Finite Group Databasebeta

← #279639

Subgroups — #279639

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29980C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic, complemented
3#22651C211yesmaximal, characteristic
4#22111C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10655C211yesmaximal, characteristic, complemented
7#10593C211yesmaximal, characteristic, complemented
8#10589C211yesmaximal, characteristic, complemented
9#9898C211yesmaximal, characteristic, complemented
10#9603C211yesmaximal, characteristic, complemented
11#9223C211yesmaximal, characteristic
12#9215C211yesmaximal, characteristic
13#8656C211yesmaximal, characteristic
14#8648C211yesmaximal, characteristic
15#8590C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#424—21no—
20#2868C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22#2293C2211yescharacteristic
23#2100—21no—
24#2100—21no—
25#2099—21no—
26#2093—21no—
27#2057—21no—
28#2023—21no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1987—21no—
32#1987—21no—
33#1987C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1941—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1934—21no—
41#1926—21no—
42#1919—21no—
43#1851—21no—
44#1850—21no—
45#1842C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1831C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1798—21no—
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76#476—22no—
77#476Q814yescomplemented
78#436—22no—
79#436—41no—
80C8×D4—22no—
81C8×D4—41no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#106—21no—
85C2×#106—21no—
86#424—44no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8D411yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391—21no—
109#390D411yescharacteristic
110#390D411yescharacteristic
111#389—21no—
112#389—21no—
113#389D411yescharacteristic
114#389D411yescharacteristic
115#388—21no—
116#388—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C43C2311yescharacteristic
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151#335—41no—
152#334—41no—
153#333—41no—
154#333—41no—
155C22×Q8—21no—
156C22×Q8—41no—
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—81no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—81no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—81no—
181C4⋊Q8—81no—
182C4⋊Q8—82no—
183C4⋊Q8C2×Q812yes—
184C4×Q8—82no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4C2×Q812yes—
188C4×D4#4112yes—
189#117—42no—
190#117—42no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C2×C42—41no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42C2411yescharacteristic
225#107—81no—
226#107—81no—
227#106—24no—
228#106—81no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—21no—
232C4⋊C8—21no—
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—42no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245C4⋊C8—81no—
246C4⋊C8C2×Q812yes—
247#102—24no—
248#102—81no—
249#102—81no—
250#102—81no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—81no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263C24—21no—
264C2×Q8—22no—
265C2×Q8—41no—
266C2×Q8—82no—
267C2×Q8—82no—
268C2×Q8—81no—
269C2×Q8—81no—
270C2×Q8—82no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4C22⋊Q812yes—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4111yescharacteristic
297M16—84no—
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—84no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4C22⋊Q812yes—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—81no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—22no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—82no—
377C42#13512yes—
378C42AGL(1,ℤ/8)12yes—
379C42C22×Q811yescharacteristic
380C42C2×#4111yescharacteristic
381C42H(5,𝔽2)11yescharacteristic
382C422−1+411yescharacteristic
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23C2×#12111yescharacteristic
391Q8—82no—
392Q8—82no—
393Q8—164no—
394D4—22no—
395D4—22no—
396D4—41no—
397D4—41no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4#46312yes—
435C2×C4#46312yes—
436C2×C4#47112yes—
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4#53711yescharacteristic
439C2×C4#54611yescharacteristic
440C2×C4D4×Q811yescharacteristic
441C2×C4#55211yescharacteristic
442C2×C4#55411yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C2×C4#57811yescharacteristic
450C2×C4#57811yescharacteristic
451C8—42no—
452C8—41no—
453C8—41no—
454C8—41no—
455C8—41no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C8—81no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22#241711yescharacteristic
469C22C2×#46311yescharacteristic
470C22#304611yescharacteristic
471C22#305911yescharacteristic
472C22#306211yescharacteristic
473C22C2×#47111yescharacteristic
474C22#306811yescharacteristic
475C4—21no—
476C4—21no—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—41no—
481C4—42no—
482C4—82no—
483C4—81no—
484C4—81no—
485C4—161no—
486C4—161no—
487C4—161no—
488C4#308211yescharacteristic
489C4#308311yescharacteristic
490C4#325811yescharacteristic
491C4#326211yescharacteristic
492C4#330211yescharacteristic
493C4#332411yescharacteristic
494C4#338111yescharacteristic
495C4#338611yescharacteristic
496C2—82no—
497C2#1919911yescharacteristic
498C2#1923411yescharacteristic
499C2#1954711yescharacteristic
500C2#1958911yescharacteristic
501C2#2481611yescharacteristic
502C2#2484911yescharacteristic
503C2#2485611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
2. #29218C2semidirect
5. #10656C2semidirect
6. #10655C2semidirect
7. #10593C2semidirect
8. #10589C2semidirect
9. #9898C2semidirect
10. #9603C2semidirect
15. #8590C2semidirect
77. #476Q8semidirect

Maximal subgroups

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

subgroupindex
#299802
#292182
#226512
#221112
#106562
#106552
#105932
#105892
#98982
#96032
#92232
#92152
#86562
#86482
#85902