Finite Group Databasebeta

← #214682

Subgroups — #214682

Order 512.

1016 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34209C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic
3#17352C211yesmaximal, characteristic
4#9104C212yesmaximal, complemented
5#9100C212yesmaximal, complemented
6#8962C212yesmaximal, complemented
7#8958C212yesmaximal, complemented
8#8731C212yesmaximal
9#8323C212yesmaximal
10C22×C4⋊Q8C411yescharacteristic
11C2×#519C2211yescharacteristic
12C2×#519C2211yescharacteristic
13#2983C2212yes—
14#2982C2212yes—
15#2365—22no—
16#2365—22no—
17#2344—22no—
18#2344—22no—
19#2327C411yescharacteristic
20#2327C411yescharacteristic
21#2327C2211yescharacteristic
22#2327C2211yescharacteristic
23#2310C411yescharacteristic
24#2310C411yescharacteristic
25C4×C4⋊Q8C411yescharacteristic
26C4×C4⋊Q8C411yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29#2285C411yescharacteristic
30C4×#117C2211yescharacteristic
31#1479C2214yes—
32#1479C2214yes—
33#1467—22no—
34#1464—22no—
35#1461—22no—
36#1461C2212yes—
37#1461C2212yes—
38#1458—22no—
39#1458C2212yes—
40#1458C2212yes—
41#1276C2214yes—
42#1274C2214yes—
43C23×Q8C2×C411yescharacteristic
44C23×Q8C2×C411yescharacteristic
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52#519—44no—
53#519—44no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D412yes—
61C2×C4×Q8D412yes—
62C2×C4×Q8C2311yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×#117—21no—
67C2×#117—21no—
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117D412yes—
72C2×#117D412yes—
73C2×#117C2311yescharacteristic
74C22×C4⋊C4—22no—
75C22×C42C2×C411yescharacteristic
76#446—44no—
77C8×Q8—44no—
78#432—24no—
79#431—24no—
80#431C2314yes—
81#431C2314yes—
82#391—22no—
83#391—22no—
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2×C411yescharacteristic
89#391C2×C411yescharacteristic
90#391C2×C411yescharacteristic
91#391C2×C411yescharacteristic
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#382C2×C411yescharacteristic
107#382C2×C411yescharacteristic
108#381—22no—
109#381—22no—
110#380—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376C2×C411yescharacteristic
121#376C2311yescharacteristic
122C43C2×C411yescharacteristic
123C43C2311yescharacteristic
124Q8⋊C8—44no—
125Q8⋊C8—44no—
126#324—24no—
127#324—22no—
128#324—22no—
129#324—24no—
130#323—22no—
131#323—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—24no—
138C22×Q8—24no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8C22⋊C412yes—
142C22×Q8C22×C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×Q8C22×C411yescharacteristic
145C22×Q8C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C22⋊C412yes—
149C23×C4C22×C411yescharacteristic
150C4⋊Q8—216no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8M1614yes—
158C4⋊Q8M1614yes—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—44no—
164C4×Q8—44no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—44no—
170C4×Q8—44no—
171#117—24no—
172#117—24no—
173#117—44no—
174#117—44no—
175#117—44no—
176#117—44no—
177#117—42no—
178#117—42no—
179C2×C4⋊C4—216no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C2×D411yescharacteristic
205C2×C22⋊C4C2×D411yescharacteristic
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C22⋊C412yes—
219C2×C42C22⋊C412yes—
220C2×C42C22⋊C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2411yescharacteristic
229C4⋊C8—24no—
230C4⋊C8—24no—
231C4⋊C8—44no—
232C4⋊C8—44no—
233C22⋊C8—24no—
234C22⋊C8—24no—
235#97—44no—
236C4×C8—24no—
237C4×C8—24no—
238C4×C8—44no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253C24C2×C22⋊C411yescharacteristic
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—48no—
273C2×Q8—48no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8C2×M1612yes—
277C2×Q8C2×M1612yes—
278C2×Q8C2×M1612yes—
279C2×Q8C2×M1612yes—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4#13114yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C2×C8—44no—
312C2×C8—44no—
313C4⋊C4—28no—
314C4⋊C4—28no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—28no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C42—24no—
340C42—22no—
341C42—24no—
342C42—24no—
343C42—21no—
344C42—21no—
345C42—22no—
346C42—22no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42C2×M1612yes—
360C42C2×M1612yes—
361C42H(5,𝔽2)14yes—
362C23—21no—
363C23—21no—
364C23—41no—
365C23—41no—
366C23#40812yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—24no—
369Q8—24no—
370Q8—24no—
371Q8—24no—
372Q8—48no—
373Q8—48no—
374Q8—42no—
375Q8—42no—
376Q8—42no—
377Q8—42no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—21no—
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—41no—
398C2×C4—41no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—48no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4#40714yes—
423C2×C4#40812yes—
424C2×C4#51812yes—
425C2×C4#53412yes—
426C2×C4#53412yes—
427C2×C4C22×M1611yescharacteristic
428C2×C4C22×M1611yescharacteristic
429C2×C4C2×#13112yes—
430C8—84no—
431C8—84no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22#230811yescharacteristic
440C22C2×#40711yescharacteristic
441C22C2×#40811yescharacteristic
442C4—24no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—24no—
447C4—42no—
448C4—41no—
449C4—41no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4#297412yes—
460C4#297412yes—
461C2—21no—
462C2—21no—
463C2—42no—
464C2#2174011yescharacteristic
465C2#2178311yescharacteristic
466C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9104C2semidirect
5. #9100C2semidirect
6. #8962C2semidirect
7. #8958C2semidirect

Maximal subgroups

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

subgroupindex
#342092
#173712
#173522
#91042
#91002
#89622
#89582
#87312
#83232