Finite Group Databasebeta

← #498299

Subgroups — #498299

Order 512.

625 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2171C211yesmaximal, characteristic, complemented
2C2×#2170C211yesmaximal, characteristic, complemented
3#33438C211yesmaximal, characteristic, complemented
4#33398C211yesmaximal, characteristic, complemented
5#21517C211yesmaximal, characteristic
6#13192C211yesmaximal, characteristic, complemented
7#13163C211yesmaximal, characteristic, complemented
8#13125C211yesmaximal, characteristic, complemented
9#13124C211yesmaximal, characteristic, complemented
10#12980C211yesmaximal, characteristic, complemented
11#12980C211yesmaximal, characteristic, complemented
12#12917C211yesmaximal, characteristic, complemented
13#12917C211yesmaximal, characteristic, complemented
14#12663C211yesmaximal, characteristic, complemented
15#12604C211yesmaximal, characteristic, complemented
16C2×#504C2211yescharacteristic
17C2×#504C2211yescharacteristic
18C2×#504C2211yescharacteristic
19C2×#504C2211yescharacteristic
20#3035C411yescharacteristic
21C2×#443C2211yescharacteristic
22C2×#443C2211yescharacteristic
23#2931C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#2171—24no—
26#2170—24no—
27#2169—21no—
28#2168—21no—
29#2168—21no—
30D4×C16—21no—
31C2×#364C2211yescharacteristic
32C2×#364C2211yescharacteristic
33#2151—21no—
34#2151—21no—
35#2151C2211yescharacteristic
36#2151C2211yescharacteristic
37#2150—21no—
38#2150C2211yescharacteristic
39C2×C4⋊C16—21no—
40C2×C4⋊C16C2211yescharacteristic
41C2×#350C2211yescharacteristic
42C2×#350C2211yescharacteristic
43C2×#350C2211yescharacteristic
44C2×#350C2211yescharacteristic
45C2×#350C2211yescharacteristic
46C2×#350C2211yescharacteristic
47#2113C2211yescharacteristic
48#2113C2211yescharacteristic
49C4×M32C2211yescharacteristic
50C2×#346C2211yescharacteristic
51C2×#346C2211yescharacteristic
52C2×C4×C16C2211yescharacteristic
53#1980C411yescharacteristic
54#1950C411yescharacteristic
55#1929C411yescharacteristic
56#1929C411yescharacteristic
57#1876C411yescharacteristic
58#1876C411yescharacteristic
59#1806C411yescharacteristic
60C8×C4⋊C4C2211yescharacteristic
61#1764C2211yescharacteristic
62#1764C2211yescharacteristic
63#1379C2211yescharacteristic
64#1379C2211yescharacteristic
65#1376C2211yescharacteristic
66#1376C2211yescharacteristic
67#1374C2211yescharacteristic
68#1374C2211yescharacteristic
69C2×#135C2×C411yescharacteristic
70C2×#135C2×C411yescharacteristic
71C2×#131C2×C411yescharacteristic
72C2×#131C2311yescharacteristic
73C2×#131C2311yescharacteristic
74C2×#131C2311yescharacteristic
75#532C2×C411yescharacteristic
76C4×#41C2×C411yescharacteristic
77#504—22no—
78#504—22no—
79#504—22no—
80#504—22no—
81#504—41no—
82#504—41no—
83#504—41no—
84#504—41no—
85C2×M32—21no—
86C2×M32—21no—
87C2×M32—21no—
88C2×M32—21no—
89C2×M32C2311yescharacteristic
90C2×M32C2311yescharacteristic
91C2×M32C2311yescharacteristic
92C2×M32C2311yescharacteristic
93C22×C16—21no—
94C22×C16—21no—
95C22×C16—21no—
96C22×C16—21no—
97C22×C16C2311yescharacteristic
98C22×C16C2311yescharacteristic
99C22×C16C2311yescharacteristic
100C22×C16C2311yescharacteristic
101#464C811yescharacteristic
102#464C811yescharacteristic
103#463C811yescharacteristic
104#463C811yescharacteristic
105#462C811yescharacteristic
106#461C811yescharacteristic
107#460C811yescharacteristic
108#459C811yescharacteristic
109#443—22no—
110#443—22no—
111#443—22no—
112#443—22no—
113#436—21no—
114#436—21no—
115C8×D4—21no—
116C8×D4—21no—
117C2×#108C2×C411yescharacteristic
118C2×#108C2×C411yescharacteristic
119#423C2×C411yescharacteristic
120C2×C4⋊C8C2×C411yescharacteristic
121C2×#104C2×C411yescharacteristic
122C2×#104C2×C411yescharacteristic
123C2×#104C2×C411yescharacteristic
124C2×#104C2×C411yescharacteristic
125C2×#104C2×C411yescharacteristic
126C2×#104C2×C411yescharacteristic
