Finite Group Databasebeta

← #225384

Subgroups — #225384

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626C211yesmaximal, characteristic
2#22603C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic
4#9597C211yesmaximal, characteristic, complemented
5#9593C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9539C211yesmaximal, characteristic, complemented
8#9538C211yesmaximal, characteristic, complemented
9#9534C211yesmaximal, characteristic, complemented
10#8710C211yesmaximal, characteristic
11#8706C211yesmaximal, characteristic
12#8598C211yesmaximal, characteristic
13#8290C211yesmaximal, characteristic, complemented
14#8283C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic
16C2×#432—21no—
17C2×#431C2211yescharacteristic
18#2285C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1948C411yescharacteristic
21#1948C411yescharacteristic
22#1948C411yescharacteristic
23#1948C411yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1917—21no—
29#1917—21no—
30#1917—21no—
31#1916—21no—
32#1853C411yescharacteristic
33#1853C411yescharacteristic
34#1853C411yescharacteristic
35#1853C411yescharacteristic
36#1817—21no—
37#1798—21no—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C4×#106C2211yescharacteristic
41C4×#106C2211yescharacteristic
42C4×#106C2211yescharacteristic
43C4×#106C2211yescharacteristic
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#322C2211yescharacteristic
54C2×C82C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×#117—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#432—44no—
73#431—22no—
74#431—22no—
75C2×#106C2×C411yescharacteristic
76C2×#106C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×#106C2×C411yescharacteristic
80C2×#106C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2×C411yescharacteristic
88C2×C4⋊C8C2×C411yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8C2×C411yescharacteristic
97C2×C22⋊C8C2×C411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C2×C22⋊C8D411yescharacteristic
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#382—21no—
113#380—21no—
114#380C2×C411yescharacteristic
115#380C2×C411yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C2×#95—21no—
126#340C2×C411yescharacteristic
127#340C2×C411yescharacteristic
128#340C2×C411yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#334—22no—
148#334—22no—
149#322—22no—
150C82—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C22×C411yescharacteristic
163C22×C8C22×C411yescharacteristic
164C22×C8C22×C411yescharacteristic
165C22×C8C22×C411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170#117—81no—
171#117—81no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208#106—22no—
209#106—22no—
210#106—41no—
211#106—41no—
212#106—41no—
213#106—41no—
214#106—41no—
215#106—41no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—42no—
234C22⋊C8—42no—
235C22⋊C8—42no—
236C22⋊C8SD1614yes—
237#97—42no—
238#97—42no—
239#97—81no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—81no—
251C4×C8#4112yes—
252C4×C8#4112yes—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C24C4×D411yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
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—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—81no—
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C2×SD1612yes—
294C22×C4#13512yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8#11712yes—
337C2×C8#11712yes—
338C2×C8C2×SD1612yes—
339C2×C8C2×SD1612yes—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4#13112yes—
362C22⋊C4—81no—
363C22⋊C4—81no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—42no—
379C42—42no—
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C422−1+411yescharacteristic
383C422−1+411yescharacteristic
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23C4×SD1612yes—
391C23C2×C4×D411yescharacteristic
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4C4×SD1612yes—
439C2×C4C2×#11711yescharacteristic
440C2×C4#51911yescharacteristic
441C2×C4#54711yescharacteristic
442C2×C4#54711yescharacteristic
443C2×C4#55211yescharacteristic
444C2×C4#55211yescharacteristic
445C2×C4C2×#13111yescharacteristic
446C2×C4#56811yescharacteristic
447C2×C4#56811yescharacteristic
448C2×C4#56811yescharacteristic
449C2×C4C22×SD1611yescharacteristic
450C2×C4C2×#13511yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57711yescharacteristic
453C8—22no—
454C8—22no—
455C8—22no—
456C8—22no—
457C8—42no—
458C8—41no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—81no—
463C8—81no—
464C8—81no—
465C8—81no—
466C8—81no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22#234411yescharacteristic
475C22#293011yescharacteristic
476C22#293311yescharacteristic
477C22C2×C4×SD1611yescharacteristic
478C22#294911yescharacteristic
479C22#295511yescharacteristic
480C22#295611yescharacteristic
481C4—21no—
482C4—21no—
483C4—22no—
484C4—42no—
485C4—41no—
486C4—41no—
487C4—81no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4#292311yescharacteristic
495C4#292511yescharacteristic
496C4#298511yescharacteristic
497C4#298711yescharacteristic
498C4#333511yescharacteristic
499C4#333511yescharacteristic
500C4#333711yescharacteristic
501C4#333711yescharacteristic
502C2—42no—
503C2#1793011yescharacteristic
504C2#1795111yescharacteristic
505C2#1799511yescharacteristic
506C2#1803011yescharacteristic
507C2#2260511yescharacteristic
508C2#2260811yescharacteristic
509C2#2260911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9597C2semidirect
5. #9593C2semidirect
6. #9539C2semidirect
7. #9539C2semidirect
8. #9538C2semidirect
9. #9534C2semidirect
13. #8290C2semidirect
14. #8283C2semidirect

Maximal subgroups

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

subgroupindex
#226262
#226032
#213592
#95972
#95932
#95392
#95382
#95342
#87102
#87062
#85982
#82902
#82832
#81482