Finite Group Databasebeta

← #164109

Subgroups — #164109

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic
2#22643C211yesmaximal, characteristic, complemented
3#22639C211yesmaximal, characteristic, complemented
4#7726C212yesmaximal, complemented
5#7712C212yesmaximal, complemented
6#7686C212yesmaximal, complemented
7#7672C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic, complemented
9#7572C211yesmaximal, characteristic
10#7572C211yesmaximal, characteristic
11#7564C211yesmaximal, characteristic, complemented
12#2860C2212yes—
13#2860C2212yes—
14#2644C412yes—
15#2644C412yes—
16#2513—21no—
17#2513—21no—
18#2512—21no—
19#2512—21no—
20#2480—22no—
21#2480—22no—
22#2317C411yescharacteristic
23#2316C411yescharacteristic
24#2309C411yescharacteristic
25#2309C411yescharacteristic
26#2296C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1924—22no—
32#1923—22no—
33#1839C2212yescomplemented
34#1839C2212yescomplemented
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1770—21no—
38C8×C4⋊C4—21no—
39#1767C2212yescomplemented
40C4×C4⋊C8C2212yescomplemented
41C4×C22⋊C8C2211yescharacteristic
42C4×C22⋊C8C2211yescharacteristic
43#1271C2211yescharacteristic, complemented
44#1271C2212yescomplemented
45#1271C2211yescharacteristic, complemented
46#1271C2212yescomplemented
47#1271C2211yescharacteristic, complemented
48#1271C2211yescharacteristic, complemented
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C23×D4C2×C411yescharacteristic
54C2×#126—22no—
55C2×#126—22no—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×#97—22no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—22no—
76#397—22no—
77#397—22no—
78#397—22no—
79#397—22no—
80#397—22no—
81#397—22no—
82#396—22no—
83#396—22no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#390—21no—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#390C2×C412yes—
94#390C2×C412yes—
95#390C2312yes—
96#389—21no—
97#389—21no—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C411yescharacteristic
103#389C2×C411yescharacteristic
104#389C2311yescharacteristic
105#389C2311yescharacteristic
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#385—21no—
113#385—21no—
114#385—22no—
115#384—21no—
116#382—21no—
117#380—22no—
118#379—22no—
119#379—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4D412yes—
127C4×C22⋊C4D412yes—
128#376C2×C412yes—
129C43C2312yes—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336C2312yes—
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C25—21no—
141C22×D4—21no—
142C22×D4—24no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4C22×C412yes—
146C23×C4—21no—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156#126—82no—
157#126—82no—
158#126—82no—
159#126—82no—
160#117—24no—
161#117—24no—
162#117—42no—
163#117—42no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—81no—
198C2×C22⋊C4—81no—
199C2×C22⋊C4—81no—
200C2×C22⋊C4—81no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—42no—
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—44no—
219C4⋊C8—44no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222C22⋊C8—81no—
223C22⋊C8—81no—
224#97—82no—
225C4×C8—42no—
226C4×C8—42no—
227C4×C8—82no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—21no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24—22no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—81no—
245C24—81no—
246C24—81no—
247C24—81no—
248C24C2×C22⋊C411yescharacteristic
249C2×D4—24no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—88no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4#13114yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—41no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—81no—
318C4⋊C4—81no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—24no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—24no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—42no—
349C42—42no—
350C42—41no—
351C42—41no—
352C42—42no—
353C42—42no—
354C42—42no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—84no—
369C23—84no—
370C23—84no—
371C23—162no—
372C23—162no—
373C23—162no—
374C23—162no—
375C23#40812yes—
376C23C22×C22⋊C411yescharacteristic
377D4—84no—
378D4—84no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—28no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—162no—
424C2×C4#40812yes—
425C2×C4#41012yes—
426C2×C4#41312yes—
427C2×C4C2×#13112yes—
428C2×C4#57611yescharacteristic
429C2×C4#57611yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57711yescharacteristic
432C8—42no—
433C8—42no—
434C8—82no—
435C8—82no—
436C8—81no—
437C8—81no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—168no—
453C22C2×#40811yescharacteristic
454C22C2×#41011yescharacteristic
455C22C2×#41311yescharacteristic
456C22#289711yescharacteristic
457C22#289711yescharacteristic
458C22#290111yescharacteristic
459C22#290111yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—81no—
472C4—81no—
473C4—82no—
474C4—81no—
475C4—81no—
476C4#147314yes—
477C2—42no—
478C2—162no—
479C2C2×#147311yescharacteristic
480C2#1746811yescharacteristic
481C2#1748311yescharacteristic
482C2#1760711yescharacteristic
483C2#1760711yescharacteristic
484C2#1769111yescharacteristic
485C2#1777611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×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. #22643C2semidirect
3. #22639C2semidirect
4. #7726C2semidirect
5. #7712C2semidirect
6. #7686C2semidirect
7. #7672C2semidirect
8. #7598C2semidirect
11. #7564C2semidirect
33. #1839C22semidirect
34. #1839C22semidirect
39. #1767C22semidirect
40. C4×C4⋊C8C22semidirect
43. #1271C22semidirect
44. #1271C22semidirect
45. #1271C22semidirect
46. #1271C22semidirect
47. #1271C22semidirect
48. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#226432
#226392
#77262
#77122
#76862
#76722
#75982
#75722
#75642