Finite Group Databasebeta

← #274238

Subgroups — #274238

Order 512.

840 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33106C211yesmaximal, characteristic
2#25718C211yesmaximal, characteristic
3#22603C211yesmaximal, characteristic, complemented
4C2×#1988C211yesmaximal, characteristic, complemented
5#10592C212yesmaximal, complemented
6#10537C212yesmaximal, complemented
7#9744C212yesmaximal, complemented
8#9569C212yesmaximal, complemented
9#9133C211yesmaximal, characteristic
10#8829C211yesmaximal, characteristic, complemented
11#8340C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#492C2212yes—
14C2×#480—21no—
15C2×#448—21no—
16C2×#448—21no—
17C2×#432—21no—
18C22×C4⋊C8C2211yescharacteristic
19#2856C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#2013—22no—
22#1988—21no—
23#1988—21no—
24#1988C2212yes—
25#1988C2212yes—
26#1982C2211yescharacteristic
27#1982C2211yescharacteristic
28#1956C2212yes—
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1939C2212yes—
32#1925—21no—
33#1925—21no—
34#1917—22no—
35#1874—21no—
36#1817—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1770C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C2×#340—21no—
42C2×#340—21no—
43C2×#335C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#331C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8—21no—
50C22×SD16—22no—
51C22×SD16—22no—
52C23×C8—21no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8C2312yes—
59C2×#127C2311yescharacteristic
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#492—24no—
70#492—42no—
71#480—44no—
72#448—44no—
73#448—44no—
74#432—44no—
75C2×#107—21no—
76C2×#107—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×#103—22no—
85C2×#103—22no—
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—21no—
89C2×#102—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8D411yescharacteristic
93C2×C22⋊C8D411yescharacteristic
94C2×#97—22no—
95C2×C4×C8C2312yes—
96#390—21no—
97#390—21no—
98#390—21no—
99#384C2311yescharacteristic
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4D411yescharacteristic
103C4×C4⋊C4D411yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105C4×C4⋊C4C2311yescharacteristic
106C2×#95—21no—
107#340—21no—
108#340—21no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#336—22no—
114#336—22no—
115#336—21no—
116#336—21no—
117#336D412yes—
118#335—42no—
119#332—24no—
120#332—42no—
121#331—42no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8—41no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—82no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—41no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—41no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4#4111yescharacteristic
138C2×SD16—44no—
139C2×SD16—44no—
140C2×SD16—44no—
141C2×SD16—44no—
142C2×SD16—82no—
143C2×SD16—82no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—42no—
149C22×C8—42no—
150C22×C8—42no—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C4⋊Q8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C4⋊Q8—42no—
160C4⋊Q8#4114yes—
161#127—22no—
162#127—22no—
163#121—42no—
164#121—42no—
165#121—42no—
166#121—42no—
167#121—82no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
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—44no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42C2411yescharacteristic
203#107—42no—
204#107—42no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—42no—
212C4⋊C8—44no—
213#103—44no—
214#103—44no—
215#102—42no—
216#102—42no—
217#102—42no—
218#102—42no—
219C22⋊C8—22no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222C22⋊C8D814yes—
223#97—44no—
224C4×C8—22no—
225C4×C8—22no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230C24—41no—
231C24#12111yescharacteristic
232C2×Q8—24no—
233C2×Q8—44no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—88no—
240C2×Q8—88no—
241C2×Q8—82no—
242C2×Q8—82no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—81no—
255C2×D4—81no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—82no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—81no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4C2×D812yes—
287C22×C4AGL(1,ℤ/8)12yes—
288C22×C4AGL(1,ℤ/8)12yes—
289C22×C4#13712yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4111yescharacteristic
293SD16—88no—
294SD16—88no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—84no—
313C2×C8C2×D814yes—
314C4⋊C4—21no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4#13712yes—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C42—21no—
339C42—21no—
340C42—22no—
341C42—22no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—42no—
347C42—42no—
348C42C2×#4112yes—
349C42H(5,𝔽2)12yes—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23#45912yes—
362C23#46812yes—
363C23#47112yes—
364C23C2×#12111yescharacteristic
365Q8—44no—
366Q8—44no—
367Q8—88no—
368Q8—82no—
369Q8—82no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—82no—
375D4—82no—
376D4—162no—
377D4—164no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—44no—
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—84no—
408C2×C4—88no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4#45912yes—
416C2×C4#46812yes—
417C2×C4#52511yescharacteristic
418C2×C4#53511yescharacteristic
419C2×C4#54612yes—
420C2×C4#55112yes—
421C2×C4C22×D811yescharacteristic
422C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
423C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
424C2×C4C2×#13711yescharacteristic
425C2×C4C2×#13711yescharacteristic
426C2×C4#57511yescharacteristic
427C2×C4#57712yes—
428C8—24no—
429C8—42no—
430C8—81no—
431C8—81no—
432C8—82no—
433C8—82no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22#168514yes—
448C22#241111yescharacteristic
449C22C2×#45911yescharacteristic
450C22#304311yescharacteristic
451C22C2×#46811yescharacteristic
452C22#305611yescharacteristic
453C22C2×#47111yescharacteristic
454C22#306911yescharacteristic
455C4—22no—
456C4—22no—
457C4—22no—
458C4—41no—
459C4—42no—
460C4—84no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—161no—
465C4#311211yescharacteristic
466C4#311711yescharacteristic
467C4#321311yescharacteristic
468C4#321811yescharacteristic
469C4#329412yes—
470C4#331012yes—
471C2—24no—
472C2—321no—
473C2C2×#168511yescharacteristic
474C2#1916311yescharacteristic
475C2#1943311yescharacteristic
476C2#1946911yescharacteristic
477C2#2447611yescharacteristic
478C2#2450911yescharacteristic
479C2#2452311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22603C2semidirect
4. C2×#1988C2semidirect
5. #10592C2semidirect
6. #10537C2semidirect
7. #9744C2semidirect
8. #9569C2semidirect
10. #8829C2semidirect
11. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#331062
#257182
#226032
C2×#19882
#105922
#105372
#97442
#95692
#91332
#88292
#83402