Finite Group Databasebeta

← #197515

Subgroups — #197515

Order 512.

1485 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic, complemented
3#17574C212yesmaximal
4C2×#1461C211yesmaximal, characteristic, complemented
5#9204C218yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×#536—21no—
8C2×#532C2211yescharacteristic
9C2×#519—21no—
10C22×C4×Q8C2211yescharacteristic
11C2×#431C2212yes—
12#2854—22no—
13#2618C2214yes—
14#2548C412yescomplemented
15#2546C412yescomplemented
16#2545—22no—
17#2538—22no—
18#2536C414yescomplemented
19#2351—24no—
20Q8×C4⋊C4—24no—
21#2316—21no—
22#2316C2214yes—
23#2311—21no—
24#2285—22no—
25#2285C2211yescharacteristic
26#2285C2212yes—
27#1519C2218yes—
28#1517—24no—
29#1502—24no—
30#1461—22no—
31#1461—22no—
32C2×#324C2214yes—
33#1289C2218yes—
34C22×#41—21no—
35C23×Q8D411yescharacteristic
36#536—44no—
37#536—44no—
38#532—22no—
39#532—22no—
40#532—22no—
41#532—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#124—22no—
46C2×#124—21no—
47C2×#123—24no—
48C2×C22⋊Q8—24no—
49C2×#121—21no—
50C2×#121C2312yes—
51#519—44no—
52#519—44no—
53C2×C4×Q8—28no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8D412yes—
61C2×C4×Q8D412yes—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×#117—21no—
65C2×#117—22no—
66C2×#117—24no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73#431—22no—
74#431—22no—
75C2×C4⋊C8—22no—
76C2×C22⋊C8C2314yes—
77C2×C4×C8—22no—
78#398C2×C418yes—
79#396—28no—
80#394C2×C418yes—
81#391—22no—
82#390—22no—
83#390—24no—
84#389—28no—
85#389—24no—
86#389—216no—
87#388—28no—
88#386—24no—
89#384—24no—
90#384—22no—
91#384—21no—
92#382—24no—
93#382C2×C412yes—
94#382D414yes—
95#380—24no—
96#380—21no—
97#380—22no—
98#380—24no—
99#380C2312yes—
100#380C2314yes—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4D414yes—
107#327—216no—
108#324—28no—
109C2×#41—42no—
110C2×#41—42no—
111C2×#41—42no—
112C2×#41—42no—
113C2×#41—42no—
114C2×#41—42no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8C22⋊C412yes—
121C22×Q8C22⋊C412yes—
122C22×Q8D814yes—
123C22×Q8SD1614yes—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4C22×C412yes—
128C23×C4—22no—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—24no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—48no—
138#127—22no—
139#127—22no—
140#124—48no—
141#124—42no—
142#124—42no—
143#123—48no—
144#123—48no—
145C22⋊Q8—416no—
146#121—24no—
147#121—24no—
148#121—41no—
149#121—41no—
150#121—41no—
151#121—41no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—432no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—42no—
164C4×D4—42no—
165#117—48no—
166#117—48no—
167#117—44no—
168#117—42no—
169#117—42no—
170#117—44no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C414yes—
189C2×C4⋊C4C2×D414yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—28no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—28no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42C22⋊C414yes—
204C2×C42C22⋊C414yes—
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4112yes—
212C2×C42#4112yes—
213C4⋊C8—42no—
214C4⋊C8—42no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C4×C8—42no—
218C4×C8—42no—
219#95—28no—
220#95—22no—
221#95—28no—
222#95—24no—
223#95C22×C414yes—
224#95C2×D418yes—
225C24—22no—
226C24C22×D411yescharacteristic
227#41—84no—
228#41—84no—
229#41—84no—
230#41—84no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—28no—
236C2×Q8—28no—
237C2×Q8—28no—
238C2×Q8—28no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8#10218yes—
243C2×Q8#10218yes—
244C2×D4—24no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—82no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—81no—
272C22×C4—82no—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C414yes—
275C22×C4C2×C22⋊C414yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12012yes—
282C22×C4#12311yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4#12312yes—
285C22×C4C2×D814yes—
286C22×C4C2×D812yes—
287C22×C4C2×SD1614yes—
288C22×C4C2×SD1612yes—
289C22×C4AGL(1,ℤ/8)12yes—
290C22×C4#13712yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4112yes—
295C2×C8—44no—
296C2×C8—84no—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—48no—
303C4⋊C4—44no—
304C4⋊C4—416no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—44no—
308C4⋊C4—416no—
309C4⋊C4—48no—
310C4⋊C4—82no—
311C4⋊C4—84no—
312C4⋊C4—88no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—416no—
321C22⋊C4—88no—
322C42—24no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—416no—
327C42—44no—
328C42—48no—
329C42C4×D412yes—
330C42C4×D412yes—
331C42C4×D412yes—
332C42C4×D412yes—
333C42#12112yes—
334C42#12112yes—
335C42#12112yes—
336C42#12112yes—
337C23—41no—
338C23—41no—
339C23—44no—
340C23—44no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23C22×C22⋊C411yescharacteristic
345C23C2×#12011yescharacteristic
346C23C2×#12311yescharacteristic
347C23C22×D811yescharacteristic
348C23C22×SD1611yescharacteristic
349C23C2×AGL(1,ℤ/8)11yescharacteristic
350C23C2×#13711yescharacteristic
351Q8—216no—
352Q8—216no—
353Q8—84no—
354D4—84no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4—82no—
359D4—82no—
360D4—84no—
361D4—82no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—48no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—88no—
392C2×C4#38611yescharacteristic
393C2×C4#38611yescharacteristic
394C2×C4#38611yescharacteristic
395C2×C4#38612yes—
396C2×C4#38612yes—
397C2×C4#38611yescharacteristic
398C2×C4C2×#10218yes—
399C2×C4C2×#10212yes—
400C2×C4C2×#10212yes—
401C2×C4#41712yes—
402C2×C4#41712yes—
403C2×C4#44714yes—
404C2×C4#44814yes—
405C2×C4#48014yes—
406C2×C4#48114yes—
407C2×C4C2×C4×D412yes—
408C2×C4C2×#12112yes—
409C8—84no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—84no—
421C22—84no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22C2×#38611yescharacteristic
426C22C22×#10211yescharacteristic
427C22C2×#41711yescharacteristic
428C22C2×#44711yescharacteristic
429C22C2×#44811yescharacteristic
430C22C2×#48011yescharacteristic
431C22C2×#48111yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—22no—
437C4—28no—
438C4—44no—
439C4—88no—
440C4—84no—
441C4—82no—
442C4#187414yes—
443C4#187414yes—
444C4#191212yes—
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—162no—
452C2C2×#187411yescharacteristic
453C2C2×#191211yescharacteristic
454C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34182C2semidirect
4. C2×#1461C2semidirect
5. #9204C2semidirect
6. #7981C2semidirect
14. #2548C4semidirect
15. #2546C4semidirect
18. #2536C4semidirect

Maximal subgroups

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

subgroupindex
#351782
#341822
#175742
C2×#14612
#92042
#79812