Finite Group Databasebeta

← #108265

Subgroups — #108265

Order 512.

757 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60290C211yesmaximal, characteristic
2#21421C211yesmaximal, characteristic
3#17399C211yesmaximal, characteristic
4#7977C211yesmaximal, characteristic, complemented
5#7956C211yesmaximal, characteristic, complemented
6#7902C211yesmaximal, characteristic, complemented
7#7856C211yesmaximal, characteristic, complemented
8C2×#529C2211yescharacteristic
9#2994—22no—
10#2993—22no—
11#2983C411yescharacteristic, complemented
12#2982C411yescharacteristic, complemented
13#2975C411yescharacteristic, complemented
14#2974—22no—
15#2972C411yescharacteristic, complemented
16#2969—22no—
17#2966—22no—
18#2924—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2283C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1996—21no—
26#1964—21no—
27#1899C414yes—
28C4×#99—22no—
29#1473C414yescomplemented
30#1472—21no—
31#1471—21no—
32#1464—22no—
33#1288—22no—
34#1286—22no—
35#1279—22no—
36#1274—22no—
37#529—21no—
38#529—21no—
39#529—21no—
40#529—21no—
41#529—21no—
42#529—21no—
43#529—21no—
44#529—21no—
45C2×#126C2×C412yes—
46C2×#125Q811yescharacteristic
47C2×#124Q811yescharacteristic
48C2×#123C2×C412yes—
49C2×C22⋊Q8D411yescharacteristic
50C2×#121D411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×#117C2×C412yes—
55C2×#117D412yes—
56C22×C42C2311yescharacteristic
57#446—22no—
58#446—22no—
59C8×Q8—22no—
60C8×Q8—22no—
61#436—24no—
62#436—22no—
63#436—22no—
64#435—24no—
65#435—24no—
66C8×D4—24no—
67C8×D4—22no—
68C8×D4—22no—
69#433—24no—
70#433—24no—
71#432—22no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432C2×C412yes—
80#431—22no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431C2×C412yes—
85#431C2×C411yescharacteristic
86#431C2×C411yescharacteristic
87#431C2×C411yescharacteristic
88#431C2×C411yescharacteristic
89#431D412yes—
90#431D412yes—
91#424—24no—
92#423—21no—
93#423—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×#99—24no—
97#407—22no—
98#407—21no—
99#407—21no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103#405—24no—
104C4×M16—21no—
105C4×M16D412yes—
106C2×#97—21no—
107C2×C4×C8D412yes—
108#383—21no—
109#382—21no—
110#381C2×C412yes—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118#376—21no—
119#376—21no—
120C43—21no—
121#342—24no—
122#324—24no—
123#323—24no—
124C22×Q8C4⋊C411yescharacteristic
125C22×D4C22⋊C412yes—
126C22×D4C4⋊C411yescharacteristic
127C23×C4C22⋊C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C2×M16—24no—
132C2×M16—22no—
133C2×M16—41no—
134C22×C8—24no—
135C22×C8—22no—
136C22×C8—41no—
137#126—22no—
138#126—22no—
139#126—22no—
140#126—22no—
141#125—21no—
142#125—21no—
143#125—21no—
144#125—21no—
145#124—21no—
146#124—21no—
147#124—21no—
148#124—21no—
149#123—22no—
150#123—22no—
151#123—22no—
152#123—22no—
153C22⋊Q8—21no—
154C22⋊Q8—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157#121—21no—
158#121—21no—
159#121—21no—
160#121—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—44no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4C4212yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C4⋊C412yes—
183C2×C4⋊C4C4⋊C411yescharacteristic
184C2×C4⋊C4C4⋊C412yes—
185C2×C4⋊C4C4⋊C411yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C4212yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C4⋊C412yes—
192C2×C22⋊C4C4⋊C412yes—
193C2×C42—22no—
194C2×C42—41no—
195C2×C42—41no—
196C2×C42—41no—
197C2×C42C22⋊C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q811yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222#99—48no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234C22⋊C8—41no—
235#97—24no—
236#97—22no—
237#97—22no—
238#97—41no—
239#97—41no—
240C4×C8—24no—
241C4×C8—22no—
242C4×C8—22no—
243C4×C8C22⋊C412yes—
244C4×C8C22⋊C412yes—
245C4×C8C22⋊C412yes—
246C4×C8C22⋊C412yes—
247#95—22no—
248#95—22no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24#9512yes—
254C24C2×C22⋊C411yescharacteristic
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—42no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—42no—
263C2×D4—44no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—81no—
271C22×C4—82no—
272C22×C4#9512yes—
273C22×C4#9512yes—
274C22×C4#9512yes—
275C22×C4C2×C4211yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C4⋊C411yescharacteristic
279C22×C4C2×C4⋊C411yescharacteristic
280C22×C4C2×C4⋊C411yescharacteristic
281C22×C4C4×D411yescharacteristic
282C22×C4C4×D411yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4#12311yescharacteristic
293M16—82no—
294M16—84no—
295M16—82no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—84no—
303C2×C8—82no—
304C2×C8—82no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—42no—
319C4⋊C4—82no—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C42—22no—
334C42—22no—
335C42—24no—
336C42—42no—
337C42—82no—
338C42—82no—
339C42—82no—
340C42C2×C22⋊C411yescharacteristic
341C42C2×C22⋊C411yescharacteristic
342C42#11711yescharacteristic
343C42#11711yescharacteristic
344C42C4×D411yescharacteristic
345C42C4×D411yescharacteristic
346C42C4×D411yescharacteristic
347C42C4×D411yescharacteristic
348C42#12011yescharacteristic
349C42#12011yescharacteristic
350C42C22⋊Q811yescharacteristic
351C42C22⋊Q811yescharacteristic
352C42C22⋊Q811yescharacteristic
353C42C22⋊Q811yescharacteristic
354C42#12311yescharacteristic
355C42#12311yescharacteristic
356C23—22no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23C2×#9511yescharacteristic
364C23#38611yescharacteristic
365C23#38611yescharacteristic
366Q8—42no—
367Q8—42no—
368D4—42no—
369D4—44no—
370D4—42no—
371D4—82no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4C4×C22⋊C411yescharacteristic
405C2×C4C4×C22⋊C411yescharacteristic
406C2×C4#37911yescharacteristic
407C2×C4#38011yescharacteristic
408C2×C4#38011yescharacteristic
409C2×C4#38111yescharacteristic
410C2×C4#38511yescharacteristic
411C2×C4#38511yescharacteristic
412C2×C4#38511yescharacteristic
413C2×C4#38511yescharacteristic
414C2×C4#38611yescharacteristic
415C2×C4#38611yescharacteristic
416C8—42no—
417C8—42no—
418C8—42no—
419C8—42no—
420C8—82no—
421C8—82no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—82no—
434C22—162no—
435C22C22⋊#9511yescharacteristic
436C22#189112yes—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—84no—
452C4—162no—
453C4#181212yes—
454C2—22no—
455C2—41no—
456C2—41no—
457C2—82no—
458C2#1074311yescharacteristic
459C2#1074611yescharacteristic
460C2#1075911yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7977C2semidirect
5. #7956C2semidirect
6. #7902C2semidirect
7. #7856C2semidirect
11. #2983C4semidirect
12. #2982C4semidirect
13. #2975C4semidirect
15. #2972C4semidirect
29. #1473C4semidirect

Maximal subgroups

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

subgroupindex
#602902
#214212
#173992
#79772
#79562
#79022
#78562