127#416C2×C411yescharacteristic
128#416C2×C411yescharacteristic
129#408C2×C411yescharacteristic
130#408C2×C411yescharacteristic
131#405—21no—
132#405—21no—
133#405C2×C411yescharacteristic
134#405C2311yescharacteristic
135C4×M16—21no—
136C4×M16C2×C411yescharacteristic
137C2×C4×C8—21no—
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142#364—22no—
143#364—22no—
144C4⋊C16—21no—
145C4⋊C16—21no—
146C4⋊C16—21no—
147C4⋊C16—21no—
148C4⋊C16—21no—
149C4⋊C16—21no—
150C4⋊C16D411yescharacteristic
151C4⋊C16D411yescharacteristic
152C4⋊C16D411yescharacteristic
153C4⋊C16D411yescharacteristic
154#350—22no—
155#350—22no—
156#350—22no—
157#350—41no—
158#350—41no—
159#350D414yescomplemented
160C22⋊C16—21no—
161C22⋊C16—21no—
162C22⋊C16—21no—
163C22⋊C16—21no—
164#346—22no—
165#346—21no—
166#346—21no—
167#346—21no—
168C4×C16—22no—
169C4×C16—21no—
170#341C2×C411yescharacteristic
171#341C2×C411yescharacteristic
172#339C2×C411yescharacteristic
173#339C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#336C2×C411yescharacteristic
176C2×#41C22×C411yescharacteristic
177C2×#41C22×C411yescharacteristic
178C2×#41C22×C411yescharacteristic
179#135—22no—
180#135—22no—
181#135—22no—
182#135—22no—
183C2×Q16C2×C811yescharacteristic
184C2×Q16C2×C811yescharacteristic
185C2×SD16C2×C811yescharacteristic
186C2×SD16C2×C811yescharacteristic
187C2×SD16C2×C811yescharacteristic
188C2×SD16C2×C811yescharacteristic
189C2×D8C2×C811yescharacteristic
190C2×D8C2×C811yescharacteristic
191#131—21no—
192#131—21no—
193#131—22no—
194#131—21no—
195#131—21no—
196#131—21no—
197#131—21no—
198#131—21no—
199#131—21no—
200#131—21no—
201#131—21no—
202#131—41no—
203#131C2×D412yes—
204#131#4112yes—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16C22×C411yescharacteristic
210C2×M16C22×C411yescharacteristic
211C2×M16C22×C411yescharacteristic
212C2×M16C22×C411yescharacteristic
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C2411yescharacteristic
221C4⋊Q8C2×C811yescharacteristic
222#127C2×C811yescharacteristic
223#124C2×C811yescharacteristic
224#124C2×C811yescharacteristic
225#121—21no—
226#121—21no—
227C4×Q8C2×C811yescharacteristic
228C4×Q8C2×C811yescharacteristic
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4C2×C811yescharacteristic
234C4×D4C2×C811yescharacteristic
235#117—21no—
236#117C22×C411yescharacteristic
237C2×C4⋊C4—21no—
238C2×C42—21no—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242M32—21no—
243M32—21no—
244M32—21no—
245M32—21no—
246M32—21no—
247M32—21no—
248M32—21no—
249M32—42no—
250M32—42no—
251M32—42no—
252M32—42no—
253M32C2×D412yes—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—21no—
262C2×C16—21no—
263C2×C16—21no—
264C2×C16—21no—
265C2×C16—21no—
266C2×C16—21no—
267C2×C16—42no—
268C2×C16—42no—
269C2×C16—42no—
270C2×C16—42no—
271C2×C16C2×D412yes—
272C2×C16C2×D411yescharacteristic
273C2×C16C2×D411yescharacteristic
274C2×C16C2×D411yescharacteristic
275C2×C16C2×D411yescharacteristic
276#108—22no—
277#108—22no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8C2×C811yescharacteristic
281C4⋊C8C2×C811yescharacteristic
282C4⋊C8C2×C811yescharacteristic
283C4⋊C8C2×C811yescharacteristic
284#104—22no—
285#104—22no—
286#104—22no—
287#104—22no—
288#104—41no—
289#104—41no—
290#103C2×C811yescharacteristic
291#103C2×C811yescharacteristic
292#103C2×C811yescharacteristic
293#103C2×C811yescharacteristic
294#102C2×C811yescharacteristic
295#102C2×C811yescharacteristic
296#102C2×C811yescharacteristic
297#102C2×C811yescharacteristic
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302#97—21no—
303#97—21no—
304#97—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8C2×D411yescharacteristic
313C4×C8C2×D411yescharacteristic
314C4×C8#4111yescharacteristic
315C4×C8#4111yescharacteristic
316#95—21no—
317#41—21no—
318#41—21no—
319#41—21no—
320#41—21no—
321#41—21no—
322#41—21no—
323#41—21no—
