Finite Group Databasebeta

← #201589

Subgroups — #201589

Order 512.

1081 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#35179C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#17491C212yesmaximal
5#9488C214yesmaximal, complemented
6#9484C214yesmaximal, complemented
7#7961C212yesmaximal
8C2×#533C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2984—24no—
11#2983—22no—
12#2983C2212yes—
13#2982C2212yes—
14Q8×M16—24no—
15#2920—22no—
16#2920C2212yes—
17#2920C2212yes—
18#2855—22no—
19#2627C414yes—
20#2582—21no—
21#2582—21no—
22#2581C2212yes—
23#2581C2212yes—
24#2568C412yes—
25#2567C412yes—
26#2565—22no—
27#2317—21no—
28#2310—21no—
29#2285C2211yescharacteristic
30#1821—21no—
31#1801C2212yes—
32#1801C2212yes—
33#1784—21no—
34C4×#101—21no—
35C4×#101—21no—
36#1497—24no—
37#1487C2218yes—
38#1485—24no—
39#1286C2218yes—
40C23×Q8D411yescharacteristic
41#533—22no—
42#533—22no—
43#533—22no—
44#533—22no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8C2312yes—
60C2×C4×Q8C2311yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×#117—21no—
63C2×#117—21no—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4D411yescharacteristic
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68#446—216no—
69#446—28no—
70C8×Q8—28no—
71#432—24no—
72#432—24no—
73#432—22no—
74#432—22no—
75#432—22no—
76#432—22no—
77#431—24no—
78#431—24no—
79#431—22no—
80#431—22no—
81#431D414yes—
82#431C2314yes—
83#423—24no—
84#423—22no—
85#423C2314yes—
86C2×#101—22no—
87C2×#101—22no—
88#407—22no—
89#407—22no—
90C4×M16—24no—
91C4×M16—22no—
92C4×M16D414yes—
93#400C2×C414yes—
94#400C2×C414yes—
95#398—24no—
96#398—24no—
97#395C2×C414yes—
98#395C2×C414yes—
99#391—22no—
100#391—22no—
101#391—22no—
102#391—22no—
103#391—22no—
104#391—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#387—24no—
110#387—24no—
111#385—24no—
112#384—22no—
113#384—21no—
114#382—22no—
115#382C2312yes—
116#381—24no—
117#380—21no—
118#380C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121#343—22no—
122#343—22no—
123Q8⋊C8—216no—
124C22×Q8—24no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—42no—
129C22×Q8C22⋊C412yes—
130C22×Q8C22×C412yes—
131C22×Q8C2×D411yescharacteristic
132C22×Q8C2×D411yescharacteristic
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—24no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144#125—42no—
145#125—42no—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—24no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—42no—
167C4×Q8—42no—
168#117—44no—
169#117—42no—
170#117—42no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42C2411yescharacteristic
213C4⋊C8—28no—
214C4⋊C8—28no—
215C4⋊C8—24no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219#101—44no—
220#101—44no—
221C22⋊C8—24no—
222C22⋊C8—24no—
223C22⋊C8—42no—
224C22⋊C8—42no—
225#97—28no—
226#97—24no—
227#97—42no—
228C4×C8—28no—
229C4×C8—42no—
230C4×C8C2×D418yes—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238#95C22×C414yes—
239C24C2×C22⋊C411yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—42no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8—44no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—82no—
255C2×Q8#13714yes—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12311yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288M16—48no—
289M16—44no—
290M16—44no—
291C2×C8—28no—
292C2×C8—44no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—82no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—44no—
325C42—44no—
326C42C4×D412yes—
327C42C4×D412yes—
328C42#12012yes—
329C42#12112yes—
330C42#12112yes—
331C42#12312yes—
332C42C22×D412yes—
333C42C2×#4112yes—
334C23—21no—
335C23—21no—
336C23—41no—
337C23—41no—
338C23#38611yescharacteristic
339C23#38611yescharacteristic
340C23C22×C22⋊C411yescharacteristic
341Q8—28no—
342Q8—48no—
343Q8—48no—
344Q8—44no—
345Q8—82no—
346Q8—82no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4#38611yescharacteristic
383C2×C4#38612yes—
384C2×C4#38612yes—
385C2×C4#38611yescharacteristic
386C2×C4#41914yes—
387C2×C4C2×C4×D412yes—
388C2×C4C2×#12011yescharacteristic
389C2×C4C2×#12112yes—
390C2×C4C2×#12311yescharacteristic
391C2×C4C2×#13712yes—
392C2×C4#57512yes—
393C8—48no—
394C8—84no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22C2×#38611yescharacteristic
406C22C2×#41911yescharacteristic
407C22#289611yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—24no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—84no—
419C4—82no—
420C4—82no—
421C4#191212yes—
422C4#305614yes—
423C2—21no—
424C2—21no—
425C2—41no—
426C2—41no—
427C2#2096411yescharacteristic
428C2#2097211yescharacteristic
429C2C2×#191211yescharacteristic
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. #9488C2semidirect
6. #9484C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#351792
#207272
#174912
#94882
#94842
#79612