Finite Group Databasebeta

← #190067

Subgroups — #190067

Order 512.

817 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34085C211yesmaximal, characteristic
2#21831C211yesmaximal, characteristic
3#21826C211yesmaximal, characteristic
4#8741C214yesmaximal
5#8740C214yesmaximal
6#7945C211yesmaximal, characteristic
7#7944C211yesmaximal, characteristic
8#7941C211yesmaximal, characteristic
9#7940C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2336—22no—
14#2332—22no—
15#2331—21no—
16#2327—21no—
17#2323—22no—
18#2298C2214yes—
19#2295C412yescomplemented
20#2295C414yescomplemented
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#2283C2211yescharacteristic
24C42×Q8C412yescomplemented
25C4×#117C2211yescharacteristic
26#1998—22no—
27#1997—21no—
28#1997—21no—
29#1996—22no—
30#1996—22no—
31#1822C2214yes—
32#1820C2212yes—
33#1820C2212yes—
34#1284C2214yes—
35#1284C2212yes—
36#1284C2212yes—
37#1284C2212yes—
38#1284C2212yes—
39#1284C2214yes—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48C2×C4⋊Q8D411yescharacteristic
49C2×C4⋊Q8D411yescharacteristic
50C2×#125D412yes—
51C2×C22⋊Q8D412yes—
52C2×C22⋊Q8D412yes—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2×C412yes—
56C2×#117—21no—
57C2×#117C2312yes—
58C22×C42C2311yescharacteristic
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#423—21no—
68#423—21no—
69#407C2312yes—
70#407C2312yes—
71C4×M16—21no—
72C4×M16—21no—
73#391—22no—
74#391—24no—
75#389—22no—
76#389—24no—
77#388—24no—
78#388—24no—
79#387—24no—
80#387—24no—
81#387—24no—
82#387—24no—
83#387—24no—
84#383—21no—
85#383—21no—
86#383—21no—
87#382—21no—
88#382—21no—
89#382—21no—
90#381—21no—
91#381C2312yes—
92#381C2311yescharacteristic
93#381C2311yescharacteristic
94#380—21no—
95#380—21no—
96#380—21no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C414yes—
105C4×C4⋊C4C2×C414yes—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4C2312yes—
110C4×C22⋊C4C2312yes—
111#376—21no—
112#376—21no—
113#376—21no—
114#376C2×C412yes—
115#376C2×C414yes—
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118C43C2×C414yes—
119#343—24no—
120#343—24no—
121#342—24no—
122#342—24no—
123C22×Q8C22⋊C412yes—
124C22×Q8C22⋊C412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C2×M16—22no—
129C2×M16—22no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138#125—22no—
139#125—22no—
140#125—22no—
141#125—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—48no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4C22⋊C414yes—
171C2×C4⋊C4C22⋊C414yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×D414yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—24no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—24no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42C22×C414yes—
191C2×C42C22×C412yes—
192C2×C42C22×C412yes—
193C2×C42C22×C414yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42#4111yescharacteristic
202C2×C42#4111yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C4⋊C8—41no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C4⋊C8—41no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214#97—41no—
215#97—41no—
216C4×C8—41no—
217C4×C8—41no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95—24no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238C24C22×D411yescharacteristic
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—44no—
244C2×Q8—82no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—81no—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C414yes—
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263C22×C4#12311yescharacteristic
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4#12311yescharacteristic
267C22×C4#12311yescharacteristic
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274M16—44no—
275M16—44no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—24no—
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—48no—
293C4⋊C4—82no—
294C4⋊C4—84no—
295C4⋊C4—82no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—44no—
299C22⋊C4—42no—
300C22⋊C4—44no—
301C22⋊C4—42no—
302C22⋊C4—82no—
303C42—24no—
304C42—24no—
305C42—24no—
306C42—42no—
307C42—42no—
308C42—44no—
309C42—42no—
310C42—48no—
311C42—82no—
312C42—81no—
313C42—81no—
314C42—84no—
315C42—82no—
316C42#11711yescharacteristic
317C42#11711yescharacteristic
318C42#11711yescharacteristic
319C42#11711yescharacteristic
320C42#11711yescharacteristic
321C42#11711yescharacteristic
322C42#11711yescharacteristic
323C42#11711yescharacteristic
324C42#12311yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C42#12311yescharacteristic
330C42#12311yescharacteristic
331C42#12311yescharacteristic
332C23—22no—
333C23—41no—
334C23—41no—
335C23C22×C22⋊C411yescharacteristic
336C23C2×#12311yescharacteristic
337C23C2×#12311yescharacteristic
338Q8—44no—
339Q8—44no—
340Q8—82no—
341Q8—82no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4#38111yescharacteristic
383C2×C4#38111yescharacteristic
384C2×C4#38111yescharacteristic
385C2×C4#38111yescharacteristic
386C2×C4#38111yescharacteristic
387C2×C4#38111yescharacteristic
388C2×C4#38111yescharacteristic
389C2×C4#38111yescharacteristic
390C2×C4C2×#11711yescharacteristic
391C2×C4C2×#11711yescharacteristic
392C2×C4C2×#12311yescharacteristic
393C2×C4C2×#12311yescharacteristic
394C8—82no—
395C8—82no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—82no—
405C22C2×#38111yescharacteristic
406C22#288412yes—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—44no—
419C4—44no—
420C4—44no—
421C4—82no—
422C4—84no—
423C4—161no—
424C4—161no—
425C4—162no—
426C4#182712yes—
427C2—22no—
428C2—41no—
429C2—41no—
430C2#2069111yescharacteristic
431C2C2×#182711yescharacteristic
432C2#2070811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
19. #2295C4semidirect
20. #2295C4semidirect
24. C42×Q8C4semidirect

Maximal subgroups

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

subgroupindex
#340852
#218312
#218262
#87412
#87402
#79452
#79442
#79412
#79402