Finite Group Databasebeta

← #503641

Subgroups — #503641

Order 512.

693 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C211yesmaximal, characteristic, complemented
2C2×#2139C211yesmaximal, characteristic
3#26067C211yesmaximal, characteristic, complemented
4#22451C211yesmaximal, characteristic
5#13528C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13476C211yesmaximal, characteristic, complemented
8#13365C211yesmaximal, characteristic, complemented
9#12938C211yesmaximal, characteristic, complemented
10#12938C211yesmaximal, characteristic, complemented
11#12920C211yesmaximal, characteristic, complemented
12#12913C211yesmaximal, characteristic, complemented
13#12836C211yesmaximal, characteristic, complemented
14#12814C211yesmaximal, characteristic, complemented
15#12733C211yesmaximal, characteristic
16C22×#135C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2442—21no—
23#2341—21no—
24#2229—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31#2139—24no—
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41#2085—21no—
42#2059—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1880—21no—
48#1880—21no—
49#1853—21no—
50#1846C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1788—21no—
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×#41—21no—
62C2×#135—22no—
63C2×#135—22no—
64C2×#135—22no—
65C2×#135—41no—
66C2×#135—41no—
67C2×#135Q814yes—
68C22×Q16C2311yescharacteristic
69C22×SD16—21no—
70C22×D8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×#125—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C22×C4⋊C4—21no—
76C2×M32—22no—
77C2×M32—22no—
78C2×M32—41no—
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83#499—22no—
84#499—41no—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×Q16—41no—
88C4×Q16—41no—
89C2×#107—22no—
90C2×#107—21no—
91C2×#107—41no—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#106—21no—
99C2×#106—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×C22⋊C8D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#400—41no—
108#398—41no—
109#390—21no—
110#390—21no—
111#390—41no—
112#390—41no—
113#389—21no—
114#389—21no—
115#385—41no—
116#382—21no—
117#380—21no—
118#380—21no—
119#380—41no—
120#380—41no—
121C4×C22⋊C4—41no—
122C4×C22⋊C4D411yescharacteristic
123#367—22no—
124#367—41no—
125#367—41no—
126#366—22no—
127C4⋊C16—22no—
128C4⋊C16—41no—
129#358—22no—
130#358—22no—
131#358—22no—
132#358D414yes—
133#357—22no—
134#357—22no—
135#357—41no—
136#357—41no—
137C22⋊C16—22no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#336D411yescharacteristic
143#336D411yescharacteristic
144C2×#41—24no—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—42no—
148C2×#41—81no—
149C2×#41—81no—
150C22×Q8—21no—
151C22×D4—21no—
152C22×D4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156#135—28no—
157#135—44no—
158#135—44no—
159#135—44no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16C2×D412yes—
164C2×Q16C2×Q812yes—
165C2×Q16#4112yes—
166C2×SD16—24no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8C2×Q812yes—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×Q812yes—
183C22×C8#4112yes—
184C22×C8C2411yescharacteristic
185#125—42no—
186#125—81no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—81no—
190C4×Q8—81no—
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—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—81no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4D811yescharacteristic
213C2×C22⋊C4D811yescharacteristic
214C2×C42—41no—
215C2×C42—41no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218M32—44no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16C2×D412yes—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—21no—
236#107—21no—
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—42no—
243#107C2×D412yes—
244#106—42no—
245#106—81no—
246#103—42no—
247#103—42no—
248#102—81no—
249#102—81no—
250C22⋊C8—22no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95D811yescharacteristic
262#95D811yescharacteristic
263C24—41no—
264C24C2×D811yescharacteristic
265#41—48no—
266#41—84no—
267#41—84no—
268#41—84no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×D4—22no—
276C2×D4—44no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—82no—
283C2×D4—82no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—44no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—161no—
301C22×C4C22⋊Q812yes—
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4C2×D811yescharacteristic
305C22×C4C2×D811yescharacteristic
306C22×C4C2×D811yescharacteristic
307C22×C4#13712yes—
308C22×C4C22×D411yescharacteristic
309Q16—22no—
310Q16—22no—
311Q16—22no—
312Q16C22⋊Q814yes—
313SD16—48no—
314D8—22no—
315D8—22no—
316D8—41no—
317D8—41no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8C22⋊Q812yes—
331C2×C8C22×D411yescharacteristic
332C2×C8C22×Q811yescharacteristic
333C2×C8C2×#4111yescharacteristic
334C2×C8C2×#4111yescharacteristic
335C2×C8H(5,𝔽2)11yescharacteristic
336C2×C8H(5,𝔽2)11yescharacteristic
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
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—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—161no—
355C4⋊C4—161no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—161no—
362C22⋊C4—161no—
363C22⋊C4—161no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—82no—
369C42—82no—
370C16—22no—
371C16—41no—
372C16—41no—
373C16—41no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—42no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23#47412yes—
388C23C22×D811yescharacteristic
389Q8—24no—
390Q8—42no—
391Q8—42no—
392Q8—42no—
393D4—42no—
394D4—42no—
395D4—88no—
396D4—81no—
397D4—81no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—88no—
421C2×C4—81no—
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—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—164no—
439C2×C4#47412yes—
440C2×C4C2×C22⋊Q811yescharacteristic
441C2×C4D4211yescharacteristic
442C2×C4#54611yescharacteristic
443C2×C4C2×#13711yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57711yescharacteristic
446C8—22no—
447C8—41no—
448C8—41no—
449C8C2×C22⋊Q811yescharacteristic
450C8#54311yescharacteristic
451C8#54611yescharacteristic
452C8#55111yescharacteristic
453C22—22no—
454C22—22no—
455C22—22no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—161no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22C2×#47411yescharacteristic
468C22D4×D811yescharacteristic
469C22#329411yescharacteristic
470C22#341411yescharacteristic
471C22#341411yescharacteristic
472C22#341611yescharacteristic
473C22#341611yescharacteristic
474C4—22no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—44no—
479C4—41no—
480C4—82no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4#243711yescharacteristic
487C4#308111yescharacteristic
488C4#326611yescharacteristic
489C4#331411yescharacteristic
490C2—42no—
491C2—162no—
492C2#2595011yescharacteristic
493C2#3363011yescharacteristic
494C2#3363411yescharacteristic
495C2#3383711yescharacteristic
496C2#3384111yescharacteristic
497C2#3388411yescharacteristic
498C2#3389311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2229C2semidirect
3. #26067C2semidirect
5. #13528C2semidirect
6. #13476C2semidirect
7. #13476C2semidirect
8. #13365C2semidirect
9. #12938C2semidirect
10. #12938C2semidirect
11. #12920C2semidirect
12. #12913C2semidirect
13. #12836C2semidirect
14. #12814C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22292
C2×#21392
#260672
#224512
#135282
#134762
#133652
#129382
#129202
#129132
#128362
#128142
#127332