Finite Group Databasebeta

← #501975

Subgroups — #501975

Order 512.

603 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×#2234C211yesmaximal, characteristic, complemented
2C2×#2234C211yesmaximal, characteristic, complemented
3#28225C211yesmaximal, characteristic
4#22658C211yesmaximal, characteristic, complemented
5#13494C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13190C211yesmaximal, characteristic, complemented
10#13178C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic, complemented
12#12921C211yesmaximal, characteristic, complemented
13#12908C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C2×#466C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2535—21no—
23#2354—21no—
24#2234—24no—
25#2234—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
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#2034—21no—
43#2034—21no—
44#1948—21no—
45#1854—21no—
46#1854—21no—
47#1846C2211yescharacteristic
48#1845C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1804—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×C4×D4—21no—
67C2×C4×D4D411yescharacteristic
68C2×#117—21no—
69C2×#117—21no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#466—22no—
75#466—22no—
76#466—41no—
77#466—41no—
78C4×D8—22no—
79C4×D8—22no—
80C8×D4—22no—
81C8×D4—22no—
82#427—22no—
83#427—22no—
84#427—41no—
85#427—41no—
86C2×#107—21no—
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×C4⋊C8D411yescharacteristic
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C22⋊C8D411yescharacteristic
102C2×C22⋊C8D411yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#397—41no—
105#392—41no—
106#390—41no—
107#388—41no—
108#388—41no—
109#386—41no—
110#386—41no—
111#384—21no—
112#382—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—41no—
119#367—22no—
120#367—22no—
121#367—41no—
122#366—22no—
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×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8C2×D412yes—
153C22×C8—21no—
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×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2411yescharacteristic
169#121—42no—
170#121—42no—
171#121—81no—
172#121—81no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—42no—
176C4×D4—42no—
177#117—42no—
178#117—42no—
179#117—81no—
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—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4D811yescharacteristic
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×C42—41no—
205C2×C42C2×D411yescharacteristic
206C2×C16—21no—
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×C16C2×D412yes—
217C2×C16#4112yes—
218#107—21no—
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—81no—
232#107C2×D412yes—
233#106—42no—
234#106—42no—
235#106—81no—
236C4⋊C8—22no—
237#102—42no—
238#102—42no—
239#102—42no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C4×C8#4112yes—
243C4×C8#4112yes—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—41no—
252C24C2×D811yescharacteristic
253C24C2×D811yescharacteristic
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—161no—
268C2×D4—161no—
269C22×C4—21no—
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—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—161no—
290C22×C4—161no—
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C22×D411yescharacteristic
296D8—22no—
297D8—22no—
298D8—41no—
299D8—41no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8#12312yes—
317C2×C8C22×D411yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8H(5,𝔽2)11yescharacteristic
322C2×C82−1+411yescharacteristic
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—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—82no—
339C4⋊C4—161no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—161no—
348C22⋊C4—161no—
349C22⋊C4—161no—
350C22⋊C4—161no—
351C42—82no—
352C42—82no—
353C42—82no—
354C42#12312yes—
355C42#13712yes—
356C16—22no—
357C16—22no—
358C16—41no—
359C16—41no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—42no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23C22×D811yescharacteristic
382D4—44no—
383D4—42no—
384D4—42no—
385D4—81no—
386D4—81no—
387D4—164no—
388D4—164no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
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—164no—
429C2×C4#48012yes—
430C2×C4#48012yes—
431C2×C4C2×#12311yescharacteristic
432C2×C4D4211yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57611yescharacteristic
436C2×C4#57811yescharacteristic
437C8—22no—
438C8—41no—
439C8—41no—
440C8#53211yescharacteristic
441C8#54211yescharacteristic
442C8#54711yescharacteristic
443C8#55311yescharacteristic
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22C2×#48011yescharacteristic
466C22D4×D811yescharacteristic
467C22#333411yescharacteristic
468C22#341411yescharacteristic
469C22#341411yescharacteristic
470C22#341511yescharacteristic
471C22#341611yescharacteristic
472C4—22no—
473C4—22no—
474C4—41no—
475C4—41no—
476C4—82no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4#242211yescharacteristic
484C4#309811yescharacteristic
485C4#327611yescharacteristic
486C4#335711yescharacteristic
487C2—81no—
488C2—81no—
489C2—162no—
490C2#2501011yescharacteristic
491C2#3364711yescharacteristic
492C2#3365011yescharacteristic
493C2#3383311yescharacteristic
494C2#3383411yescharacteristic
495C2#3391411yescharacteristic
496C2#3391711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28225C2
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×#2234C2semidirect
2. C2×#2234C2semidirect
4. #22658C2semidirect
5. #13494C2semidirect
6. #13492C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13190C2semidirect
10. #13178C2semidirect
11. #12921C2semidirect
12. #12921C2semidirect
13. #12908C2semidirect
14. #12811C2semidirect
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×#22342
#282252
#226582
#134942
#134922
#133812
#133802
#131902
#131782
#129212
#129082
#128112
#126372