Finite Group Databasebeta

← #226566

Subgroups — #226566

Order 512.

748 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22799C211yesmaximal, characteristic, complemented
2#22650C211yesmaximal, characteristic, complemented
3C42×D8C211yesmaximal, characteristic
4#9554C212yesmaximal, complemented
5#9537C212yesmaximal, complemented
6#8951C212yesmaximal
7#8784C211yesmaximal, characteristic, complemented
8#8563C211yesmaximal, characteristic, complemented
9#8563C211yesmaximal, characteristic, complemented
10C8×C4⋊C8C211yesmaximal, characteristic, complemented
11#8206C212yesmaximal, complemented
12C2×C4×D8C411yescharacteristic
13C2×C4×D8C411yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C2×#436C2212yes—
16C2×#432—21no—
17C2×#423—21no—
18C42×D4C2211yescharacteristic
19C42×D4C2211yescharacteristic
20#1954C2212yes—
21#1939C2212yes—
22#1924—22no—
23#1851—22no—
24#1842C2211yescharacteristic
25#1842C2211yescharacteristic
26#1797—21no—
27#1797—21no—
28C4×#107C411yescharacteristic
29C4×#107C411yescharacteristic
30C4×#107C2211yescharacteristic
31#1770C2212yes—
32C4×C4⋊C8C2211yescharacteristic
33C4×C4⋊C8C2211yescharacteristic
34C4×#102C411yescharacteristic
35C4×#102C411yescharacteristic
36C4×#102C411yescharacteristic
37C4×#102C411yescharacteristic
38C4×#102C2211yescharacteristic
39C4×#102C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×C82C2212yes—
44#1279C2212yes—
45#1279C2212yes—
46#1277C2212yes—
47#1277C2212yes—
48#1273C2212yes—
49C22×D8C2×C411yescharacteristic
50C22×M16—22no—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2311yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—21no—
59C4×D8—22no—
60C4×D8—22no—
61C4×D8—24no—
62C4×D8—24no—
63#436—24no—
64#436—42no—
65#432—44no—
66C2×#107C2×C411yescharacteristic
67C2×#107C2×C411yescharacteristic
68C2×#107C2×C411yescharacteristic
69#423—44no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8D411yescharacteristic
77C2×C4⋊C8D411yescharacteristic
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#102C2×C411yescharacteristic
81C2×#102C2×C411yescharacteristic
82C2×#102C2×C411yescharacteristic
83C2×#102C2×C411yescharacteristic
84C2×#102C2×C411yescharacteristic
85C2×#102C2×C411yescharacteristic
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×#97—22no—
90C2×C4×C8C2×C411yescharacteristic
91C2×C4×C8C2×C411yescharacteristic
92C2×C4×C8C2×C411yescharacteristic
93C2×C4×C8C2×C411yescharacteristic
94C2×C4×C8C2312yes—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2312yes—
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4C2×C411yescharacteristic
101C4×C4⋊C4C2×C411yescharacteristic
102C4×C4⋊C4C2311yescharacteristic
103C4×C4⋊C4C2311yescharacteristic
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C43C2311yescharacteristic
111#340C2×C411yescharacteristic
112#340C2×C411yescharacteristic
113#340C2×C411yescharacteristic
114#340C2×C411yescharacteristic
115#340C2×C411yescharacteristic
116#340C2×C411yescharacteristic
117#336—22no—
118#336—22no—
119#336—21no—
120#336D412yes—
121D4⋊C8—24no—
122D4⋊C8—42no—
123C82—42no—
124C22×D4C22×C411yescharacteristic
125C22×D4C22×C411yescharacteristic
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C2×D8—22no—
133C2×D8—22no—
134C2×D8—22no—
135C2×M16—44no—
136C2×M16—44no—
137C2×M16—82no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C22×C411yescharacteristic
142C22×C8C22×C411yescharacteristic
143C22×C8C22×C411yescharacteristic
144C22×C8C22×C411yescharacteristic
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—21no—
154C4×D4—21no—
155C4×D4—21no—
156C4×D4—21no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—41no—
160C4×D4—41no—
161C4×D4#4112yes—
162C4×D4#4112yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42C2411yescharacteristic