324#41—21no—
325#41—21no—
326#41—21no—
327#41C4×D412yes—
328#41#13112yes—
329C2×Q8C22×C811yescharacteristic
330C2×Q8C22×C811yescharacteristic
331C2×Q8C22×C811yescharacteristic
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4C22×C811yescharacteristic
338C2×D4C22×C811yescharacteristic
339C2×D4C22×C811yescharacteristic
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4#13111yescharacteristic
348C22×C4#13111yescharacteristic
349C22×C4C23×C411yescharacteristic
350Q16—22no—
351Q16—22no—
352SD16—22no—
353SD16—22no—
354SD16—22no—
355SD16—22no—
356D8—22no—
357D8—22no—
358M16—21no—
359M16—22no—
360M16—22no—
361M16—21no—
362M16—21no—
363M16—21no—
364M16—21no—
365M16—21no—
366M16—21no—
367M16—42no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16C4×D412yes—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8C4×D411yescharacteristic
391C2×C8C4×D411yescharacteristic
392C2×C8C22×C811yescharacteristic
393C2×C8C22×C811yescharacteristic
394C2×C8C22×C811yescharacteristic
395C2×C8C22×C811yescharacteristic
396C2×C8C22×D411yescharacteristic
397C2×C8C22×D411yescharacteristic
398C2×C8C2×#4111yescharacteristic
399C2×C8C2×#4111yescharacteristic
400C2×C8H(5,𝔽2)11yescharacteristic
401C2×C82−1+411yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—42no—
407C4⋊C4C22×C811yescharacteristic
408C4⋊C4C22×C811yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—21no—
415C22⋊C4—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42C4×D411yescharacteristic
424C42C4×D411yescharacteristic
425C42C22×C811yescharacteristic
426C42C22×C811yescharacteristic
427C16—21no—
428C16—21no—
429C16—21no—
430C16—21no—
431C16—22no—
432C16—22no—
433C16—41no—
434C16—41no—
435C16—41no—
436C16—41no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23C2×#13111yescharacteristic
441Q8—21no—
442Q8—21no—
443Q8—21no—
444Q8—21no—
445Q8—21no—
446Q8C8×D412yes—
447D4—22no—
448D4—22no—
449D4—21no—
450D4—21no—
451D4—21no—
452D4—21no—
453D4—21no—
454D4—42no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—81no—
460D4C8×D412yes—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4C8×D411yescharacteristic
483C2×C4C8×D411yescharacteristic
484C2×C4C8×D411yescharacteristic
485C2×C4C8×D411yescharacteristic
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4#52011yescharacteristic
489C2×C4C23×C811yescharacteristic
490C2×C4C2×#13111yescharacteristic
491C2×C4#56811yescharacteristic
492C8—22no—
493C8—21no—
494C8—21no—
495C8—21no—
496C8—21no—
497C8—21no—
498C8—21no—
499C8—41no—
500C8—41no—
501C8—41no—
502C8D4211yescharacteristic
503C8#54711yescharacteristic
504C8#54811yescharacteristic
505C8#55311yescharacteristic
506C22—21no—
507C22—21no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22C2×C8×D411yescharacteristic
518C22#293511yescharacteristic
519C22#299411yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—22no—
525C4—42no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—81no—
531C4#236111yescharacteristic
532C4C2×C8×D411yescharacteristic
533C4#293311yescharacteristic
534C4#298811yescharacteristic
535C2—21no—
536C2—21no—
537C2—42no—
538C2—81no—
539C2—81no—
540C2#2298511yescharacteristic
541C2#3344111yescharacteristic
542C2#3344311yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radical#3035C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4⋊C16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2171C2semidirect
2. C2×#2170C2semidirect
3. #33438C2semidirect
4. #33398C2semidirect
6. #13192C2semidirect
7. #13163C2semidirect
8. #13125C2semidirect
9. #13124C2semidirect
10. #12980C2semidirect
11. #12980C2semidirect
12. #12917C2semidirect
13. #12917C2semidirect
14. #12663C2semidirect
15. #12604C2semidirect
159. #350D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21712
C2×#21702
#334382
#333982
#215172
#131922
#131632
#131252
#131242
#129802
#129172
#126632
#126042