Finite Group Databasebeta

← #193501

Subgroups — #193501

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2990C211yesmaximal, characteristic
2#34971C211yesmaximal, characteristic
3#20718C211yesmaximal, characteristic
4C2×#1487C211yesmaximal, characteristic, complemented
5C2×#1479C211yesmaximal, characteristic, complemented
6#8941C214yesmaximal, complemented
7#8937C214yesmaximal, complemented
8#7899C211yesmaximal, characteristic, complemented
9#7894C211yesmaximal, characteristic, complemented
10C2×C4×#41C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2990—24no—
13#2990—24no—
14C2×#446—21no—
15C2×#435—22no—
16C2×C8×D4—21no—
17C2×#432C2212yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2212yes—
23#2599—22no—
24#2581C412yes—
25#2536C412yes—
26#2486C412yes—
27#2484—21no—
28#2409C412yes—
29#2406—21no—
30#2399—21no—
31#2398—21no—
32C4×C22⋊Q8C2214yes—
33C4×#123—22no—
34#2283C2211yescharacteristic
35#1818—21no—
36#1798—22no—
37#1780—21no—
38C4×C22⋊C8C2214yes—
39#1487—24no—
40#1479—24no—
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2214yes—
44#1279C2212yes—
45#1279C2212yes—
46#1279C2214yes—
47C22×#41D411yescharacteristic
48C23×Q8C2×C411yescharacteristic
49C22×M16—21no—
50C23×C8—22no—
51C2×#123—22no—
52C2×C22⋊Q8C2×C414yes—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C4×#41—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8C2312yes—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D412yes—
71C2×#117D412yes—
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#446—28no—
79#435—216no—
80C8×D4—28no—
81#432—24no—
82#432—24no—
83#431—22no—
84#431—22no—
85#431—22no—
86#431—22no—
87#431—22no—
88#431D414yes—
89#423—22no—
90#423D414yes—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2311yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×#97—22no—
106C2×C4×C8—21no—
107C2×C4×C8C2312yes—
108C2×C4×C8C2312yes—
109#400—22no—
110#398—22no—
111#398—22no—
112#397—22no—
113#395—22no—
114#395—22no—
115#394—22no—
116#393—22no—
117#391C2×C414yes—
118#389C2×C414yes—
119#388—24no—
120#387—24no—
121#387—24no—
122#386—22no—
123#385—24no—
124#385—22no—
125#383—21no—
126#382C2×C412yes—
127#381—21no—
128#381—22no—
129#380C2×C412yes—
130#380C2×C414yes—
131C4×C4⋊C4—24no—
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—21no—
135#376—21no—
136#336—24no—
137Q8⋊C8—24no—
138Q8⋊C8—24no—
139Q8⋊C8—28no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—44no—
143C22×Q8—21no—
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22×C412yes—
150C22×D4C22⋊C411yescharacteristic
151C22×D4C22⋊C412yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×M16—24no—
161C2×M16—82no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—28no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—84no—
169#123—48no—
170C22⋊Q8—28no—
171C22⋊Q8—44no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194#117—22no—
195#117—22no—
196#117—24no—
197#117—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C414yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42C22×C412yes—
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42C2411yescharacteristic
239C4⋊C8—28no—
240C4⋊C8—24no—
241C4⋊C8—21no—
242C4⋊C8—22no—
243C4⋊C8—21no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8C2×D412yes—
247C4⋊C8C2×D414yes—
248C4⋊C8C2×D412yes—
249C22⋊C8—28no—
250C22⋊C8—24no—
251C22⋊C8—48no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—28no—
255C4×C8—24no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—22no—
260#95—22no—
261#95—22no—
262#95—24no—
263#95—22no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—22no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C412yes—
270#41—48no—
271#41—48no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—28no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—48no—
282C2×Q8—44no—
283C2×Q8—42no—
284C2×Q8#13114yes—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—42no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C4×D412yes—
321C22×C4C4×D412yes—
322C22×C4#12012yes—
323C22×C4#12112yes—
324C22×C4#12112yes—
325C22×C4#12312yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329M16—84no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—44no—
335C2×C8—42no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—88no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—48no—
359C22⋊C4—44no—
360C22⋊C4—48no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—44no—
372C42C4×D412yes—
373C42C4×D412yes—
374C42#12012yes—
375C42#12112yes—
376C42#12112yes—
377C42#12312yes—
378C42C22×D411yescharacteristic
379C42C22×D411yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C23—24no—
383C23—22no—
384C23—24no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23#38612yes—
390C23#42114yes—
391C23C22×C22⋊C411yescharacteristic
392Q8—24no—
393Q8—24no—
394Q8—48no—
395Q8—44no—
396Q8—42no—
397Q8—42no—
398D4—28no—
399D4—48no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—84no—
437C2×C4#38612yes—
438C2×C4#38612yes—
439C2×C4#38612yes—
440C2×C4C2×C4×D411yescharacteristic
441C2×C4C2×C4×D411yescharacteristic
442C2×C4C2×#12011yescharacteristic
443C2×C4C2×#12111yescharacteristic
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×#12311yescharacteristic
446C2×C4C2×#13112yes—
447C2×C4#56812yes—
448C2×C4#57712yes—
449C2×C4#57812yes—
450C8—42no—
451C8—44no—
452C8—42no—
453C8—82no—
454C8—82no—
455C22—28no—
456C22—22no—
457C22—22no—
458C22—24no—
459C22—22no—
460C22—44no—
461C22—44no—
462C22—42no—
463C22—42no—
464C22C2×#38611yescharacteristic
465C22#288012yes—
466C22#289812yes—
467C22C2×#42112yes—
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—22no—
473C4—42no—
474C4—44no—
475C4—42no—
476C4—42no—
477C4—84no—
478C4—82no—
479C4#292911yescharacteristic
480C4#293011yescharacteristic
481C4#293312yes—
482C4#300911yescharacteristic
483C4#301111yescharacteristic
484C4#309712yes—
485C2—44no—
486C2—42no—
487C2#1752414yes—
488C2#2089911yescharacteristic
489C2#2090211yescharacteristic
490C2#2091311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1487C2semidirect
5. C2×#1479C2semidirect
6. #8941C2semidirect
7. #8937C2semidirect
8. #7899C2semidirect
9. #7894C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29902
#349712
#207182
C2×#14872
C2×#14792
#89412
#89372
#78992
#78942