Finite Group Databasebeta

← #504484

Subgroups — #504484

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#25034C211yesmaximal, characteristic, complemented
2#23025C211yesmaximal, characteristic, complemented
3#13537C211yesmaximal, characteristic, complemented
4#13474C211yesmaximal, characteristic, complemented
5#13380C211yesmaximal, characteristic, complemented
6#13304C211yesmaximal, characteristic
7#13147C211yesmaximal, characteristic, complemented
8#13120C211yesmaximal, characteristic, complemented
9#13109C211yesmaximal, characteristic, complemented
10#13012C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic, complemented
12#12991C211yesmaximal, characteristic, complemented
13#12866C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2422—21no—
23#2361—21no—
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#2071—21no—
36#2033—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1925—21no—
42#1874—21no—
43#1862—21no—
44#1862—21no—
45#1845C2211yescharacteristic
46#1808C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#123—41no—
63C2×#121—21no—
64C2×#121—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4—41no—
68C22×C4⋊C4—21no—
69C2×D16—22no—
70C2×D16—22no—
71C2×D16—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#466—22no—
77#466—22no—
78#466—22no—
79#466—41no—
80C4×D8—22no—
81C4×D8—22no—
82C4×D8—41no—
83C4×D8—41no—
84C2×#107—22no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—41no—
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#397—41no—
104#394—41no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#388—41no—
110#386—41no—
111#385—41no—
112#385—41no—
113#384—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C22⋊C4D411yescharacteristic
117#366—22no—
118C4⋊C16—41no—
119#357—22no—
120#357—22no—
121#357—22no—
122#357—41no—
123#357—41no—
124#357—41no—
125C22⋊C16—41no—
126C22⋊C16—41no—
127C4×C16—41no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—41no—
137C22×D4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8#4112yes—
153C2×D8#4112yes—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2411yescharacteristic
164#123—84no—
165#121—24no—
166#121—42no—
167#121—42no—
168#121—81no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—81no—
172C4×D4—81no—
173C4×D4—84no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—81no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
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⋊C4D811yescharacteristic
197C2×C22⋊C4D811yescharacteristic
198C2×C42—41no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201D16—44no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—42no—
225#107—81no—
226#107#4112yes—
227#107#4112yes—
228#102—24no—
229#102—42no—
230#102—42no—
231#102—42no—
232#102—42no—
233#102—81no—
234C22⋊C8—41no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95D811yescharacteristic
244#95D811yescharacteristic
245C24—41no—
246C24—41no—
247C24C2×D811yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—161no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—161no—
285C22×C4—161no—
286C22×C4—161no—
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C22×D411yescharacteristic
293D8—22no—
294D8—22no—
295D8—22no—
296D8—22no—
297D8—41no—
298D8—41no—
299D8—41no—
300D8—41no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8C2×#4111yescharacteristic
315C2×C8C2×#4111yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8C2×#4111yescharacteristic
318C2×C8H(5,𝔽2)11yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—161no—
337C22⋊C4—41no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—161no—
347C22⋊C4D1614yescomplemented
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—82no—
353C16—41no—
354C16—41no—
355C16—41no—
356C16—41no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—42no—
364C23—42no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23C2×D1612yes—
378C23C22×D811yescharacteristic
379D4—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—88no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—81no—
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—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
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—164no—
425C2×C4—164no—
426C2×C4—164no—
427C2×C4C2×D1612yes—
428C2×C4C2×D1612yes—
429C2×C4#52511yescharacteristic
430C2×C4#54611yescharacteristic
431C2×C4#54611yescharacteristic
432C2×C4#57511yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C8—41no—
436C8—41no—
437C8—41no—
438C8#52911yescharacteristic
439C8#54011yescharacteristic
440C8#55111yescharacteristic
441C8#55311yescharacteristic
442C22—22no—
443C22—22no—
444C22—22no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22#310011yescharacteristic
464C22#329411yescharacteristic
465C22#329411yescharacteristic
466C22C22×D1611yescharacteristic
467C22#341411yescharacteristic
468C22#341611yescharacteristic
469C22#341611yescharacteristic
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—82no—
476C4—82no—
477C4—161no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4#245711yescharacteristic
482C4#312511yescharacteristic
483C4#330711yescharacteristic
484C4#332611yescharacteristic
485C2—42no—
486C2—162no—
487C2—162no—
488C2#2665611yescharacteristic
489C2#3366011yescharacteristic
490C2#3368511yescharacteristic
491C2#3386111yescharacteristic
492C2#3386111yescharacteristic
493C2#3387411yescharacteristic
494C2#3389311yescharacteristic
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. #25034C2semidirect
2. #23025C2semidirect
3. #13537C2semidirect
4. #13474C2semidirect
5. #13380C2semidirect
7. #13147C2semidirect
8. #13120C2semidirect
9. #13109C2semidirect
10. #13012C2semidirect
11. #13012C2semidirect
12. #12991C2semidirect
13. #12866C2semidirect
14. #12811C2semidirect
15. C4×#357C2semidirect
347. C22⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
#250342
#230252
#135372
#134742
#133802
#133042
#131472
#131202
#131092
#130122
#129912
#128662
#128112
C4×#3572