Finite Group Databasebeta

← #190049

Subgroups — #190049

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34109C211yesmaximal, characteristic
2#20694C211yesmaximal, characteristic
3C2×#1820C211yesmaximal, characteristic
4#8741C214yesmaximal, complemented
5#8740C214yesmaximal, complemented
6#7943C212yesmaximal
7C2×#1284C212yesmaximal
8C22×#117C2211yescharacteristic
9C2×#407C2212yes—
10#2347—21no—
11#2346—21no—
12#2345—21no—
13#2344—21no—
14#2343—22no—
15#2342—22no—
16#2298C2212yes—
17#2298C2212yes—
18#2296C412yescomplemented
19#2283C412yescomplemented
20#2280C2211yescharacteristic
21C2×#381C2212yes—
22C4×#117C412yescomplemented
23C4×#117C412yescomplemented
24C2×C4×C22⋊C4C2211yescharacteristic
25#1822C2214yes—
26#1821—21no—
27#1821—21no—
28#1820—21no—
29#1820—21no—
30#1820C2214yes—
31#1742—22no—
32C2×#342—22no—
33#1284C2218yes—
34#1284C2214yes—
35#1284C2214yes—
36C24×C4C2311yescharacteristic
37C22×M16—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117C2×C412yes—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117D412yes—
47C2×#117D411yescharacteristic
48C2×#117D411yescharacteristic
49C2×#117C2312yes—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C22⋊C4—21no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4D412yes—
55C22×C42C2×C412yes—
56C22×C42C2×C412yes—
57#407—22no—
58#407—22no—
59#407—22no—
60#407C2314yes—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63#389—22no—
64#389—22no—
65#388—22no—
66#388—22no—
67#388—22no—
68#388—22no—
69#387—22no—
70#387—22no—
71#387—22no—
72#387—22no—
73#385—22no—
74#385—22no—
75#383—22no—
76#382—22no—
77#381—22no—
78#381—21no—
79#381—22no—
80#381—22no—
81#381—22no—
82#381C2314yes—
83#380—22no—
84#380—21no—
85#380—21no—
86#379—21no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4C2×C412yescomplemented
91C4×C4⋊C4C2×C412yescomplemented
92C4×C4⋊C4C2×C412yescomplemented
93C4×C4⋊C4C2×C412yescomplemented
94C4×C22⋊C4—22no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4C2×C411yescharacteristic
100C4×C22⋊C4C2×C411yescharacteristic
101C4×C22⋊C4C2×C412yes—
102C4×C22⋊C4C2312yes—
103C4×C22⋊C4C2311yescharacteristic
104C4×C22⋊C4C2311yescharacteristic
105#376—22no—
106#376C2×C412yescomplemented
107#376C2×C412yescomplemented
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C43C2×C412yescomplemented
112C43C2×C412yescomplemented
113#343—24no—
114#343—24no—
115#342—24no—
116#342—24no—
117C25C2×D411yescharacteristic
118C23×C4—22no—
119C23×C4—44no—
120C23×C4—42no—
121C23×C4C22⋊C412yes—
122C23×C4C22⋊C412yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4112yes—
129C23×C4#4112yes—
130C23×C4#4112yes—
131C23×C4#4112yes—
132C23×C4C2411yescharacteristic
133C2×M16—24no—
134C2×M16—42no—
135C2×M16—48no—
136C22×C8—42no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—24no—
146#117—24no—
147#117—24no—
148#117—24no—
149#117—48no—
150#117—44no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C2×D412yes—
182C2×C22⋊C4C2×D412yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—44no—
193C2×C42C22⋊C412yes—
194C2×C42C22⋊C412yes—
195C2×C42C22×C412yes—
196C2×C42C22×C412yes—
197C2×C42C22×C412yes—
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—42no—
204C22⋊C8—42no—
205#95—21no—
206#95—21no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—21no—
218#95—21no—
219#95—24no—
220#95—24no—
221#95—42no—
222#95—42no—
223C24—24no—
224C24—42no—
225C24—81no—
226C24C2×C22⋊C411yescharacteristic
227C24C2×C22⋊C411yescharacteristic
228C24#12312yes—
229C24#12312yes—
230C24C22×D411yescharacteristic
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4#11712yes—
265C22×C4#11712yes—
266C22×C4#11711yescharacteristic
267C22×C4#11711yescharacteristic
268C22×C4#11711yescharacteristic
269C22×C4#11711yescharacteristic
270C22×C4#12312yes—
271C22×C4#12312yes—
272C22×C4#12312yes—
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4#12312yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4112yes—
280M16—48no—
281M16—84no—
282C2×C8—44no—
283C2×C8—84no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—44no—
301C22⋊C4—42no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—48no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—84no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42—44no—
321C42—44no—
322C42—44no—
323C42—82no—
324C42—82no—
325C23—24no—
326C23—22no—
327C23—24no—
328C23—22no—
329C23—24no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—41no—
336C23—41no—
337C23—42no—
338C23—42no—
339C23—88no—
340C23#38111yescharacteristic
341C23#38112yes—
342C23#38111yescharacteristic
343C23#41014yes—
344C23C22×C22⋊C411yescharacteristic
345C23C2×#12312yes—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—88no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—168no—
393C2×C4#38111yescharacteristic
394C2×C4#38111yescharacteristic
395C2×C4#38112yes—
396C2×C4C2×#11711yescharacteristic
397C2×C4C2×#11711yescharacteristic
398C2×C4C2×#12312yes—
399C8—84no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—21no—
412C22—44no—
413C22—48no—
414C22—44no—
415C22—82no—
416C22—84no—
417C22#182014yes—
418C22#182712yes—
419C22C2×#38111yescharacteristic
420C22C2×#41012yes—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—44no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—82no—
433C4—82no—
434C4—82no—
435C4—164no—
436C2—22no—
437C2—22no—
438C2—24no—
439C2—42no—
440C2—41no—
441C2C2×#182011yescharacteristic
442C2C2×#182711yescharacteristic
443C2#2070811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8741C2semidirect
5. #8740C2semidirect
18. #2296C4semidirect
19. #2283C4semidirect
22. C4×#117C4semidirect
23. C4×#117C4semidirect
90. C4×C4⋊C4C2×C4semidirect
91. C4×C4⋊C4C2×C4semidirect
92. C4×C4⋊C4C2×C4semidirect
93. C4×C4⋊C4C2×C4semidirect
106. #376C2×C4semidirect
107. #376C2×C4semidirect
111. C43C2×C4semidirect
112. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341092
#206942
C2×#18202
#87412
#87402
#79432
C2×#12842