Finite Group Databasebeta

← #195256

Subgroups — #195256

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic
2#34070C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5#14274C211yesmaximal, characteristic, complemented
6#9043C218yesmaximal, complemented
7#8026C211yesmaximal, characteristic, complemented
8#7906C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41—21no—
12#3226—24no—
13#3226C418yescomplemented
14C2×#500—21no—
15C2×#486—21no—
16C2×#480C2214yes—
17C2×#474—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2328—21no—
21#2327—21no—
22#2326—22no—
23#2325—22no—
24#2325C2214yes—
25#2297—21no—
26C2×#391—21no—
27C2×#391—21no—
28#2290—22no—
29#2290—22no—
30#2285C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1990—22no—
34#1988—22no—
35#1823C2214yes—
36#1517—24no—
37C2×#327C2214yes—
38#1445—22no—
39#1445C2214yes—
40#1294C2214yes—
41#1280C2214yes—
42C22×#41—21no—
43C23×Q8—21no—
44#532—22no—
45#532—22no—
46#532C2×C414yes—
47#532D414yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yescomplemented
56C2×C4⋊Q8D411yescharacteristic
57C2×#127D411yescharacteristic
58C2×#124—21no—
59C2×#121—21no—
60C2×#121C2312yes—
61C4×#41—44no—
62C4×#41—44no—
63C2×C4×Q8—22no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4—21no—
67C2×C4×D4D412yes—
68C2×#117—22no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2312yes—
73C22×C42—22no—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77#500—28no—
78#486—28no—
79#480—28no—
80#480C2×C4116yescomplemented
81#474—216no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431C2×C414yescomplemented
86C2×#107—24no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×#102—24no—
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2312yes—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#391—22no—
95#391—44no—
96#391—44no—
97#390—22no—
98#388—24no—
99#388—24no—
100#387—24no—
101#387—24no—
102#386—24no—
103#386—24no—
104#386—24no—
105#385—24no—
106#385—24no—
107#384D412yes—
108#382—21no—
109#382—22no—
110#382—22no—
111#380—21no—
112#380C2312yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117#376—21no—
118#376—21no—
119#376—21no—
120#376—21no—
121C2×#95—21no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95D412yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127#340—28no—
128#327—28no—
129C2×#41—24no—
130C2×#41—24no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—42no—
135C22×Q8—24no—
136C22×Q8—21no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C2×D412yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—22no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8C22⋊C414yes—
169#127—22no—
170#127C22⋊C414yes—
171#124—24no—
172#124—24no—
173#121—24no—
174#121—24no—
175#121—22no—
176#121—22no—
177#121C22×C414yes—
178#121C2×D414yes—
179C4×Q8—48no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—48no—
184C4×D4—48no—
185C4×D4—48no—
186C4×D4C22⋊C418yes—
187#117—48no—
188#117—42no—
189#117—42no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D414yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—48no—
225C2×C42D814yes—
226C2×C42C22×C412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4112yes—
233C2×C42#4112yes—
234#107—216no—
235C4⋊C8—24no—
236C4⋊C8—42no—
237#102—216no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8C22×C414yes—
242C4×C8—24no—
243C4×C8—42no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—48no—
255#95—42no—
256#95—48no—
257C24—22no—
258C24C22×D411yescharacteristic
259#41—48no—
260#41—48no—
261#41—48no—
262#41—48no—
263C2×Q8—24no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—82no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—84no—
275C2×Q8—84no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—44no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—82no—
289C2×D4C2×C22⋊C414yes—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—28no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—48no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4C4×D412yes—
322C22×C4C4×D412yes—
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4C2×D814yes—
330C22×C4C2×D812yes—
331C22×C4#13512yes—
332C22×C4AGL(1,ℤ/8)12yes—
333C22×C4AGL(1,ℤ/8)12yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C2×C8—24no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8—42no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—82no—
356C4⋊C4—84no—
357C4⋊C4C2×C22⋊C414yes—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—28no—
361C22⋊C4—48no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—84no—
365C22⋊C4—84no—
366C42—24no—
367C42—24no—
368C42—48no—
369C42—44no—
370C42—42no—
371C42—42no—
372C42C2×C22⋊C412yes—
373C42C2×C22⋊C412yes—
374C42C4×D412yes—
375C42C4×D412yes—
376C42#12112yes—
377C42#12112yes—
378C42#12312yes—
379C42#12312yes—
380C23—22no—
381C23—22no—
382C23—24no—
383C23—42no—
384C23—42no—
385C23—44no—
386C23—44no—
387C23C2×C4×D411yescharacteristic
388C23C2×#12011yescharacteristic
389C23C2×#12111yescharacteristic
390C23C22×D811yescharacteristic
391C23C2×#13511yescharacteristic
392C23C2×AGL(1,ℤ/8)11yescharacteristic
393C23C2×AGL(1,ℤ/8)11yescharacteristic
394Q8—48no—
395Q8—88no—
396Q8—88no—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—48no—
403D4—82no—
404D4—84no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—416no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—82no—
434C2×C4—816no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#38612yes—
446C2×C4#38612yes—
447C2×C4#38612yes—
448C2×C4#38612yes—
449C2×C4C4×D814yes—
450C2×C4#44214yes—
451C2×C4#44712yes—
452C2×C4#44712yes—
453C2×C4#44912yes—
454C2×C4#44912yes—
455C2×C4#45914yes—
456C2×C4#46314yes—
457C2×C4C22×C22⋊C411yescharacteristic
458C2×C4C2×C4×D411yescharacteristic
459C2×C4C2×#12111yescharacteristic
460C2×C4C2×#12311yescharacteristic
461C8—44no—
462C8—82no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—44no—
472C22—44no—
473C22—44no—
474C22—84no—
475C22—84no—
476C22C2×#38611yescharacteristic
477C22C2×C4×D811yescharacteristic
478C22C2×#44211yescharacteristic
479C22C2×#44711yescharacteristic
480C22C2×#44911yescharacteristic
481C22#302212yes—
482C22#302812yes—
483C22C2×#45911yescharacteristic
484C22C2×#46311yescharacteristic
485C4—22no—
486C4—22no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—44no—
492C4—44no—
493C4—84no—
494C4—82no—
495C4—84no—
496C4—162no—
497C4#187914yes—
498C4#187914yes—
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—84no—
504C2#1830812yes—
505C2#1831212yes—
506C2C2×#187911yescharacteristic
507C2#2094411yescharacteristic
508C2#2096211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21833C2semidirect
5. #14274C2semidirect
6. #9043C2semidirect
7. #8026C2semidirect
8. #7906C2semidirect
13. #3226C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
80. #480C2×C4semidirect
85. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#340702
#218332
C2×#15172
#142742
#90432
#80262
#79062