Finite Group Databasebeta

← #107906

Subgroups — #107906

Order 512.

917 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34222C211yesmaximal, characteristic
2#17717C211yesmaximal, characteristic
3#17583C211yesmaximal, characteristic
4#8072C211yesmaximal, characteristic, complemented
5#8013C211yesmaximal, characteristic, complemented
6#7944C211yesmaximal, characteristic, complemented
7#7857C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2372—22no—
13#2364—22no—
14#2353—22no—
15#2339—22no—
16#2331—21no—
17#2328—21no—
18#2319C412yescomplemented
19C4×#121—22no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22#2275C412yescomplemented
23C2×#376C2211yescharacteristic
24#1553C414yescomplemented
25#1549—21no—
26#1545—21no—
27#1540—22no—
28#1517C414yescomplemented
29#1514—22no—
30#1507—22no—
31#1300—22no—
32#1294—22no—
33#1284—22no—
34#1274—22no—
35C22×#41—21no—
36#532—21no—
37#532—21no—
38#532—21no—
39#532—21no—
40#532C2×C412yes—
41#532C2×C412yes—
42#532D412yes—
43#532Q812yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×#127D411yescharacteristic
46C2×#124—21no—
47C2×#121—21no—
48C2×#121D412yes—
49C2×C4×D4—22no—
50C2×C4×D4—24no—
51C2×C4×D4—21no—
52C2×C4×D4C2×C412yes—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432C2×C412yescomplemented
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431C2×C412yescomplemented
66#431C2×C412yescomplemented
67#423—21no—
68#423—21no—
69C2×C4⋊C8—21no—
70#407—21no—
71#407—21no—
72C2×C22⋊C8—21no—
73C4×M16—21no—
74C4×M16—21no—
75C2×C4×C8—21no—
76#391—24no—
77#390—22no—
78#389—22no—
79#388—24no—
80#388—24no—
81#387—24no—
82#386—24no—
83#386—24no—
84#386—24no—
85#383—21no—
86#383—21no—
87#382—21no—
88#382C2×C412yescomplemented
89#381—21no—
90#381—22no—
91#381—21no—
92#380—21no—
93#380—22no—
94#380—21no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4C2×C412yes—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—24no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4D412yes—
103#376—21no—
104#376—21no—
105#376—21no—
106#376C2×C411yescharacteristic
107#376C2×C411yescharacteristic
108#376D412yes—
109#376D412yes—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95D412yes—
113#332—24no—
114#331—24no—
115#327—28no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—84no—
121C22×Q8—21no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4C22⋊C412yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—41no—
133C2×M16—41no—
134C22×C8—41no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8C22⋊C412yes—
138C4⋊Q8C4⋊C412yes—
139#127—21no—
140#127—21no—
141#127C22⋊C412yes—
142#127C4⋊C412yes—
143#124—22no—
144#124—22no—
145#124—22no—
146#124—22no—
147#121—22no—
148#121—22no—
149#121—22no—
150#121—22no—
151#121—22no—
152#121—22no—
153#121C22⋊C414yes—
154#121C4⋊C414yes—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—48no—
162C4×D4—88no—
163C4×D4C4214yes—
164C4×D4C4⋊C414yes—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—41no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—44no—
182C2×C42—41no—
183C2×C42C22⋊C414yes—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q811yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208#97—22no—
209#97—41no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—41no—
213C4×C8—41no—
214#95—24no—
215#95—24no—
216#95—21no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—41no—
227#95—41no—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230C24—22no—
231C24#12011yescharacteristic
232#41—84no—
233#41—84no—
234#41—84no—
235#41—84no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—82no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4#9514yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—48no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—82no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C4⋊C411yescharacteristic
273C22×C4C4×D411yescharacteristic
274C22×C4C4×D411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4C4×D411yescharacteristic
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4C22⋊Q811yescharacteristic
283C22×C4C22⋊Q811yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12311yescharacteristic
286M16—82no—
287M16—82no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—82no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—44no—
296C4⋊C4—82no—
297C4⋊C4—84no—
298C4⋊C4#9514yes—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—48no—
304C22⋊C4—44no—
305C22⋊C4—84no—
306C22⋊C4—84no—
307C22⋊C4—82no—
308C42—24no—
309C42—44no—
310C42—42no—
311C42—44no—
312C42—82no—
313C42—84no—
314C42C2×C4211yescharacteristic
315C42C2×C22⋊C411yescharacteristic
316C42C2×C4⋊C411yescharacteristic
317C42C2×C4⋊C411yescharacteristic
318C42#11711yescharacteristic
319C42#11711yescharacteristic
320C42C4×D411yescharacteristic
321C42C4×D411yescharacteristic
322C42C4×D411yescharacteristic
323C42C4×D411yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42#12311yescharacteristic
329C42#12311yescharacteristic
330C23—22no—
331C23—22no—
332C23—24no—
333C23—41no—
334C23—44no—
335C23—44no—
336C23—42no—
337C23—42no—
338C23#37911yescharacteristic
339C23#38511yescharacteristic
340C23#38611yescharacteristic
341Q8—82no—
342Q8—82no—
343D4—44no—
344D4—48no—
345D4—48no—
346D4—44no—
347D4—82no—
348D4—82no—
349D4—82no—
350D4—82no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—88no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—168no—
387C2×C4C2×#9511yescharacteristic
388C2×C4C4×C22⋊C411yescharacteristic
389C2×C4C4×C22⋊C411yescharacteristic
390C2×C4#38011yescharacteristic
391C2×C4#38011yescharacteristic
392C2×C4#38111yescharacteristic
393C2×C4#38511yescharacteristic
394C2×C4#38511yescharacteristic
395C2×C4#38511yescharacteristic
396C2×C4#38611yescharacteristic
397C2×C4#38611yescharacteristic
398C2×C4#38611yescharacteristic
399C8—82no—
400C8—82no—
401C8—82no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—44no—
410C22—44no—
411C22—44no—
412C22—44no—
413C22—84no—
414C22—84no—
415C22C22⋊#9511yescharacteristic
416C22#179412yes—
417C22#190412yes—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—44no—
430C4—44no—
431C4—82no—
432C4—84no—
433C4—162no—
434C4#182712yes—
435C2—22no—
436C2—22no—
437C2—41no—
438C2—84no—
439C2#1075811yescharacteristic
440C2#1077911yescharacteristic
441C2#1078511yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C42 ▷ 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. #8072C2semidirect
5. #8013C2semidirect
6. #7944C2semidirect
7. #7857C2semidirect
18. #2319C4semidirect
22. #2275C4semidirect
24. #1553C4semidirect
28. #1517C4semidirect
58. #432C2×C4semidirect
65. #431C2×C4semidirect
66. #431C2×C4semidirect
88. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342222
#177172
#175832
#80722
#80132
#79442
#78572