Finite Group Databasebeta

← #279608

Subgroups — #279608

Order 512.

591 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29998C211yesmaximal, characteristic
2#29207C211yesmaximal, characteristic, complemented
3#22651C211yesmaximal, characteristic
4#22111C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10652C211yesmaximal, characteristic, complemented
7#10597C211yesmaximal, characteristic, complemented
8#10597C211yesmaximal, characteristic, complemented
9#9900C211yesmaximal, characteristic, complemented
10#9602C211yesmaximal, characteristic, complemented
11#9223C211yesmaximal, characteristic
12#9215C211yesmaximal, characteristic
13#8652C211yesmaximal, characteristic
14#8642C211yesmaximal, characteristic
15#8587C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#424—21no—
20#2859C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22#2293C2211yescharacteristic
23#2088—21no—
24#2088—21no—
25#2065—21no—
26#2065—21no—
27#2059—21no—
28#2059—21no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1987C2211yescharacteristic
32#1949—21no—
33#1949—21no—
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1941—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1934—21no—
41#1926—21no—
42#1919—21no—
43#1851—21no—
44#1850—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1842C2211yescharacteristic
48#1842C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1798—21no—
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#334C2211yescharacteristic
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⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×C4×Q8—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74#476—22no—
75#476Q814yes—
76#436—22no—
77#436—41no—
78C8×D4—22no—
79C8×D4—41no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88#424—44no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8D411yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391—21no—
113#390D411yescharacteristic
114#390D411yescharacteristic
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389D411yescharacteristic
122#389D411yescharacteristic
123#388—21no—
124#388—21no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#382—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C43C2311yescharacteristic
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#340—21no—
145#340—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151#334—41no—
152#334—41no—
153#333—41no—
154#333—41no—
155C22×Q8—21no—
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C2×M16—42no—
159C2×M16—42no—
160C2×M16—81no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—81no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8C2×Q812yes—
178#125—81no—
179#125—81no—
180C4×Q8—82no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4C2×Q812yes—
184C4×D4#4112yes—
185#117—42no—
186#117—42no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
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—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C42—21no—
212C2×C42—41no—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#107—81no—
221#107—81no—
222#106—24no—
223#106—81no—
224#106—81no—
225#106—81no—
226#106—81no—
227#106—81no—
228C4⋊C8—22no—
229C4⋊C8—21no—
230C4⋊C8—21no—
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—42no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8—81no—
244C4⋊C8—81no—
245C4⋊C8—81no—
246C4⋊C8SD1614yes—
247C4⋊C8C2×Q812yes—
248#102—24no—
249#102—81no—
250#102—81no—
251#102—81no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—81no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266C24—21no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—81no—
270C2×Q8—81no—
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—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—41no—
310C2×C8—41no—
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—84no—
327C2×C8C2×SD1612yes—
328C2×C8C2×SD1612yes—
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⋊C4C22⋊Q812yes—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—81no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—82no—
376C42C2×SD1612yes—
377C42#13712yes—
378C42C22×Q811yescharacteristic
379C42C2×#4111yescharacteristic
380C42H(5,𝔽2)11yescharacteristic
381C422−1+411yescharacteristic
382C23—22no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23C2×#12111yescharacteristic
390Q8—82no—
391Q8—82no—
392D4—22no—
393D4—22no—
394D4—41no—
395D4—41no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—81no—
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#46112yes—
434C2×C4#46112yes—
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4#53711yescharacteristic
437C2×C4#54611yescharacteristic
438C2×C4D4×Q811yescharacteristic
439C2×C4#55211yescharacteristic
440C2×C4#55411yescharacteristic
441C2×C4C22×SD1611yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4#57611yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C8—22no—
450C8—22no—
451C8—41no—
452C8—41no—
453C8—42no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22#241711yescharacteristic
467C22C2×#46111yescharacteristic
468C22#304511yescharacteristic
469C22#306011yescharacteristic
470C22#306311yescharacteristic
471C22#306711yescharacteristic
472C22#306811yescharacteristic
473C4—21no—
474C4—21no—
475C4—22no—
476C4—22no—
477C4—41no—
478C4—41no—
479C4—42no—
480C4—82no—
481C4—81no—
482C4—81no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4#308211yescharacteristic
487C4#308411yescharacteristic
488C4#325811yescharacteristic
489C4#326011yescharacteristic
490C4#329511yescharacteristic
491C4#333111yescharacteristic
492C4Q8×SD1611yescharacteristic
493C4#338511yescharacteristic
494C2—82no—
495C2#1919411yescharacteristic
496C2#1923311yescharacteristic
497C2#1954011yescharacteristic
498C2#1957711yescharacteristic
499C2#2481511yescharacteristic
500C2#2485411yescharacteristic
501C2#2485911yescharacteristic
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. #29207C2semidirect
5. #10654C2semidirect
6. #10652C2semidirect
7. #10597C2semidirect
8. #10597C2semidirect
9. #9900C2semidirect
10. #9602C2semidirect
15. #8587C2semidirect

Maximal subgroups

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

subgroupindex
#299982
#292072
#226512
#221112
#106542
#106522
#105972
#99002
#96022
#92232
#92152
#86522
#86422
#85872