192#107—22no—
193#107—22no—
194#107—22no—
195C4⋊C8—24no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—44no—
202C4⋊C8—42no—
203C4⋊C8—42no—
204C4⋊C8—81no—
205C4⋊C8—81no—
206C4⋊C8D814yescomplemented
207#102—22no—
208#102—22no—
209#102—22no—
210#102—22no—
211#102—22no—
212#102M1614yescomplemented
213C22⋊C8—44no—
214C22⋊C8—44no—
215C22⋊C8—82no—
216#97—44no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—22no—
223C4×C8—21no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—21no—
227C4×C8—41no—
228C4×C8M1614yescomplemented
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235C24—21no—
236C24—21no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4#11712yes—
248C2×D4C2×M1612yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C2×#4111yescharacteristic
286D8—28no—
287M16—88no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8C2×M1612yes—
312C2×C8C2×M1612yes—
313C2×C8C2×M1612yes—
314C2×C8C2×D814yes—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4#11712yes—
326C4⋊C4C2×M1612yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42C2×M1612yes—
350C42#13112yes—
351C42C2×D812yes—
352C42#13512yes—
353C42C2×#4111yescharacteristic
354C42C2×#4111yescharacteristic
355C42H(5,𝔽2)11yescharacteristic
356C422−1+411yescharacteristic
357C23—22no—
358C23—22no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23C2×C4×D411yescharacteristic
372D4—22no—
373D4—22no—
374D4—22no—
375D4—22no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—41no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
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—84no—
412C2×C4—84no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4#43612yes—
420C2×C4#43612yes—
421C2×C4C4×D812yes—
422C2×C4C4×D812yes—
423C2×C4#44312yes—
424C2×C4C2×#11711yescharacteristic
425C2×C4#52011yescharacteristic
426C2×C4#54611yescharacteristic
427C2×C4#54811yescharacteristic
428C2×C4#55211yescharacteristic
429C2×C4#55311yescharacteristic
430C2×C4C22×M1611yescharacteristic
431C2×C4C22×M1611yescharacteristic
432C2×C4C2×#13111yescharacteristic
433C2×C4#56811yescharacteristic
434C2×C4C22×D811yescharacteristic
435C2×C4C2×#13511yescharacteristic
436C2×C4#57711yescharacteristic
437C2×C4#57711yescharacteristic
438C8—22no—
439C8—22no—
440C8—22no—
441C8—24no—
442C8—44no—
443C8—41no—
444C8—42no—
445C8—82no—
446C8—82no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22#235211yescharacteristic
462C22C2×#43611yescharacteristic
463C22D4×M1611yescharacteristic
464C22C2×C4×D811yescharacteristic
465C22#295211yescharacteristic
466C22C2×#44311yescharacteristic
467C22#295711yescharacteristic
468C4—22no—
469C4—22no—
470C4—21no—
471C4—21no—
472C4—42no—
473C4—41no—
474C4—42no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4#158914yes—
480C4#292011yescharacteristic
481C4#292511yescharacteristic
482C4#298911yescharacteristic
483C4#299011yescharacteristic
484C4#329411yescharacteristic
485C4#330311yescharacteristic
486C4#335811yescharacteristic
487C4#336011yescharacteristic
488C2—82no—
489C2—82no—
490C2C2×#158911yescharacteristic
491C2#1789911yescharacteristic
492C2M16×D811yescharacteristic
493C2#1805711yescharacteristic
494C2#2275111yescharacteristic
495C2#2276611yescharacteristic
496C2#2278311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22799C2semidirect
2. #22650C2semidirect
4. #9554C2semidirect
5. #9537C2semidirect
7. #8784C2semidirect
8. #8563C2semidirect
9. #8563C2semidirect
10. C8×C4⋊C8C2semidirect
11. #8206C2semidirect
206. C4⋊C8D8semidirect
212. #102M16semidirect
228. C4×C8M16semidirect

Maximal subgroups

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

subgroupindex
#227992
#226502
C42×D82
#95542
#95372
#89512
#87842
#85632
C8×C4⋊C82
#82062