Finite Group Databasebeta

← #503643

Subgroups — #503643

Order 512.

677 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2227C211yesmaximal, characteristic, complemented
2C2×#2141C211yesmaximal, characteristic
3#26064C211yesmaximal, characteristic
4C22⋊C4×Q16C211yesmaximal, characteristic
5#13529C211yesmaximal, characteristic, complemented
6#13477C211yesmaximal, characteristic, complemented
7#13475C211yesmaximal, characteristic, complemented
8#13364C211yesmaximal, characteristic, complemented
9#12938C211yesmaximal, characteristic, complemented
10#12938C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic
12#12913C211yesmaximal, characteristic
13#12836C211yesmaximal, characteristic
14#12814C211yesmaximal, characteristic, complemented
15#12732C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2439—21no—
23Q8×C22⋊C4—21no—
24#2227—24no—
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31#2141—24no—
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41#2085—21no—
42#2058—21no—
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1878—21no—
48#1878—21no—
49#1854—21no—
50#1846C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1787—21no—
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C23×Q8—21no—
62C22×Q16—22no—
63C22×Q16—22no—
64C22×Q16—21no—
65C22×Q16—22no—
66C22×Q16—41no—
67C22×Q16—41no—
68C22×Q16Q814yescomplemented
69C22×Q16C2311yescharacteristic
70C22×Q16C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×C4⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×#117—21no—
76C2×M32—22no—
77C2×M32—22no—
78C2×M32—41no—
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83#500—22no—
84#500—41no—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×Q16—41no—
88C4×Q16—41no—
89#427—22no—
90#427—41no—
91C2×#107—21no—
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×#103—21no—
103C2×#103—21no—
104C2×C22⋊C8D411yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#396—41no—
108#393—41no—
109#391—21no—
110#391—21no—
111#391—41no—
112#391—41no—
113#391—41no—
114#389—21no—
115#389—21no—
116#387—41no—
117#382—21no—
118#380—21no—
119#380—21no—
120#380—41no—
121C4×C22⋊C4—41no—
122C4×C22⋊C4D411yescharacteristic
123#367—41no—
124#367—41no—
125#366—22no—
126#366—22no—
127C4⋊C16—22no—
128C4⋊C16—41no—
129#358—22no—
130#358—22no—
131#358—22no—
132#358—22no—
133#358—22no—
134#358—41no—
135#358—41no—
136#358D414yes—
137C22⋊C16—22no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#336D411yescharacteristic
143#336D411yescharacteristic
144C22×Q8—24no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×Q8—42no—
152C22×Q8—81no—
153C22×Q8—81no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C2×Q16—28no—
157C2×Q16—24no—
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—44no—
167C2×Q16—44no—
168C2×Q16—44no—
169C2×Q16—42no—
170C2×Q16—42no—
171C2×Q16C2×D412yes—
172C2×Q16C2×Q812yes—
173C2×Q16C2×Q812yes—
174C2×Q16#4112yes—
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
185C4⋊Q8—42no—
186C4⋊Q8—81no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—81no—
190C4×Q8—81no—
191#117—42no—
192#117—81no—
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—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4D811yescharacteristic
208C2×C22⋊C4D811yescharacteristic
209C2×C42—41no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214M32—44no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16C2×D412yes—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—21no—
236#107—21no—
237#107—21no—
238#107—42no—
239#107C2×D412yes—
240#106—42no—
241#106—81no—
242#103—42no—
243#103—42no—
244#103—81no—
245#103—81no—
246C22⋊C8—22no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95D811yescharacteristic
262#95D811yescharacteristic
263C24C2×D811yescharacteristic
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—48no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—44no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—82no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—82no—
285C2×Q8—161no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—44no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—82no—
300C22×C4—82no—
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—
312Q16—22no—
313Q16—22no—
314Q16—48no—
315Q16—41no—
316Q16—41no—
317Q16C22⋊Q814yes—
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—161no—
354C22⋊C4—22no—
355C22⋊C4—41no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—161no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—82no—
364C42—82no—
365C42—82no—
366C16—22no—
367C16—41no—
368C16—41no—
369C16—41no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23#47412yes—
377C23C22×D811yescharacteristic
378Q8—24no—
379Q8—42no—
380Q8—42no—
381Q8—42no—
382Q8—42no—
383Q8—42no—
384Q8—88no—
385Q8—81no—
386Q8—81no—
387Q8—164no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—88no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
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#47412yes—
431C2×C4#51012yes—
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4D4211yescharacteristic
434C2×C4#54611yescharacteristic
435C2×C4C2×#13711yescharacteristic
436C2×C4#57611yescharacteristic
437C2×C4#57711yescharacteristic
438C8—22no—
439C8—41no—
440C8—41no—
441C8C2×C22⋊Q811yescharacteristic
442C8#54311yescharacteristic
443C8#54611yescharacteristic
444C8#55111yescharacteristic
445C22—22no—
446C22—22no—
447C22—22no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22C2×#47411yescharacteristic
453C22D4×D811yescharacteristic
454C22#329411yescharacteristic
455C22C2×#51011yescharacteristic
456C22#341411yescharacteristic
457C22#341611yescharacteristic
458C22#341711yescharacteristic
459C4—22no—
460C4—22no—
461C4—41no—
462C4—41no—
463C4—44no—
464C4—41no—
465C4—82no—
466C4—82no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4#243711yescharacteristic
473C4#308111yescharacteristic
474C4#326611yescharacteristic
475C4#331411yescharacteristic
476C2—42no—
477C2#2595011yescharacteristic
478C2#3362811yescharacteristic
479C2#3363211yescharacteristic
480C2#3383611yescharacteristic
481C2#3384111yescharacteristic
482C2#3388011yescharacteristic
483C2#3389611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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×#2227C2semidirect
5. #13529C2semidirect
6. #13477C2semidirect
7. #13475C2semidirect
8. #13364C2semidirect
9. #12938C2semidirect
10. #12938C2semidirect
14. #12814C2semidirect
68. C22×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22272
C2×#21412
#260642
C22⋊C4×Q162
#135292
#134772
#134752
#133642
#129382
#129212
#129132
#128362
#128142
#127322