Finite Group Databasebeta

← #202307

Subgroups — #202307

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21362C211yesmaximal, characteristic
2#21356C211yesmaximal, characteristic
3#20441C211yesmaximal, characteristic
4#9540C211yesmaximal, characteristic, complemented
5#9539C211yesmaximal, characteristic, complemented
6#9538C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#9536C211yesmaximal, characteristic, complemented
9#9535C211yesmaximal, characteristic, complemented
10#9534C211yesmaximal, characteristic, complemented
11#9533C211yesmaximal, characteristic, complemented
12#7900C211yesmaximal, characteristic
13#7891C211yesmaximal, characteristic
14#7879C211yesmaximal, characteristic
15#7850C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#405—21no—
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1948C411yescharacteristic
25#1948C411yescharacteristic
26#1947C411yescharacteristic
27#1947C411yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1854C411yescharacteristic
33#1854C411yescharacteristic
34#1854C411yescharacteristic
35#1854C411yescharacteristic
36#1851—21no—
37#1851—21no—
38C4×#107C2211yescharacteristic
39C4×#107C2211yescharacteristic
40C4×#106C2211yescharacteristic
41C4×#106C2211yescharacteristic
42#1771—21no—
43#1771—21no—
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48#1767—21no—
49#1767—21no—
50#1729—21no—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53#1279C2211yescharacteristic
54#1279C2211yescharacteristic
55#1279C2211yescharacteristic
56#1279C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16D411yescharacteristic
68C22×M16D411yescharacteristic
69C23×C8—21no—
70C2×#117—21no—
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×#106C2×C411yescharacteristic
85C2×#106C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2×C411yescharacteristic
96C2×C4⋊C8C2×C411yescharacteristic
97C2×C4⋊C8Q814yes—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8C2×C411yescharacteristic
105C2×C22⋊C8C2×C411yescharacteristic
106#405—44no—
107C4×M16—22no—
108C4×M16—22no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97D411yescharacteristic
113C2×#97D411yescharacteristic
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#384C2×C411yescharacteristic
123#384C2×C411yescharacteristic
124#382—21no—
125#380—21no—
126#380C2×C411yescharacteristic
127#380C2×C411yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138C43—21no—
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—41no—
163C2×M16—41no—
164C22×C8—24no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181#117—22no—
182#117—22no—
183#117—41no—
184#117—41no—
185#117—42no—
186#117—42no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q812yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219#107—22no—
220#107—22no—
221#107—41no—
222#107—41no—
223#106—22no—
224#106—22no—
225#106—41no—
226#106—41no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—24no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8C2×Q812yes—
242C4⋊C8C2×Q812yes—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—42no—
246C22⋊C8—42no—
247#97—22no—
248#97—22no—
249#97—42no—
250#97—42no—
251#97—42no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—42no—
265C4×C8—42no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274C24C4×D411yescharacteristic
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4#11712yes—
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×Q812yes—
298C22×C4C22⋊Q812yes—
299C22×C4#12312yes—
300C22×C4#12512yes—
301C22×C4#12612yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305M16—28no—
306M16—44no—
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—22no—
316C2×C8—22no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
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—21no—
331C2×C8—48no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8AGL(1,ℤ/8)12yes—
349C2×C8#13712yes—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4#13112yes—
369C4⋊C4#13112yes—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C42—42no—
375C42—42no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—81no—
385C42—81no—
386C42#11712yes—
387C42C4×Q812yes—
388C42C22⋊Q812yes—
389C42#12312yes—
390C42#12512yes—
391C42#12612yes—
392C42C22×Q811yescharacteristic
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23#38712yes—
403C23C2×C4×D411yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
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—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4#38712yes—
447C2×C4#38712yes—
448C2×C4#38712yes—
449C2×C4C2×#11711yescharacteristic
450C2×C4C2×C4×Q811yescharacteristic
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4C2×#12311yescharacteristic
453C2×C4C2×#12511yescharacteristic
454C2×C4C2×#12611yescharacteristic
455C2×C4C2×#13111yescharacteristic
456C2×C4C2×#13111yescharacteristic
457C2×C4#56811yescharacteristic
458C2×C4#56811yescharacteristic
459C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
460C2×C4C2×#13711yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57511yescharacteristic
463C8—22no—
464C8—22no—
465C8—42no—
466C8—42no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22C2×#38711yescharacteristic
485C22#293311yescharacteristic
486C22#293311yescharacteristic
487C22C4×AGL(1,ℤ/8)11yescharacteristic
488C22C4×#13711yescharacteristic
489C22#295711yescharacteristic
490C22#295711yescharacteristic
491C4—22no—
492C4—22no—
493C4—21no—
494C4—21no—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4#292511yescharacteristic
505C4#292511yescharacteristic
506C4#296011yescharacteristic
507C4#296111yescharacteristic
508C4#307711yescharacteristic
509C4#307811yescharacteristic
510C4#309211yescharacteristic
511C4#309311yescharacteristic
512C2—42no—
513C2#1800711yescharacteristic
514C2#1800911yescharacteristic
515C2#1801111yescharacteristic
516C2#1801311yescharacteristic
517C2#2102711yescharacteristic
518C2#2103311yescharacteristic
519C2#2104311yescharacteristic
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 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9540C2semidirect
5. #9539C2semidirect
6. #9538C2semidirect
7. #9537C2semidirect
8. #9536C2semidirect
9. #9535C2semidirect
10. #9534C2semidirect
11. #9533C2semidirect

Maximal subgroups

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

subgroupindex
#213622
#213562
#204412
#95402
#95392
#95382
#95372
#95362
#95352
#95342
#95332
#79002
#78912
#78792
#78502