Finite Group Databasebeta

← #163890

Subgroups — #163890

Order 512.

918 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34253C211yesmaximal, characteristic
2#22639C211yesmaximal, characteristic, complemented
3#22632C211yesmaximal, characteristic, complemented
4#7597C211yesmaximal, characteristic
5#7596C212yesmaximal, complemented
6#7580C212yesmaximal, complemented
7#7576C211yesmaximal, characteristic, complemented
8#7574C212yesmaximal, complemented
9#7573C211yesmaximal, characteristic, complemented
10#7563C211yesmaximal, characteristic
11#7562C212yesmaximal, complemented
12C2×#520C411yescharacteristic
13#2353—22no—
14#2352—22no—
15#2347—21no—
16#2346—21no—
17#2345—21no—
18#2344—21no—
19#2320C412yes—
20#2319C412yes—
21#2316C411yescharacteristic
22#2311C411yescharacteristic
23#2297C2211yescharacteristic
24#2295C2212yes—
25#2293C2212yes—
26C2×#389C2211yescharacteristic
27#2274C411yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1923—22no—
32#1923—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37C8×C4⋊C4—21no—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1744C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41C2×C411yescharacteristic
54#520—24no—
55#520C818yescomplemented
56C2×C4×Q8C2×C412yes—
57C2×C4×D4—22no—
58C2×C4×D4C2×C412yes—
59C2×#117—21no—
60C2×#117—22no—
61C2×#117C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C4×C8—22no—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81#391C2×C412yes—
82#391C2×C411yescharacteristic
83#391C2311yescharacteristic
84#390C2×C412yes—
85#390C2×C411yescharacteristic
86#390C2311yescharacteristic
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2312yes—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#386—21no—
104#386—21no—
105#386—22no—
106#383—22no—
107#382—22no—
108#381—22no—
109#380—21no—
110#380—21no—
111#380—22no—
112#380—22no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C22⋊C4—22no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4D412yes—
127C4×C22⋊C4D412yes—
128#376—22no—
129#376C2×C412yes—
130C2×#95—21no—
131C43C2312yes—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C2×#41—22no—
143C2×#41—42no—
144C2×#41—42no—
145C2×#41C2×C814yes—
146C22×Q8C22×C411yescharacteristic
147C22×D4—21no—
148C22×D4C22×C411yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C4×Q8—24no—
163C4×Q8C2×C818yes—
164C4×D4—216no—
165C4×D4—24no—
166C4×D4C2×C818yes—
167#117—28no—
168#117—42no—
169#117—42no—
170#117—82no—
171#117—82no—
172#117C2×C814yes—
173#117M1614yes—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—81no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C2×C814yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42C22×C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2411yescharacteristic
220C4⋊C8—42no—
221C4⋊C8—42no—
222C4⋊C8—44no—
223C4⋊C8—82no—
224C22⋊C8—42no—
225C22⋊C8—42no—
226C22⋊C8—81no—
227C22⋊C8—81no—
228C4×C8—42no—
229C4×C8—42no—
230C4×C8—44no—
231C4×C8—82no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248C24—21no—
249C24C2×C22⋊C411yescharacteristic
250#41—48no—
251#41—48no—
252C2×Q8—21no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8C22×C812yes—
256C2×D4—24no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4C22×C812yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—28no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C22×C812yes—
298C22×C4C2×M1612yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—41no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4C22⋊C814yes—
335C4⋊C4C22⋊C814yes—
336C4⋊C4C22×C814yes—
337C22⋊C4—28no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—82no—
345C22⋊C4—81no—
346C22⋊C4—81no—
347C22⋊C4—82no—
348C22⋊C4C22⋊C818yes—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—28no—
354C42—22no—
355C42—22no—
356C42—42no—
357C42—42no—
358C42—41no—
359C42—41no—
360C42—42no—
361C42—42no—
362C42—82no—
363C42—82no—
364C42C22×C814yes—
365C42C2×M1614yes—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—22no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C2×C22⋊C812yes—
380C23C22×C22⋊C411yescharacteristic
381Q8—42no—
382Q8—42no—
383D4—42no—
384D4—42no—
385D4—48no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—48no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—88no—
431C2×C4—84no—
432C2×C4—162no—
433C2×C4C2×C22⋊C814yes—
434C2×C4C2×C22⋊C814yes—
435C2×C4C2×C22⋊C812yes—
436C2×C4#41012yes—
437C2×C4#41312yes—
438C2×C4C23×C811yescharacteristic
439C2×C4C22×M1611yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57811yescharacteristic
444C8—42no—
445C8—42no—
446C8—81no—
447C8—82no—
448C8—82no—
449C8—81no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22C22×C22⋊C811yescharacteristic
465C22C2×#41011yescharacteristic
466C22C2×#41311yescharacteristic
467C22#289711yescharacteristic
468C22#289811yescharacteristic
469C22#290111yescharacteristic
470C22#290111yescharacteristic
471C4—24no—
472C4—24no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—42no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—41no—
484C4—41no—
485C4—81no—
486C4—81no—
487C4—82no—
488C4#145814yes—
489C2—42no—
490C2—82no—
491C2C2×#145811yescharacteristic
492C2#1743811yescharacteristic
493C2#1743911yescharacteristic
494C2#1761011yescharacteristic
495C2#1761211yescharacteristic
496C2#1768711yescharacteristic
497C2#1778111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22639C2semidirect
3. #22632C2semidirect
5. #7596C2semidirect
6. #7580C2semidirect
7. #7576C2semidirect
8. #7574C2semidirect
9. #7573C2semidirect
11. #7562C2semidirect
55. #520C8semidirect

Maximal subgroups

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

subgroupindex
#342532
#226392
#226322
#75972
#75962
#75802
#75762
#75742
#75732
#75632
#75622