Finite Group Databasebeta

← #214092

Subgroups — #214092

Order 512.

1130 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2323C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic, complemented
3#21818C211yesmaximal, characteristic, complemented
4#9086C212yesmaximal, complemented
5#9084C212yesmaximal
6#9082C212yesmaximal, complemented
7#9080C212yesmaximal
8#8701C212yesmaximal, complemented
9#8300C212yesmaximal
10C22×C22⋊Q8C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C2×#432C2212yes—
14C2×#431C2212yes—
15#2323—24no—
16#2323—24no—
17C2×#391C411yescharacteristic
18C2×#389C411yescharacteristic
19C2×#389C411yescharacteristic
20C2×#389C411yescharacteristic
21C2×#388C411yescharacteristic
22C2×#388C411yescharacteristic
23C2×#388C2211yescharacteristic
24C2×#388C2211yescharacteristic
25C2×#387C2211yescharacteristic
26C2×#387C2211yescharacteristic
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1985—24no—
30#1984—24no—
31#1817C2212yes—
32#1817C2212yes—
33#1817C2212yes—
34#1817C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×Q8C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—24no—
44C2×C4×Q8—44no—
45C2×C4×Q8D414yes—
46C2×C4×Q8D414yes—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—44no—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#432—44no—
74#431—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8C2314yes—
78C2×C22⋊C8C2314yes—
79C2×#97—22no—
80C2×C4×C8—22no—
81#391—24no—
82#389—24no—
83#389—24no—
84#389—24no—
85#388—24no—
86#388—24no—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#380—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2311yescharacteristic
107C2×#95C2311yescharacteristic
108#336—24no—
109#336—24no—
110#336—24no—
111#336—24no—
112#324—44no—
113#323—44no—
114C25C22×C411yescharacteristic
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—24no—
118C22×Q8—42no—
119C22×Q8C22⋊C414yes—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—42no—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141C22⋊Q8—232no—
142C4×Q8—44no—
143C4×Q8—44no—
144C4×Q8—44no—
145C4×Q8—44no—
146#117—48no—
147#117—48no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
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—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—81no—
169C2×C4⋊C4—81no—
170C2×C4⋊C4C22⋊C414yes—
171C2×C4⋊C4C22⋊C414yes—
172C2×C4⋊C4C22⋊C414yes—
173C2×C4⋊C4M1614yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—81no—
187C2×C22⋊C4—81no—
188C2×C22⋊C4M1614yes—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42—81no—
197C2×C42—81no—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C4⋊C8—44no—
205C4⋊C8—44no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C22⋊C8—24no—
210#97—44no—
211C4×C8—44no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224C24—21no—
225C24—21no—
226C24—22no—
227C24—22no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24C2×M1612yes—
233C24C23×C411yescharacteristic
234C2×Q8—28no—
235C2×Q8—28no—
236C2×Q8—44no—
237C2×Q8—48no—
238C2×Q8—48no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—28no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—48no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—48no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×M1612yes—
299C22×C4C2×M1612yes—
300C22×C4C2×M1612yes—
301C22×C4C2×M1612yes—
302C22×C4C2×M1612yes—
303C22×C4#13114yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4H(5,𝔽2)14yes—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—44no—
313C4⋊C4—28no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—44no—
341C42—82no—
342C42—82no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—24no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—42no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23C22×C22⋊C411yescharacteristic
365C23#51812yes—
366C23C22×M1611yescharacteristic
367C23C22×M1611yescharacteristic
368C23C2×#13112yes—
369Q8—416no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—44no—
399C2×C4—416no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#40714yes—
433C2×C4#40814yes—
434C2×C4#53412yes—
435C2×C4#53412yes—
436C8—84no—
437C8—84no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—84no—
450C22—84no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22#230811yescharacteristic
456C22C2×#40711yescharacteristic
457C22C2×#40811yescharacteristic
458C22#288411yescharacteristic
459C22#288411yescharacteristic
460C22#288811yescharacteristic
461C22#288911yescharacteristic
462C22#297412yes—
463C22#297412yes—
464C4—24no—
465C4—24no—
466C4—22no—
467C4—22no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C2—22no—
480C2—22no—
481C2—82no—
482C2#1737111yescharacteristic
483C2#1737211yescharacteristic
484C2#1739011yescharacteristic
485C2#1739111yescharacteristic
486C2#2174011yescharacteristic
487C2#2175111yescharacteristic
488C2#2175711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21822C2semidirect
3. #21818C2semidirect
4. #9086C2semidirect
6. #9082C2semidirect
8. #8701C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23232
#218222
#218182
#90862
#90842
#90822
#90802
#87012
#83002