Finite Group Databasebeta

← #214467

Subgroups — #214467

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2#21822C211yesmaximal, characteristic
3#13942C211yesmaximal, characteristic
4#9152C212yesmaximal
5#9144C212yesmaximal
6#9116C214yesmaximal
7#8716C212yesmaximal
8#8333C212yesmaximal
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#432C2212yes—
12#2855C2212yes—
13#2855C2212yes—
14#2627—21no—
15#2627—21no—
16#2582C414yescomplemented
17#2581—22no—
18#2568C412yescomplemented
19#2567C412yescomplemented
20#2565—22no—
21#2317—21no—
22#2310—21no—
23C2×#391—21no—
24#2285C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1985—22no—
28#1985—22no—
29#1817C2212yes—
30#1817C2212yes—
31C2×#328C2214yes—
32C2×#328C2214yes—
33#1441—21no—
34#1441—21no—
35C4⋊#95—21no—
36#1434C2212yes—
37#1434C2212yes—
38#1294C2214yes—
39#1294C2214yes—
40C23×Q8D411yescharacteristic
41#533—42no—
42#533—42no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2312yes—
48C2×C4×Q8—21no—
49C2×C4×Q8—42no—
50C2×C4×Q8—42no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8D412yes—
56C2×#117—21no—
57C2×#117—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67#432—42no—
68#432—42no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2314yes—
71C2×#97—22no—
72#400—22no—
73#400—22no—
74#398—24no—
75#398—24no—
76#395—22no—
77#395—22no—
78#391—22no—
79#391—22no—
80#391—24no—
81#391—44no—
82#391C2×C414yes—
83#391C2×C414yes—
84#389—24no—
85#389C2×C414yes—
86#389C2×C414yes—
87#387—24no—
88#387—24no—
89#385—22no—
90#385—22no—
91#384—22no—
92#384C2×C412yes—
93#382—22no—
94#382—21no—
95#382—42no—
96#381—22no—
97#381—22no—
98#380—21no—
99#380C2312yes—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—42no—
102C4×C4⋊C4C2×C412yes—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95C2312yes—
109C2×#95C2312yes—
110#336—22no—
111#336—22no—
112#336—24no—
113#328—44no—
114#328—44no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—41no—
121C22×Q8—41no—
122C22×Q8C22⋊C412yes—
123C22×Q8C22⋊C412yes—
124C22×Q8C22×C412yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—24no—
135C4⋊Q8—42no—
136C4⋊Q8—42no—
137#125—44no—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140C22⋊Q8—42no—
141C22⋊Q8—42no—
142C4×Q8—44no—
143C4×Q8—44no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×Q8—42no—
148C4×Q8—44no—
149C4×Q8—44no—
150#117—42no—
151#117—42no—
152#117—42no—
153#117—42no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C414yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—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—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C412yes—
198C2×C42C22×C412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C4⋊C8—44no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206#97—44no—
207#95—22no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—22no—
215#95—24no—
216#95—24no—
217#95—44no—
218#95—44no—
219#95—42no—
220#95—44no—
221#95—44no—
222C24C22×D411yescharacteristic
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—44no—
231C2×Q8—44no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—82no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4AGL(1,ℤ/8)12yes—
287C22×C4#13712yes—
288C22×C4#13712yes—
289C22×C4#13712yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4H(5,𝔽2)12yes—
293C22×C4H(5,𝔽2)12yes—
294C2×C8—48no—
295C2×C8—44no—
296C4⋊C4—24no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—48no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—82no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C42—24no—
320C42—24no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—44no—
330C42—44no—
331C42—84no—
332C23—22no—
333C23—22no—
334C23C22×C22⋊C411yescharacteristic
335C23#53411yescharacteristic
336C23#53411yescharacteristic
337C23C2×AGL(1,ℤ/8)11yescharacteristic
338C23C2×#13711yescharacteristic
339C23C2×#13711yescharacteristic
340C23C2×#13711yescharacteristic
341Q8—48no—
342Q8—48no—
343Q8—84no—
344Q8—84no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#41712yes—
406C2×C4#41712yes—
407C2×C4#41914yes—
408C2×C4#51812yes—
409C2×C4#53412yes—
410C8—84no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22#230811yescharacteristic
420C22#288412yes—
421C22C2×#41911yescharacteristic
422C22C2×#41711yescharacteristic
423C22#290112yes—
424C22#319311yescharacteristic
425C22#319311yescharacteristic
426C22#319511yescharacteristic
427C22#319511yescharacteristic
428C4—22no—
429C4—22no—
430C4—24no—
431C4—22no—
432C4—22no—
433C4—42no—
434C4—42no—
435C4—41no—
436C4—42no—
437C4—41no—
438C4—84no—
439C4—82no—
440C4—84no—
441C4—82no—
442C4—162no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2—21no—
447C2#1762812yes—
448C2#1763212yes—
449C2#2175111yescharacteristic
450C2#2176511yescharacteristic
451C2#2177511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
16. #2582C4semidirect
18. #2568C4semidirect
19. #2567C4semidirect

Maximal subgroups

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

subgroupindex
#351792
#218222
#139422
#91522
#91442
#91162
#87162
#83332