Finite Group Databasebeta

← #279243

Subgroups — #279243

Order 512.

779 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33195C211yesmaximal, characteristic, complemented
2#28508C211yesmaximal, characteristic, complemented
3D4×C4⋊C8C211yesmaximal, characteristic, complemented
4C4×#459C211yesmaximal, characteristic
5#10649C212yesmaximal, complemented
6#10591C212yesmaximal, complemented
7#9898C211yesmaximal, characteristic, complemented
8#9601C211yesmaximal, characteristic, complemented
9#9221C212yesmaximal
10#8644C212yesmaximal, complemented
11#8585C211yesmaximal, characteristic, complemented
12C2×#474—21no—
13C2×#474—21no—
14C2×#474—21no—
15C2×#459C2211yescharacteristic
16C2×C4×D8—21no—
17C2×C8×D4C2212yes—
18C22×C4⋊C8—21no—
19#2868C2211yescharacteristic
20C4×#127C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#2093—22no—
23#1988—21no—
24#1982C2211yescharacteristic
25#1957C2212yes—
26#1947—21no—
27#1947C2212yes—
28#1947C2211yescharacteristic
29#1939C2212yes—
30#1925—21no—
31#1879—21no—
32#1850—22no—
33#1845C2212yes—
34#1842C2211yescharacteristic
35#1842C2211yescharacteristic
36#1802—21no—
37#1802C2212yes—
38#1802C2211yescharacteristic
39#1797—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#102—21no—
43C2×#335C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C22×D8—21no—
50C23×C8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×#127C2311yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2311yescharacteristic
61C22×C42—21no—
62#474—44no—
63#474—44no—
64#474—44no—
65#459—22no—
66#459Q814yescomplemented
67C4×D8—44no—
68C8×D4—24no—
69C8×D4—42no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—21no—
73C2×#107—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—44no—
80C2×C4⋊C8D411yescharacteristic
81C2×C4⋊C8D411yescharacteristic
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×#102—21no—
87C2×#102—22no—
88C2×#102—21no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C22⋊C8—22no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96#390—21no—
97#390—21no—
98#390—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384C2311yescharacteristic
103#384C2312yes—
104C4×C4⋊C4D411yescharacteristic
105C4×C4⋊C4D411yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C43C2311yescharacteristic
111#340—22no—
112#340—21no—
113#340—22no—
114#340—21no—
115#336—22no—
116#336—21no—
117#336D412yes—
118#335—42no—
119#331—42no—
120C22×Q8—41no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4C2×D411yescharacteristic
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—41no—
129C2×D8—24no—
130C2×D8—42no—
131C2×D8—42no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—44no—
137C22×C8—44no—
138C22×C8—82no—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C4⋊Q8—42no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—82no—
146#127—21no—
147#127—21no—
148#127—21no—
149#127C2×Q812yes—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—21no—
153C4×D4—21no—
154C4×D4—41no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—82no—
160C4×D4D814yes—
161C4×D4C2×Q812yes—
162C4×D4#4112yes—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42C2411yescharacteristic
193#107—44no—
194#107—44no—
195#107—42no—
196#107—44no—
197C4⋊C8—22no—
198C4⋊C8—21no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—21no—
203C4⋊C8—22no—
204C4⋊C8—21no—
205C4⋊C8—42no—
206C4⋊C8—44no—
207C4⋊C8—42no—
208C4⋊C8—42no—
209C4⋊C8—81no—
210C4⋊C8D814yescomplemented
211C4⋊C8C2×Q812yes—
212#102—24no—
213#102—42no—
214#102—44no—
215#102—42no—
216#102—44no—
217#102—44no—
218#102—42no—
219C22⋊C8—44no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—42no—
223#95—21no—
224#95—21no—
225#95—21no—
226C24—21no—
227C24—41no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×D4—22no—
232C2×D4—21no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—81no—
249C2×D4—81no—
250C2×D4C22⋊Q812yes—
251C2×D4C2×D812yes—
252C2×D4#13712yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—81no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4#12111yescharacteristic
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4111yescharacteristic
280D8—48no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—88no—
299C2×C8C2×D814yes—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4C22⋊Q812yes—
318C4⋊C4C2×D812yes—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—24no—
322C22⋊C4—42no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—22no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—42no—
335C42—42no—
336C42C2×D812yes—
337C42C2×D812yes—
338C42#13512yes—
339C42AGL(1,ℤ/8)12yes—
340C42C22×Q811yescharacteristic
341C42C2×#4111yescharacteristic
342C42H(5,𝔽2)11yescharacteristic
343C422−1+411yescharacteristic
344C23—22no—
345C23—22no—
346C23—22no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23C2×#12111yescharacteristic
359Q8—164no—
360D4—22no—
361D4—22no—
362D4—22no—
363D4—42no—
364D4—42no—
365D4—42no—
366D4—42no—
367D4—82no—
368D4—82no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4#47414yescomplemented
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—48no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—84no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—164no—
405C2×C4#45912yes—
406C2×C4#45912yes—
407C2×C4#46612yes—
408C2×C4#46612yes—
409C2×C4#47212yes—
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4#53711yescharacteristic
412C2×C4#54611yescharacteristic
413C2×C4D4×Q811yescharacteristic
414C2×C4#55211yescharacteristic
415C2×C4#55411yescharacteristic
416C2×C4C22×D811yescharacteristic
417C2×C4C22×D811yescharacteristic
418C2×C4C2×#13511yescharacteristic
419C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
420C2×C4C2×#13711yescharacteristic
421C2×C4#57511yescharacteristic
422C2×C4#57711yescharacteristic
423C2×C4#57811yescharacteristic
424C8—24no—
425C8—42no—
426C8—42no—
427C8—81no—
428C8—82no—
429C8—82no—
430C22—24no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22#241711yescharacteristic
445C22C2×#45911yescharacteristic
446C22#304611yescharacteristic
447C22C2×#46611yescharacteristic
448C22#305611yescharacteristic
449C22C2×#47211yescharacteristic
450C22#306911yescharacteristic
451C4—21no—
452C4—21no—
453C4—22no—
454C4—22no—
455C4—41no—
456C4—42no—
457C4—42no—
458C4—81no—
459C4—161no—
460C4—162no—
461C4#166714yes—
462C4C2×#47411yescharacteristic
463C4#307611yescharacteristic
464C4#324511yescharacteristic
465C4#324811yescharacteristic
466C4#329411yescharacteristic
467C4#333011yescharacteristic
468C4Q8×D811yescharacteristic
469C4#338411yescharacteristic
470C2—44no—
471C2—162no—
472C2C2×#166711yescharacteristic
473C2#1916311yescharacteristic
474C2#1941611yescharacteristic
475C2#1949111yescharacteristic
476C2#2481111yescharacteristic
477C2#2483511yescharacteristic
478C2#2485511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33195C2semidirect
2. #28508C2semidirect
3. D4×C4⋊C8C2semidirect
5. #10649C2semidirect
6. #10591C2semidirect
7. #9898C2semidirect
8. #9601C2semidirect
10. #8644C2semidirect
11. #8585C2semidirect
66. #459Q8semidirect
210. C4⋊C8D8semidirect
373. D4#474semidirect

Maximal subgroups

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

subgroupindex
#331952
#285082
D4×C4⋊C82
C4×#4592
#106492
#105912
#98982
#96012
#92212
#86442
#85852