Finite Group Databasebeta

← #197377

Subgroups — #197377

Order 512.

1145 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#35248C211yesmaximal, characteristic
3C2×C4×#101C211yesmaximal, characteristic
4#17648C212yesmaximal
5#9190C218yesmaximal, complemented
6#7973C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2984—24no—
10#2983—21no—
11#2983C2214yes—
12#2982—21no—
13Q8×M16—24no—
14#2920—22no—
15#2920C2212yes—
16C22×#101—21no—
17C2×C4×M16C2212yes—
18#2855—22no—
19#2634C2214yes—
20#2581—22no—
21#2568C411yescharacteristic
22#2568C411yescharacteristic
23#2546C411yescharacteristic
24#2546C411yescharacteristic
25#2503—21no—
26#2484C414yes—
27#2434—21no—
28#2327—21no—
29#2327—21no—
30#2283C2211yescharacteristic
31C4×#101—22no—
32C4×#101C2214yes—
33C2×#343—21no—
34#1531—24no—
35#1528C2218yes—
36#1526—24no—
37#1288C2218yes—
38C23×Q8D411yescharacteristic
39C22×M16—22no—
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#125C2312yes—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D412yes—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8C2312yes—
63C2×#117C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4D411yescharacteristic
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68#446—28no—
69#446—216no—
70C8×Q8—28no—
71#432—24no—
72#432—24no—
73#432D414yes—
74#432C2314yes—
75#431—24no—
76#431—24no—
77#431—22no—
78#431—22no—
79#423—24no—
80#423—22no—
81C2×#101—28no—
82C2×#101—24no—
83#407—22no—
84C4×M16—24no—
85C4×M16—22no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16D414yes—
89C4×M16C2314yes—
90C2×#97—22no—
91C2×C4×C8—22no—
92#400—24no—
93#400C2×C414yes—
94#400C2×C414yes—
95#398C2×C414yes—
96#398C2×C414yes—
97#395—24no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391—22no—
103#391—22no—
104#389—22no—
105#389—22no—
106#387—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#385—24no—
111#383C2312yes—
112#382—21no—
113#382—21no—
114#382—21no—
115#381—24no—
116#381C2×C412yes—
117#380—21no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C22⋊C4—21no—
121#376—21no—
122#343—24no—
123#330—216no—
124C22×Q8—24no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—42no—
130C22×Q8C22⋊C411yescharacteristic
131C22×Q8C22⋊C411yescharacteristic
132C22×Q8C2×D412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—28no—
140C2×M16—24no—
141C2×M16—44no—
142C2×M16—42no—
143C22×C8—42no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—44no—
149C4⋊Q8—44no—
150#125—24no—
151#125—24no—
152C22⋊Q8—41no—
153C22⋊Q8—41no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C22⋊Q8—41no—
157C22⋊Q8—41no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174#117—24no—
175#117—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42C22⋊C412yes—
207C2×C42C22⋊C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—28no—
219C4⋊C8—28no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222#101—216no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225#97—28no—
226#97—24no—
227#97—42no—
228#97C2×D418yes—
229C4×C8—28no—
230C4×C8—24no—
231C4×C8—24no—
232C4×C8—42no—
233#95—22no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—22no—
241#95—22no—
242#95C22×C412yes—
243#95C22×C412yes—
244C24C2×C22⋊C411yescharacteristic
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—42no—
253C2×Q8—48no—
254C2×Q8—48no—
255C2×Q8—44no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8#10414yes—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4#12311yescharacteristic
291C22×C4#12311yescharacteristic
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295M16—28no—
296M16—48no—
297M16—44no—
298M16—48no—
299C2×C8—28no—
300C2×C8—48no—
301C2×C8—44no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—48no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—48no—
321C22⋊C4—44no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—44no—
333C42—42no—
334C42—42no—
335C42C4×D412yes—
336C42C4×D412yes—
337C42#12012yes—
338C42#12112yes—
339C42#12112yes—
340C42#12312yes—
341C42C22×D412yes—
342C42C2×#4112yes—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—41no—
347C23#38611yescharacteristic
348C23#38611yescharacteristic
349C23C22×C22⋊C411yescharacteristic
350Q8—28no—
351Q8—48no—
352Q8—44no—
353Q8—82no—
354Q8—84no—
355Q8—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—82no—
394C2×C4#38611yescharacteristic
395C2×C4#38612yes—
396C2×C4#38612yes—
397C2×C4#38611yescharacteristic
398C2×C4C2×#10414yes—
399C2×C4C2×#10412yes—
400C2×C4#42112yes—
401C2×C4C2×C4×D412yes—
402C2×C4C2×#12011yescharacteristic
403C2×C4C2×#12112yes—
404C2×C4C2×#12311yescharacteristic
405C8—48no—
406C8—48no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22#191512yes—
417C22C2×#38611yescharacteristic
418C22C22×#10411yescharacteristic
419C22C2×#42111yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—24no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—82no—
434C4#187114yes—
435C4#191212yes—
436C2—21no—
437C2—21no—
438C2—22no—
439C2—41no—
440C2C2×#187111yescharacteristic
441C2C2×#191211yescharacteristic
442C2C2×#191511yescharacteristic
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. #9190C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#352482
C2×C4×#1012
#176482
#91902
#79732