semidirect
12. #12920C2semidirect
13. #12908C2semidirect
14. #12813C2semidirect
15. #12637C2semidirect
129. #357D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22332
C2×#22342
#282272
#226562
#134942
#134922
#133812
#133802
#131902
#131772
#129212
#129202
#129082
#128132
#126372