Finite Group Databasebeta

← #225261

Subgroups — #225261

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22633C211yesmaximal, characteristic
2#22600C211yesmaximal, characteristic
3#20819C211yesmaximal, characteristic
4#9596C211yesmaximal, characteristic, complemented
5#9593C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#9536C211yesmaximal, characteristic, complemented
9#9534C211yesmaximal, characteristic, complemented
10#8711C211yesmaximal, characteristic
11#8710C211yesmaximal, characteristic
12#8597C211yesmaximal, characteristic
13#8284C211yesmaximal, characteristic, complemented
14#8282C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic
16C2×#433—21no—
17C2×C4×M16C2211yescharacteristic
18#2285C2211yescharacteristic
19#2283C2211yescharacteristic
20#1939C2211yescharacteristic
21#1939C2211yescharacteristic
22#1939C2211yescharacteristic
23#1939C2211yescharacteristic
24#1924—21no—
25#1923—21no—
26#1917—21no—
27#1916—21no—
28#1847C411yescharacteristic
29#1847C411yescharacteristic
30#1836C411yescharacteristic
31#1836C411yescharacteristic
32#1832—21no—
33#1818—21no—
34#1809C411yescharacteristic
35#1809C411yescharacteristic
36#1809C411yescharacteristic
37#1809C411yescharacteristic
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#107C2211yescharacteristic
42C4×#106C2211yescharacteristic
43C4×#106C2211yescharacteristic
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49C4×C22⋊C8C2211yescharacteristic
50#1746—21no—
51C2×#335C2211yescharacteristic
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
67C22×M16C2×C411yescharacteristic
68C22×M16C2×C411yescharacteristic
69C2×#117—21no—
70C22×C42C2311yescharacteristic
71#433—44no—
72C2×#107C2×C411yescharacteristic
73C2×#107C2×C411yescharacteristic
74C2×#107C2×C411yescharacteristic
75C2×#107C2×C411yescharacteristic
76C2×#106C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×#106C2×C411yescharacteristic
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8D411yescharacteristic
92C2×C22⋊C8D411yescharacteristic
93C4×M16—22no—
94C4×M16—22no—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97C2×C411yescharacteristic
98C2×#97C2×C411yescharacteristic
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#384C2×C411yescharacteristic
108#384C2×C411yescharacteristic
109#383—21no—
110#382—21no—
111#382C2×C411yescharacteristic
112#382C2×C411yescharacteristic
113#381—21no—
114#380—21no—
115#380C2×C411yescharacteristic
116#380C2×C411yescharacteristic
117#380C2×C411yescharacteristic
118#380C2×C411yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2311yescharacteristic
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125#376—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#335—22no—
148#334—22no—
149#322—22no—
150C82—22no—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—41no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C22×C411yescharacteristic
167C22×C8C22×C411yescharacteristic
168C22×C8C22×C411yescharacteristic
169C22×C8C22×C411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174#117—42no—
175#117—42no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
189C2×C22⋊C4—21no—
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×C42C22×C411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42C2411yescharacteristic
204#107—22no—
205#107—22no—
206#107—22no—
207#107—41no—
208#106—22no—
209#106—22no—
210#106—22no—
211#106—41no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—42no—
221C4⋊C8—42no—
222C4⋊C8—81no—
223C4⋊C8—81no—
224C22⋊C8—22no—
225C22⋊C8—41no—
226C22⋊C8—42no—
227C22⋊C8—42no—
228C22⋊C8—81no—
229#97—22no—
230#97—22no—
231#97—42no—
232#97—42no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—42no—
244C4×C8—42no—
245C4×C8#4112yes—
246C4×C8#4112yes—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259C24C4×D411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
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×C4C4×D411yescharacteristic
285C22×C4C4×D411yescharacteristic
286C22×C4C4×D411yescharacteristic
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290M16—44no—
291M16—44no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
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—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
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#11712yes—
329C2×C8#11712yes—
330C2×C8#13112yes—
331C2×C8#13112yes—
332C2×C8#13112yes—
333C2×C8AGL(1,ℤ/8)12yes—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
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—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—81no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—81no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42C2×#4111yescharacteristic
373C42C2×#4111yescharacteristic
374C422−1+411yescharacteristic
375C422−1+411yescharacteristic
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23C2×C4×D411yescharacteristic
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
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—81no—
428C2×C4—81no—
429C2×C4C2×#11711yescharacteristic
430C2×C4#51911yescharacteristic
431C2×C4#54711yescharacteristic
432C2×C4#54711yescharacteristic
433C2×C4#55211yescharacteristic
434C2×C4#55211yescharacteristic
435C2×C4C2×#13111yescharacteristic
436C2×C4C2×#13111yescharacteristic
437C2×C4C2×#13111yescharacteristic
438C2×C4#56811yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57811yescharacteristic
443C8—22no—
444C8—22no—
445C8—22no—
446C8—42no—
447C8—41no—
448C8—41no—
449C8—41no—
450C8—41no—
451C8—81no—
452C8—81no—
453C8—81no—
454C8—81no—
455C8—81no—
456C8#43314yescomplemented
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22#234411yescharacteristic
465C22#293111yescharacteristic
466C22#293511yescharacteristic
467C22C4×AGL(1,ℤ/8)11yescharacteristic
468C22#294711yescharacteristic
469C22#295511yescharacteristic
470C22#295611yescharacteristic
471C4—21no—
472C4—21no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4C2×#43311yescharacteristic
485C4#292411yescharacteristic
486C4#298511yescharacteristic
487C4#298611yescharacteristic
488C4#334311yescharacteristic
489C4#334811yescharacteristic
490C4#335211yescharacteristic
491C4#335411yescharacteristic
492C2—81no—
493C2#1796611yescharacteristic
494C2#1796911yescharacteristic
495C2#1807811yescharacteristic
496C2#1808111yescharacteristic
497C2#2260011yescharacteristic
498C2#2260811yescharacteristic
499C2#2262411yescharacteristic
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. #9596C2semidirect
5. #9593C2semidirect
6. #9539C2semidirect
7. #9537C2semidirect
8. #9536C2semidirect
9. #9534C2semidirect
13. #8284C2semidirect
14. #8282C2semidirect
456. C8#433semidirect

Maximal subgroups

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

subgroupindex
#226332
#226002
#208192
#95962
#95932
#95392
#95372
#95362
#95342
#87112
#87102
#85972
#82842
#82822
#81462