Finite Group Databasebeta

← #498505

Subgroups — #498505

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2218C211yesmaximal, characteristic
2#23027C211yesmaximal, characteristic, complemented
3#23025C211yesmaximal, characteristic, complemented
4#13149C211yesmaximal, characteristic, complemented
5#13147C211yesmaximal, characteristic, complemented
6#13111C211yesmaximal, characteristic, complemented
7#13111C211yesmaximal, characteristic, complemented
8#13012C211yesmaximal, characteristic, complemented
9#13012C211yesmaximal, characteristic, complemented
10#12992C211yesmaximal, characteristic, complemented
11#12991C211yesmaximal, characteristic, complemented
12#12637C211yesmaximal, characteristic, complemented
13C4×#357C211yesmaximal, characteristic, complemented
14#12612C211yesmaximal, characteristic, complemented
15#12557C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C22×#107C2211yescharacteristic
24#2363—21no—
25#2361—21no—
26#2218C418yescomplemented
27C2×#367C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1926—21no—
42#1925—21no—
43#1875—21no—
44#1874—21no—
45#1864—21no—
46#1862—21no—
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C8×C22⋊C4C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×D8C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#121—21no—
64C2×#121—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—41no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C2×D16—41no—
71C2×D16—41no—
72C2×D16C2×C414yescomplemented
73C2×M32—41no—
74C2×M32—41no—
75C2×M32C2×C414yescomplemented
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80#466—22no—
81#466—22no—
82#466C2×C414yescomplemented
83#466C2×C414yescomplemented
84C4×D8—22no—
85C4×D8—22no—
86C4×D8—41no—
87C4×D8—41no—
88#427—22no—
89#427—41no—
90C2×#107—22no—
91C2×#107—21no—
92C2×#107—41no—
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#106—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×C22⋊C8D411yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#390—21no—
110#390—21no—
111#390—41no—
112#389—21no—
113#389—21no—
114#388—41no—
115#388—41no—
116#386—41no—
117#386—41no—
118#385—41no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C22⋊C4—41no—
122C4×C22⋊C4D411yescharacteristic
123#367C2×C414yescomplemented
124#357—22no—
125#357—22no—
126#357—41no—
127#357—41no—
128#357C2×C414yescomplemented
129#357C2×C414yescomplemented
130C22⋊C16—22no—
131#346—41no—
132C4×C16—41no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#336D411yescharacteristic
138#336D411yescharacteristic
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—41no—
142C22×D4—41no—
143C23×C4—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×D8C22×C412yes—
158C2×D8C22×C412yes—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C22×C412yes—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8#4112yes—
168C22×C8C2411yescharacteristic
169#121—24no—
170#121—24no—
171#121—42no—
172#121—42no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—81no—
176C4×D4—81no—
177C4×D4—84no—
178#117—42no—
179#117—81no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—81no—
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×C42C2×D411yescharacteristic
207D16—44no—
208M32—44no—
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×C16—21no—
218C2×C16—21no—
219C2×C16C22×C412yes—
220C2×C16C22×C412yes—
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—42no—
230#107C22×C412yes—
231#107C22×C412yes—
232#107#4112yes—
233#107#4112yes—
234#106—42no—
235#102—24no—
236#102—24no—
237#102—42no—
238#102—42no—
239#102—81no—
240#102—81no—
241C22⋊C8—22no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246#95—41no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95D811yescharacteristic
251#95D811yescharacteristic
252C24—41no—
253C24—41no—
254C24C2×D811yescharacteristic
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—161no—
272C2×D4—161no—
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—82no—
288C22×C4—82no—
289C22×C4—161no—
290C22×C4—161no—
291C22×C4—161no—
292C22×C4C4×D412yes—
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4#13512yes—
299C22×C4C22×D411yescharacteristic
300D8—22no—
301D8—22no—
302D8—22no—
303D8—22no—
304D8—41no—
305D8—41no—
306D8—41no—
307D8—41no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8C4×D412yes—
321C2×C8C23×C411yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8H(5,𝔽2)11yescharacteristic
326C2×C8H(5,𝔽2)11yescharacteristic
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
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—82no—
340C4⋊C4—82no—
341C4⋊C4—161no—
342C22⋊C4—22no—
343C22⋊C4—41no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—161no—
352C22⋊C4—161no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—82no—
358C42—82no—
359C42—161no—
360C16—22no—
361C16—22no—
362C16—41no—
363C16—41no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—42no—
371C23—42no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23C4×D812yes—
385C23C22×D811yescharacteristic
386D4—42no—
387D4—42no—
388D4—42no—
389D4—42no—
390D4—88no—
391D4—81no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—164no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
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—164no—
434C2×C4C4×D812yes—
435C2×C4#50912yes—
436C2×C4C2×C4×D411yescharacteristic
437C2×C4#54611yescharacteristic
438C2×C4#54611yescharacteristic
439C2×C4C2×#13511yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57711yescharacteristic
442C8—22no—
443C8—41no—
444C8—41no—
445C8C4×#4111yescharacteristic
446C8#51811yescharacteristic
447C8#55311yescharacteristic
448C8#55311yescharacteristic
449C22—22no—
450C22—22no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22—161no—
470C22C2×C4×D811yescharacteristic
471C22#329411yescharacteristic
472C22#329411yescharacteristic
473C22C2×#50911yescharacteristic
474C22#341411yescharacteristic
475C22#341511yescharacteristic
476C22#341611yescharacteristic
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—41no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—161no—
487C4—161no—
488C4#236411yescharacteristic
489C4#295211yescharacteristic
490C4#330111yescharacteristic
491C4#330111yescharacteristic
492C2—42no—
493C2—162no—
494C2—162no—
495C2#2312711yescharacteristic
496C2C4×#50911yescharacteristic
497C2#3345411yescharacteristic
498C2#3387311yescharacteristic
499C2#3387611yescharacteristic
500C2#3389211yescharacteristic
501C2#3389311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2218C2
LCSG ▷ C2×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
2. #23027C2semidirect
3. #23025C2semidirect
4. #13149C2semidirect
5. #13147C2semidirect
6. #13111C2semidirect
7. #13111C2semidirect
8. #13012C2semidirect
9. #13012C2semidirect
10. #12992C2semidirect
11. #12991C2semidirect
12. #12637C2semidirect
13. C4×#357C2semidirect
14. #12612C2semidirect
15. #12557C2semidirect
26. #2218C4semidirect
72. C2×D16C2×C4semidirect
75. C2×M32C2×C4semidirect
82. #466C2×C4semidirect
83. #466C2×C4semidirect
123. #367C2×C4semidirect
128. #357C2×C4semidirect
129. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22182
#230272
#230252
#131492
#131472
#131112
#130122
#129922
#129912
#126372
C4×#3572
#126122
#125572