Finite Group Databasebeta

← #510674

Subgroups — #510674

Order 512.

613 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27929C211yesmaximal, characteristic
2#25415C211yesmaximal, characteristic, complemented
3#13537C211yesmaximal, characteristic, complemented
4#13481C211yesmaximal, characteristic, complemented
5#13449C211yesmaximal, characteristic, complemented
6#13327C211yesmaximal, characteristic, complemented
7#13305C211yesmaximal, characteristic
8#13305C211yesmaximal, characteristic
9#13203C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic, complemented
11#13120C211yesmaximal, characteristic, complemented
12#13013C211yesmaximal, characteristic, complemented
13#12993C211yesmaximal, characteristic, complemented
14#12993C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×SD32C2211yescharacteristic
18C2×#494C2211yescharacteristic
19C2×#480—21no—
20C2×#467C2211yescharacteristic
21C2×#467C2211yescharacteristic
22C2×#466C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C22×#107C2211yescharacteristic
25#2501—21no—
26#2429—21no—
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39#2093—21no—
40#2093—21no—
41#2043—21no—
42#2001—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1878—21no—
48#1845C2211yescharacteristic
49#1845C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16—21no—
60C22×Q16C2311yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C4⋊Q8—21no—
65C2×#123—41no—
66C2×#123—41no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×C4×Q8—21no—
71C22×C4⋊C4—21no—
72C2×SD32—22no—
73C2×SD32—41no—
74C2×SD32—41no—
75C2×SD32—41no—
76C2×SD32—41no—
77C2×SD32—41no—
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#494—22no—
83#494—41no—
84#480—44no—
85#467—22no—
86#467—22no—
87#467—41no—
88#467—41no—
89#466—41no—
90#466—41no—
91C4×Q16—22no—
92C4×Q16—41no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107—41no—
97C2×#107—41no—
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×C22⋊C8D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#399—41no—
114#399—41no—
115#398—41no—
116#396—41no—
117#391—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#389—41no—
122#388—41no—
123#384—21no—
124#384—21no—
125C4×C22⋊C4D411yescharacteristic
126C4⋊C16—41no—
127#358—41no—
128#358—41no—
129#358—41no—
130#358—41no—
131#358—41no—
132#358—41no—
133#357—22no—
134#357—41no—
135C22⋊C16—41no—
136C22⋊C16—41no—
137C4×C16—41no—
138#340—21no—
139#336D411yescharacteristic
140#336D411yescharacteristic
141C22×Q8—21no—
142C22×Q8—21no—
143C22×D4—21no—
144C22×D4—41no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—42no—
159C2×Q16—42no—
160C2×Q16—81no—
161C2×Q16#4112yes—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2411yescharacteristic
178C4⋊Q8—42no—
179C4⋊Q8—81no—
180#123—84no—
181#123—84no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C22⋊Q8—81no—
185C22⋊Q8—81no—
186#121—42no—
187#121—42no—
188C4×Q8—42no—
189C4×Q8—81no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4D811yescharacteristic
216C2×C22⋊C4D811yescharacteristic
217C2×C42—41no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220SD32—44no—
221SD32—44no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234#107—21no—
235#107—21no—
236#107—21no—
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—42no—
244#107—42no—
245#107—42no—
246#107#4112yes—
247#103—42no—
248#103—42no—
249#103—42no—
250#103—81no—
251#102—42no—
252#102—42no—
253#102—42no—
254C22⋊C8—22no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95D811yescharacteristic
263#95D811yescharacteristic
264C24—41no—
265C24C2×D811yescharacteristic
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4—82no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4C2×D811yescharacteristic
304C22×C4C2×D811yescharacteristic
305C22×C4C2×D811yescharacteristic
306C22×C4C2×D811yescharacteristic
307C22×C4C2×D811yescharacteristic
308C22×C4C22×D411yescharacteristic
309Q16—22no—
310Q16—22no—
311Q16—41no—
312Q16—41no—
313Q16—41no—
314Q16—41no—
315Q16—41no—
316Q16—41no—
317Q16—84no—
318D8—41no—
319D8—41no—
320D8—41no—
321D8—41no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8AGL(1,ℤ/8)12yes—
335C2×C8C2×#4111yescharacteristic
336C2×C8C2×#4111yescharacteristic
337C2×C8H(5,𝔽2)11yescharacteristic
338C2×C8H(5,𝔽2)11yescharacteristic
339C2×C8H(5,𝔽2)11yescharacteristic
340C2×C8H(5,𝔽2)11yescharacteristic
341C4⋊C4—22no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C22⋊C4—22no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4SD3214yescomplemented
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—82no—
376C16—41no—
377C16—41no—
378C16—41no—
379C16—41no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23C2×SD3212yes—
394C23C22×D811yescharacteristic
395Q8—42no—
396Q8—42no—
397Q8—81no—
398Q8—81no—
399Q8—81no—
400Q8—81no—
401Q8—81no—
402Q8—81no—
403D4—81no—
404D4—81no—
405D4—81no—
406D4—81no—
407D4—164no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—164no—
446C2×C4C2×SD3212yes—
447C2×C4C2×SD3212yes—
448C2×C4#50912yes—
449C2×C4#53511yescharacteristic
450C2×C4#54611yescharacteristic
451C2×C4#54611yescharacteristic
452C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57711yescharacteristic
455C8—22no—
456C8—41no—
457C8—41no—
458C8#53811yescharacteristic
459C8#54011yescharacteristic
460C8#55311yescharacteristic
461C8#56011yescharacteristic
462C22—22no—
463C22—22no—
464C22—22no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22—161no—
476C22#320511yescharacteristic
477C22#329411yescharacteristic
478C22#329411yescharacteristic
479C22C22×SD3211yescharacteristic
480C22C2×#50911yescharacteristic
481C22#341711yescharacteristic
482C22#341711yescharacteristic
483C4—22no—
484C4—22no—
485C4—22no—
486C4—41no—
487C4—41no—
488C4—82no—
489C4—82no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4—161no—
495C4#270311yescharacteristic
496C4#322111yescharacteristic
497C4#330611yescharacteristic
498C4#333211yescharacteristic
499C2—42no—
500C2—321no—
501C2#3183411yescharacteristic
502C2#3377411yescharacteristic
503C2#3378911yescharacteristic
504C2#3386511yescharacteristic
505C2#3386511yescharacteristic
506C2#3387511yescharacteristic
507C2#3389711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
2. #25415C2semidirect
3. #13537C2semidirect
4. #13481C2semidirect
5. #13449C2semidirect
6. #13327C2semidirect
9. #13203C2semidirect
10. #13147C2semidirect
11. #13120C2semidirect
12. #13013C2semidirect
13. #12993C2semidirect
14. #12993C2semidirect
15. #12812C2semidirect
370. C22⋊C4SD32semidirect

Maximal subgroups

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

subgroupindex
#279292
#254152
#135372
#134812
#134492
#133272
#133052
#132032
#131472
#131202
#130132
#129932
#128122