Finite Group Databasebeta

← #196253

Subgroups — #196253

Order 512.

1175 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60276C211yesmaximal, characteristic
2#35200C211yesmaximal, characteristic
3#17479C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic
5#9100C214yesmaximal, complemented
6#9099C214yesmaximal, complemented
7#7857C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2988—24no—
11#2982C2212yes—
12#2972—22no—
13#2972C2212yes—
14C8×#41—24no—
15#2920C2212yes—
16C2×#431—22no—
17C2×#431C2212yes—
18#2851—22no—
19#2638—22no—
20#2599—22no—
21#2567C411yescharacteristic
22#2563C411yescharacteristic
23#2547C411yescharacteristic
24#2546C411yescharacteristic
25#2499C412yes—
26#2485—22no—
27#2432C412yes—
28#2327C2212yes—
29#2326C2212yes—
30#2283C2211yescharacteristic
31#1497—24no—
32#1493—24no—
33#1479C2218yes—
34#1465—22no—
35#1464—22no—
36#1461C2212yes—
37#1460C2212yes—
38#1274C2218yes—
39C22×#41C2×C411yescharacteristic
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×#124C2×C412yes—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2×C412yes—
51C4×#41—22no—
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2312yes—
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D412yes—
61C2×C4×D4D412yes—
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117D412yes—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68C8×Q8—28no—
69C8×D4—216no—
70C8×D4—28no—
71#432—22no—
72#432—22no—
73#431—24no—
74#431—24no—
75#431—22no—
76#431—24no—
77#431—22no—
78#431—22no—
79#431D414yes—
80#431C2314yes—
81#431C2314yes—
82#423—22no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85#407—22no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—22no—
88C4×M16—22no—
89C2×C4×C8—24no—
90C2×C4×C8D414yes—
91#398—22no—
92#397—22no—
93#397—24no—
94#395—22no—
95#394—24no—
96#393—22no—
97#391C2×C412yes—
98#391C2×C412yes—
99#391C2×C412yes—
100#390C2×C412yes—
101#389—22no—
102#389—22no—
103#388—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#385—24no—
108#385—24no—
109#384C2×C412yes—
110#383—21no—
111#382—21no—
112#382C2×C412yes—
113#381—21no—
114#380—22no—
115#380—22no—
116#380—21no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C412yes—
120#376C2312yes—
121Q8⋊C8—216no—
122#324—24no—
123#323—24no—
124C2×#41—24no—
125C2×#41—24no—
126C2×#41—22no—
127C2×#41—22no—
128C22×Q8—21no—
129C22×Q8C22×C412yes—
130C22×Q8C22×C411yescharacteristic
131C22×D4C22⋊C412yes—
132C22×D4C22×C411yescharacteristic
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—42no—
140C22×C8—28no—
141C22×C8—24no—
142C4⋊Q8—28no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147#125—41no—
148#125—41no—
149#125—41no—
150#125—41no—
151#124—28no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172#117—22no—
173#117—22no—
174#117—24no—
175#117—24no—
176#117—42no—
177#117—42no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C2×D412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42C2411yescharacteristic
217C4⋊C8—28no—
218C4⋊C8—24no—
219C4⋊C8—24no—
220C4⋊C8—24no—
221C4⋊C8—42no—
222C22⋊C8—28no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226C22⋊C8—42no—
227#97—42no—
228C4×C8—28no—
229C4×C8—24no—
230C4×C8—24no—
231C4×C8C2×D418yes—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—22no—
240#95—22no—
241C24C2×C22⋊C411yescharacteristic
242C24C2×C22⋊C412yes—
243#41—216no—
244#41—216no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—42no—
251C2×Q8—82no—
252C2×Q8#13114yes—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4#13114yes—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—81no—
276C22×C4—82no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4#12012yes—
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4#12312yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292M16—84no—
293C2×C8—28no—
294C2×C8—44no—
295C2×C8—84no—
296C2×C8—88no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—48no—
314C4⋊C4—84no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—44no—
319C22⋊C4—48no—
320C22⋊C4—82no—
321C22⋊C4—84no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—24no—
329C42—24no—
330C42—44no—
331C42C4×D412yes—
332C42C4×D412yes—
333C42#12011yescharacteristic
334C42#12011yescharacteristic
335C42#12112yes—
336C42#12112yes—
337C42#12311yescharacteristic
338C42#12311yescharacteristic
339C42C22×D412yes—
340C42C2×#4112yes—
341C23—22no—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23#38612yes—
350C23#40814yes—
351C23C22×C22⋊C411yescharacteristic
352Q8—28no—
353Q8—48no—
354Q8—84no—
355D4—216no—
356D4—28no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4#38611yescharacteristic
393C2×C4#38611yescharacteristic
394C2×C4#38611yescharacteristic
395C2×C4#38612yes—
396C2×C4#38611yescharacteristic
397C2×C4#40814yes—
398C2×C4C2×C4×D412yes—
399C2×C4C2×#12011yescharacteristic
400C2×C4C2×#12112yes—
401C2×C4C2×#12311yescharacteristic
402C2×C4C2×#13112yes—
403C2×C4C2×#13112yes—
404C8—48no—
405C8—84no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—24no—
412C22—44no—
413C22—42no—
414C22—42no—
415C22—81no—
416C22—81no—
417C22—82no—
418C22C2×#38611yescharacteristic
419C22C2×#40811yescharacteristic
420C22C2×#40811yescharacteristic
421C4—22no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—22no—
426C4—21no—
427C4—21no—
428C4—24no—
429C4—42no—
430C4—42no—
431C4—82no—
432C4—82no—
433C4—82no—
434C4—82no—
435C4#190912yes—
436C4#292914yes—
437C2—21no—
438C2—21no—
439C2—42no—
440C2—44no—
441C2#2089511yescharacteristic
442C2C2×#190911yescharacteristic
443C2#2101211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ 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
3. #17479C2semidirect
5. #9100C2semidirect
6. #9099C2semidirect
7. #7857C2semidirect

Maximal subgroups

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

subgroupindex
#602762
#352002
#174792
#173662
#91002
#90992
#78572