Finite Group Databasebeta

← #201239

Subgroups — #201239

Order 512.

1067 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
1C2×#2983C211yesmaximal, characteristic
2#35179C211yesmaximal, characteristic
3#17755C212yesmaximal
4C2×#1467C211yesmaximal, characteristic
5#9456C214yesmaximal, complemented
6#9454C214yesmaximal, complemented
7#7952C212yesmaximal
8C2×#533C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2983—24no—
11#2983—22no—
12#2983C2214yes—
13C2×#446—24no—
14C2×#432—21no—
15C2×#432C2212yes—
16C2×#432C2212yes—
17C2×#431—21no—
18#2855—21no—
19#2855—21no—
20#2627—21no—
21#2627—21no—
22#2582—22no—
23#2581—22no—
24#2568C2212yes—
25#2567C2212yes—
26#2565C414yes—
27#2317C412yes—
28#2310C412yes—
29#2285C2211yescharacteristic
30#1556C2214yes—
31#1556C2214yes—
32#1552—24no—
33#1552—24no—
34#1467—22no—
35#1467C2214yes—
36C2×#324—21no—
37C2×#324—21no—
38#1285C2218yes—
39C23×Q8C2×C411yescharacteristic
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×C4⋊Q8C2311yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8D412yes—
58C2×C4×Q8C2312yes—
59C2×#117C2×C412yes—
60C2×#117C2×C412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42C2311yescharacteristic
67#446—216no—
68#446—216no—
69#432—24no—
70#432—24no—
71#432—22no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432—22no—
76#432D414yes—
77#432C2314yes—
78#432C2314yes—
79#431—24no—
80#431—24no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8D414yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×#97—22no—
88C2×#97—22no—
89C2×#97—22no—
90C2×C4×C8—22no—
91#400—22no—
92#400—22no—
93#398—24no—
94#398—24no—
95#395—22no—
96#395—22no—
97#391—24no—
98#391—22no—
99#391—22no—
100#391C2×C414yes—
101#391C2×C414yes—
102#389—24no—
103#389—22no—
104#389—22no—
105#387—24no—
106#387—24no—
107#385—22no—
108#385—22no—
109#384—21no—
110#384C2×C414yes—
111#382—21no—
112#382C2×C414yes—
113#381—22no—
114#381—22no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2312yes—
119#332—28no—
120#332—28no—
121#324—24no—
122#324—24no—
123C22×Q8—24no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—42no—
129C22×Q8C22⋊C412yes—
130C22×Q8C22×C412yes—
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—24no—
139C22×C8—42no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144#125—24no—
145#125—24no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—42no—
164#117—24no—
165#117—24no—
166#117—44no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22×C414yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42C2411yescharacteristic
214C4⋊C8—28no—
215C4⋊C8—28no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—42no—
219C4⋊C8C2×D418yes—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8—24no—
223C22⋊C8—42no—
224#97—28no—
225#97—24no—
226#97—24no—
227#97—24no—
228#97—42no—
229C4×C8—28no—
230#95—22no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238C24C2×C22⋊C411yescharacteristic
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—48no—
246C2×Q8—48no—
247C2×Q8—44no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—41no—
251C2×Q8—82no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4#10114yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C4×D412yes—
281C22×C4C4×D412yes—
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12112yes—
285C22×C4#12112yes—
286C22×C4#12311yescharacteristic
287C22×C4#12311yescharacteristic
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C2×C8—28no—
292C2×C8—48no—
293C2×C8—44no—
294C2×C8—84no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—82no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—44no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—44no—
326C42—44no—
327C42C4×D412yes—
328C42C4×D412yes—
329C42#12012yes—
330C42#12112yes—
331C42#12112yes—
332C42#12312yes—
333C42C22×D412yes—
334C42C2×#4112yes—
335C23—41no—
336C23—41no—
337C23#38611yescharacteristic
338C23#38611yescharacteristic
339C23C2×#10112yes—
340C23#41312yes—
341C23C22×C22⋊C411yescharacteristic
342Q8—48no—
343Q8—48no—
344Q8—82no—
345Q8—84no—
346Q8—82no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4#38611yescharacteristic
388C2×C4#38611yescharacteristic
389C2×C4#38612yes—
390C2×C4#38612yes—
391C2×C4C2×#10114yes—
392C2×C4C2×C4×D412yes—
393C2×C4C2×#12011yescharacteristic
394C2×C4C2×#12112yes—
395C2×C4C2×#12311yescharacteristic
396C8—48no—
397C8—84no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—42no—
405C22—42no—
406C22#190112yes—
407C22#190112yes—
408C22C2×#38611yescharacteristic
409C22C22×#10111yescharacteristic
410C22C2×#41311yescharacteristic
411C4—22no—
412C4—22no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—22no—
417C4—22no—
418C4—24no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—84no—
423C4—82no—
424C4—82no—
425C4#302012yes—
426C2—21no—
427C2—21no—
428C2—22no—
429C2—22no—
430C2#2095411yescharacteristic
431C2C2×#190111yescharacteristic
432C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9456C2semidirect
6. #9454C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#351792
#177552
C2×#14672
#94562
#94542
#79522