Finite Group Databasebeta

← #108146

Subgroups — #108146

Order 512.

934 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34086C211yesmaximal, characteristic
2#21816C211yesmaximal, characteristic
3#17714C211yesmaximal, characteristic
4#8070C211yesmaximal, characteristic, complemented
5#7980C211yesmaximal, characteristic, complemented
6#7940C211yesmaximal, characteristic, complemented
7#7869C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2337C414yescomplemented
13#2333—22no—
14#2331—21no—
15#2328—21no—
16#2324—22no—
17#2297—22no—
18#2293—21no—
19#2290—22no—
20#2283C2211yescharacteristic
21#2283C2211yescharacteristic
22C42×D4—21no—
23C4×#117C2211yescharacteristic
24#1980—22no—
25#1963C414yescomplemented
26#1789—22no—
27#1551C414yescomplemented
28#1548—21no—
29#1545—21no—
30#1543—22no—
31#1300—22no—
32#1289—22no—
33#1284—22no—
34#1276—21no—
35#1276—21no—
36C22×#41D412yes—
37#532—21no—
38#532—21no—
39#532—21no—
40#532—21no—
41#532C2×C412yes—
42#532C2×C412yes—
43#532D412yes—
44#532Q812yes—
45C2×C4⋊Q8D411yescharacteristic
46C2×#127D411yescharacteristic
47C2×#124C2×C412yes—
48C2×#121—21no—
49C2×#121—21no—
50C2×C4×D4—22no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×#117—21no—
54C2×#117—22no—
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432C2×C412yescomplemented
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#423—21no—
70#423C2×C412yescomplemented
71C2×C4⋊C8C2×C412yescomplemented
72C2×#104—28no—
73#407—21no—
74#407—21no—
75C2×C22⋊C8—21no—
76C4×M16—21no—
77C4×M16—21no—
78C2×C4×C8—21no—
79#391—24no—
80#390—22no—
81#390—24no—
82#389—22no—
83#388—24no—
84#388—24no—
85#387—24no—
86#386—24no—
87#386—24no—
88#386—24no—
89#385—24no—
90#383—21no—
91#383C2×C412yes—
92#382—21no—
93#382—21no—
94#381—21no—
95#381—21no—
96#380—21no—
97#380D412yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×C412yes—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—24no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4D412yes—
110#376—21no—
111#376C2×C412yescomplemented
112#376D412yes—
113C43D411yescharacteristic
114C43D411yescharacteristic
115#333—24no—
116#331—24no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—84no—
120C2×#41C22⋊C414yes—
121C2×#41C4⋊C414yes—
122C22×Q8C22⋊C412yes—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4C22⋊C412yes—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—22no—
133C2×M16—41no—
134C22×C8—22no—
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#124C4214yes—
146#124C4⋊C414yes—
147#121—22no—
148#121—22no—
149#121—22no—
150#121—22no—
151#121—22no—
152#121—22no—
153#121—22no—
154#121—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—48no—
164#117—44no—
165#117—84no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—41no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—41no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42—84no—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×Q811yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—41no—
208#104—88no—
209C22⋊C8—22no—
210C22⋊C8—41no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215#97—22no—
216#97—41no—
217C4×C8—41no—
218C4×C8—41no—
219C4×C8—41no—
220C4×C8—41no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—41no—
231#95—41no—
232#95C22⋊C414yes—
233C24—22no—
234C24#12011yescharacteristic
235#41—48no—
236#41—84no—
237#41—84no—
238#41—84no—
239C2×Q8—22no—
240C2×Q8—82no—
241C2×Q8#9514yes—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—44no—
250C2×D4—82no—
251C2×D4—82no—
252C2×D4—82no—
253C2×D4—82no—
254C2×D4#9514yes—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—42no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C4⋊C411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4C4×D411yescharacteristic
277C22×C4C4×D411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4C22⋊Q811yescharacteristic
285C22×C4C22⋊Q811yescharacteristic
286C22×C4#12311yescharacteristic
287C22×C4#12311yescharacteristic
288M16—44no—
289M16—82no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—82no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—82no—
301C4⋊C4—84no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—44no—
307C22⋊C4—48no—
308C22⋊C4—82no—
309C22⋊C4—84no—
310C22⋊C4—84no—
311C22⋊C4—82no—
312C42—24no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—44no—
317C42—42no—
318C42—42no—
319C42—44no—
320C42—84no—
321C42C2×C4211yescharacteristic
322C42C2×C22⋊C411yescharacteristic
323C42C2×C4⋊C411yescharacteristic
324C42C2×C4⋊C411yescharacteristic
325C42#11711yescharacteristic
326C42#11711yescharacteristic
327C42C4×D411yescharacteristic
328C42C4×D411yescharacteristic
329C42C4×D411yescharacteristic
330C42C4×D411yescharacteristic
331C42#12111yescharacteristic
332C42#12111yescharacteristic
333C42C22⋊Q811yescharacteristic
334C42C22⋊Q811yescharacteristic
335C42#12311yescharacteristic
336C42#12311yescharacteristic
337C23—22no—
338C23—24no—
339C23—41no—
340C23—41no—
341C23—44no—
342C23—42no—
343C23—42no—
344C23—44no—
345C23#37911yescharacteristic
346C23#38511yescharacteristic
347C23#38611yescharacteristic
348Q8—44no—
349Q8—82no—
350D4—44no—
351D4—44no—
352D4—44no—
353D4—82no—
354D4—82no—
355D4—82no—
356D4—84no—
357D4—84no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—48no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—816no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4C2×#9511yescharacteristic
395C2×C4C4×C22⋊C411yescharacteristic
396C2×C4C4×C22⋊C411yescharacteristic
397C2×C4#38011yescharacteristic
398C2×C4#38011yescharacteristic
399C2×C4#38111yescharacteristic
400C2×C4#38511yescharacteristic
401C2×C4#38511yescharacteristic
402C2×C4#38511yescharacteristic
403C2×C4#38611yescharacteristic
404C2×C4#38611yescharacteristic
405C2×C4#38611yescharacteristic
406C8—82no—
407C8—82no—
408C8—82no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22—82no—
421C22—84no—
422C22—84no—
423C22C22⋊#9511yescharacteristic
424C22#179412yes—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—44no—
436C4—44no—
437C4—82no—
438C4—84no—
439C4—162no—
440C4#190612yes—
441C4#191112yes—
442C2—22no—
443C2—41no—
444C2—41no—
445C2—84no—
446C2#1074011yescharacteristic
447C2#1077211yescharacteristic
448C2#1077911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22C22⋊#95
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8070C2semidirect
5. #7980C2semidirect
6. #7940C2semidirect
7. #7869C2semidirect
12. #2337C4semidirect
25. #1963C4semidirect
27. #1551C4semidirect
60. #432C2×C4semidirect
70. #423C2×C4semidirect
71. C2×C4⋊C8C2×C4semidirect
111. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340862
#218162
#177142
#80702
#79802
#79402
#78692