Finite Group Databasebeta

← #190881

Subgroups — #190881

Order 512.

983 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
1#34125C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#8711C212yesmaximal, complemented
5#8567C212yesmaximal, complemented
6#8564C212yesmaximal, complemented
7#8312C212yesmaximal
8#8183C211yesmaximal, characteristic, complemented
9#8177C211yesmaximal, characteristic, complemented
10#8175C211yesmaximal, characteristic, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#433—21no—
14#2357—22no—
15#2357C412yes—
16#2351C412yes—
17#2347—22no—
18#2347C412yes—
19#2341C412yes—
20#2293—21no—
21#2293—21no—
22#2293C2211yescharacteristic
23#2293C2211yescharacteristic
24#2283—21no—
25#2283C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27C42×Q8C2211yescharacteristic
28C4×#117—21no—
29C4×#117C2211yescharacteristic
30#1923—22no—
31#1916—22no—
32#1842C2212yes—
33#1832—21no—
34#1798—21no—
35#1797C2212yes—
36#1768—21no—
37#1767C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1744C2211yescharacteristic
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×#41C2×C411yescharacteristic
52C4×#41—24no—
53C4×#41—24no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117C2×C411yescharacteristic
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#433—44no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81C2×C4×C8C2312yes—
82C2×C4×C8C2312yes—
83#390—22no—
84#390—22no—
85#390C2×C412yes—
86#390C2×C412yes—
87#389—22no—
88#389—22no—
89#389C2×C412yes—
90#389C2×C412yes—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#383—21no—
98#383—22no—
99#382—22no—
100#382C2×C412yes—
101#381—21no—
102#381—22no—
103#380—22no—
104#380C2×C412yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2311yescharacteristic
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4D412yes—
124#376—21no—
125#376D411yescharacteristic
126#376D411yescharacteristic
127C43—21no—
128C43—21no—
129C43C2311yescharacteristic
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137Q8⋊C8—24no—
138Q8⋊C8—24no—
139D4⋊C8—24no—
140D4⋊C8—24no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—24no—
144C2×#41—42no—
145C22×Q8C22×C411yescharacteristic
146C22×D4—21no—
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8#4114yes—
166C4×D4—28no—
167C4×D4—28no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4#4114yes—
172#117—28no—
173#117—22no—
174#117—22no—
175#117—24no—
176#117—22no—
177#117—22no—
178#117—44no—
179#117—42no—
180#117—42no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C42—28no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2411yescharacteristic
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—44no—
232C4⋊C8—44no—
233C4⋊C8—44no—
234C22⋊C8—44no—
235C22⋊C8—44no—
236#97—42no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—44no—
242C4×C8—42no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—21no—
257C24C2×C22⋊C411yescharacteristic
258#41—48no—
259#41—44no—
260#41—44no—
261C2×Q8—21no—
262C2×Q8—22no—
263C2×Q8#11712yes—
264C2×Q8#13114yes—
265C2×D4—24no—
266C2×D4—21no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—44no—
270C2×D4#11712yes—
271C22×C4—21no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—82no—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C23×C411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—24no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4#11712yes—
331C4⋊C4#11712yes—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C42—24no—
343C42—24no—
344C42—21no—
345C42—21no—
346C42—22no—
347C42—24no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—21no—
355C42—21no—
356C42—42no—
357C42—42no—
358C42—44no—
359C42—44no—
360C42—42no—
361C42—41no—
362C42—41no—
363C42—82no—
364C42—82no—
365C42#13514yes—
366C42AGL(1,ℤ/8)12yes—
367C42#13712yes—
368C42C2×#4112yes—
369C42C2×#4112yes—
370C23—22no—
371C23—21no—
372C23—21no—
373C23—22no—
374C23—21no—
375C23—22no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23C22×C22⋊C411yescharacteristic
381Q8—22no—
382Q8—24no—
383Q8#43314yescomplemented
384D4—24no—
385D4—48no—
386D4—41no—
387D4—41no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—48no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4#41614yes—
430C2×C4#41714yes—
431C2×C4C2×#11711yescharacteristic
432C2×C4C2×#11711yescharacteristic
433C2×C4#52512yes—
434C2×C4#52512yes—
435C2×C4C2×#13112yes—
436C2×C4#56812yes—
437C2×C4C2×#13512yes—
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C8—44no—
441C8—82no—
442C8—82no—
443C8—82no—
444C22—41no—
445C22—42no—
446C22—41no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—41no—
452C22—82no—
453C22—82no—
454C22#228011yescharacteristic
455C22#288012yes—
456C22C2×#41611yescharacteristic
457C22C2×#41711yescharacteristic
458C22#290112yes—
459C4—22no—
460C4—24no—
461C4—24no—
462C4—24no—
463C4—41no—
464C4—42no—
465C4—41no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—82no—
471C4—82no—
472C4C2×#43311yescharacteristic
473C4#292111yescharacteristic
474C4#292412yes—
475C4#310612yes—
476C4#310712yes—
477C2—81no—
478C2—82no—
479C2#1751012yes—
480C2#1751512yes—
481C2#2071611yescharacteristic
482C2#2073711yescharacteristic
483C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21430C2semidirect
3. #20812C2semidirect
4. #8711C2semidirect
5. #8567C2semidirect
6. #8564C2semidirect
8. #8183C2semidirect
9. #8177C2semidirect
10. #8175C2semidirect
11. C4×Q8⋊C8C2semidirect
383. Q8#433semidirect

Maximal subgroups

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

subgroupindex
#341252
#214302
#208122
#87112
#85672
#85642
#83122
#81832
#81772
#81752
C4×Q8⋊C82