Finite Group Databasebeta

← #503642

Subgroups — #503642

Order 512.

729 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2226C211yesmaximal, characteristic, complemented
2C2×#2139C211yesmaximal, characteristic
3#26065C211yesmaximal, characteristic, complemented
4#22448C211yesmaximal, characteristic
5#13531C211yesmaximal, characteristic, complemented
6#13477C211yesmaximal, characteristic, complemented
7#13475C211yesmaximal, characteristic, complemented
8#13364C211yesmaximal, characteristic, complemented
9#12937C211yesmaximal, characteristic, complemented
10#12935C211yesmaximal, characteristic, complemented
11#12920C211yesmaximal, characteristic, complemented
12#12913C211yesmaximal, characteristic, complemented
13#12836C211yesmaximal, characteristic, complemented
14#12811C211yesmaximal, characteristic, complemented
15#12733C211yesmaximal, characteristic
16C22×#135C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2440—21no—
23#2339—21no—
24#2226—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#2083—21no—
42#2057—21no—
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1877—21no—
48#1877—21no—
49#1852—21no—
50#1845C2211yescharacteristic
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×#135Q814yescomplemented
68C22×Q16C2311yescharacteristic
69C22×SD16—21no—
70C22×D8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×C4⋊Q8—21no—
73C2×C22⋊Q8—41no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C22×C4⋊C4—21no—
77C2×M32—22no—
78C2×M32—22no—
79C2×M32—41no—
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83C22×C16C2311yescharacteristic
84#500—22no—
85#500—41no—
86C4×D8—22no—
87C4×D8—22no—
88C4×D8—41no—
89C4×D8—41no—
90C2×#107—22no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#107—41no—
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C22⋊C8D411yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#398—41no—
109#395—41no—
110#391—21no—
111#391—21no—
112#389—21no—
113#389—21no—
114#386—41no—
115#386—41no—
116#385—41no—
117#384—21no—
118#381—41no—
119#380—21no—
120#380—21no—
121C4×C22⋊C4D411yescharacteristic
122C2×#95—41no—
123#367—22no—
124#367—22no—
125#367—41no—
126#366—41no—
127C4⋊C16—22no—
128C4⋊C16—41no—
129#358—22no—
130#358—22no—
131#358—41no—
132#358—41no—
133#357—22no—
134#357—22no—
135#357—22no—
136#357D414yes—
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×Q8—41no—
152C22×D4—21no—
153C22×D4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159#135—28no—
160#135—44no—
161#135—44no—
162#135—44no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16C2×Q812yes—
169C2×SD16—24no—
170C2×SD16—42no—
171C2×SD16—42no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8C2×D412yes—
176C2×D8C2×Q812yes—
177C2×D8#4112yes—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×Q812yes—
186C22×C8#4112yes—
187C22×C8C2411yescharacteristic
188C4⋊Q8—42no—
189C4⋊Q8—81no—
190C22⋊Q8—84no—
191C4×D4—42no—
192C4×D4—42no—
193C4×D4—81no—
194C4×D4—81no—
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—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—81no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4D811yescharacteristic
214C2×C22⋊C4D811yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217M32—44no—
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×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16C2×D412yes—
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—21no—
239#107—21no—
240#107—21no—
241#107—42no—
242#107—42no—
243#107—81no—
244#107C2×D412yes—
245#103—81no—
246#103—81no—
247#102—42no—
248#102—42no—
249C22⋊C8—22no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95D811yescharacteristic
263#95D811yescharacteristic
264C24—41no—
265C24C2×D811yescharacteristic
266#41—48no—
267#41—84no—
268#41—84no—
269#41—84no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—82no—
277C2×Q8—82no—
278C2×Q8—161no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—82no—
287C2×D4—82no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—161no—
309C22×C4—161no—
310C22×C4—161no—
311C22×C4C22⋊Q812yes—
312C22×C4C2×D811yescharacteristic
313C22×C4C2×D811yescharacteristic
314C22×C4C2×D811yescharacteristic
315C22×C4C2×D811yescharacteristic
316C22×C4C2×D811yescharacteristic
317C22×C4#13712yes—
318C22×C4C22×D411yescharacteristic
319Q16—22no—
320Q16—22no—
321Q16—41no—
322Q16—41no—
323SD16—48no—
324D8—22no—
325D8—22no—
326D8—22no—
327D8C22⋊Q814yescomplemented
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8C22⋊Q812yes—
341C2×C8C22×D411yescharacteristic
342C2×C8C22×Q811yescharacteristic
343C2×C8C2×#4111yescharacteristic
344C2×C8C2×#4111yescharacteristic
345C2×C8H(5,𝔽2)11yescharacteristic
346C2×C8H(5,𝔽2)11yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—161no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—161no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C16—22no—
376C16—41no—
377C16—41no—
378C16—41no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23#47412yes—
393C23C22×D811yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—81no—
397Q8—81no—
398Q8—164no—
399D4—24no—
400D4—42no—
401D4—42no—
402D4—42no—
403D4—88no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—88no—
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—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—164no—
443C2×C4—164no—
444C2×C4—164no—
445C2×C4#47412yes—
446C2×C4#50912yes—
447C2×C4#51012yes—
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4D4211yescharacteristic
450C2×C4#54611yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4#57611yescharacteristic
453C2×C4#57711yescharacteristic
454C8—22no—
455C8—41no—
456C8—41no—
457C8C2×C22⋊Q811yescharacteristic
458C8#54311yescharacteristic
459C8#54611yescharacteristic
460C8#55111yescharacteristic
461C22—22no—
462C22—22no—
463C22—22no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—44no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22C2×#47411yescharacteristic
476C22D4×D811yescharacteristic
477C22#329411yescharacteristic
478C22C2×#50911yescharacteristic
479C22C2×#51011yescharacteristic
480C22#341611yescharacteristic
481C22#341611yescharacteristic
482C4—22no—
483C4—22no—
484C4—22no—
485C4—41no—
486C4—41no—
487C4—82no—
488C4—82no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4#243711yescharacteristic
495C4#308111yescharacteristic
496C4#326611yescharacteristic
497C4#331411yescharacteristic
498C2—42no—
499C2—84no—
500C2#2595011yescharacteristic
501C2#3362911yescharacteristic
502C2#3363411yescharacteristic
503C2#3383211yescharacteristic
504C2#3383811yescharacteristic
505C2#3387311yescharacteristic
506C2#3389811yescharacteristic
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×#2226C2semidirect
3. #26065C2semidirect
5. #13531C2semidirect
6. #13477C2semidirect
7. #13475C2semidirect
8. #13364C2semidirect
9. #12937C2semidirect
10. #12935C2semidirect
11. #12920C2semidirect
12. #12913C2semidirect
13. #12836C2semidirect
14. #12811C2semidirect
67. C2×#135Q8semidirect
327. D8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22262
C2×#21392
#260652
#224482
#135312
#134772
#134752
#133642
#129372
#129352
#129202
#129132
#128362
#128112
#127332