Finite Group Databasebeta

← #190969

Subgroups — #190969

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34145C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#8707C212yesmaximal
5#8570C212yesmaximal, complemented
6#8566C212yesmaximal, complemented
7#8314C212yesmaximal
8#8180C211yesmaximal, characteristic, complemented
9#8178C211yesmaximal, characteristic, complemented
10#8173C211yesmaximal, characteristic, complemented
11#8171C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13#2370—22no—
14#2369C412yes—
15#2368—22no—
16#2365C412yes—
17#2360C412yes—
18#2360C412yes—
19#2296—21no—
20#2295—21no—
21#2295—21no—
22#2295C2211yescharacteristic
23#2295C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25C42×Q8C2211yescharacteristic
26C4×#117—21no—
27C4×#117C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1840—21no—
32#1831C2212yes—
33#1818—21no—
34#1797C2212yes—
35#1767C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38#1750—21no—
39C4×C22⋊C8C2212yes—
40#1746—21no—
41#1744C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#533—24no—
52#533—24no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2311yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×#117—21no—
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70C2×C4⋊C8—22no—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×#97—22no—
76C2×#97—21no—
77C2×C4×C8—21no—
78C2×C4×C8C2312yes—
79C2×C4×C8C2312yes—
80#391—22no—
81#391—22no—
82#391C2×C412yes—
83#391C2×C412yes—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389C2×C412yes—
89#389C2×C412yes—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#385—22no—
97#385—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2311yescharacteristic
114C4×C4⋊C4C2311yescharacteristic
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4D412yes—
119#376—21no—
120#376—21no—
121#376D411yescharacteristic
122#376D411yescharacteristic
123C2×#95—21no—
124C43—21no—
125C43—21no—
126C43C2311yescharacteristic
127C43C2311yescharacteristic
128#336—22no—
129#336—22no—
130#336—21no—
131#336—21no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136Q8⋊C8—24no—
137Q8⋊C8—24no—
138Q8⋊C8—24no—
139Q8⋊C8—24no—
140C22×Q8—21no—
141C22×Q8C22×C411yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C23×C4—22no—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155#125—28no—
156C22⋊Q8—28no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C4×Q8—28no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8#4114yes—
169C4×Q8#4114yes—
170#117—28no—
171#117—24no—
172#117—41no—
173#117—41no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C412yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C22×C412yes—
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2411yescharacteristic
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—44no—
228C22⋊C8—44no—
229C22⋊C8—44no—
230#97—42no—
231#97—44no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—42no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—21no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252C24C2×C22⋊C411yescharacteristic
253C2×Q8—24no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—82no—
261C2×Q8#11712yes—
262C2×Q8#11712yes—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—82no—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4#13714yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4#11712yes—
324C4⋊C4#11712yes—
325C4⋊C4#13114yes—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C42—21no—
331C42—21no—
332C42—22no—
333C42—22no—
334C42—24no—
335C42—24no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—42no—
347C42—42no—
348C42—44no—
349C42—44no—
350C42—44no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—82no—
356C42—82no—
357C42C2×#4112yes—
358C42C2×#4112yes—
359C23—22no—
360C23—21no—
361C23—22no—
362C23—21no—
363C23#41912yes—
364C23C22×C22⋊C411yescharacteristic
365Q8—22no—
366Q8—22no—
367Q8—24no—
368Q8—41no—
369Q8—41no—
370Q8—42no—
371Q8—84no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—88no—
412C2×C4#41912yes—
413C2×C4C2×#11711yescharacteristic
414C2×C4C2×#11711yescharacteristic
415C2×C4#52512yes—
416C2×C4#52512yes—
417C2×C4C2×#13112yes—
418C2×C4#56812yes—
419C2×C4C2×#13712yes—
420C2×C4#57511yescharacteristic
421C2×C4#57511yescharacteristic
422C8—44no—
423C8—82no—
424C8—82no—
425C8—82no—
426C22—22no—
427C22—24no—
428C22—41no—
429C22—42no—
430C22—41no—
431C22#228011yescharacteristic
432C22#288012yes—
433C22C2×#41911yescharacteristic
434C22#289611yescharacteristic
435C22#290112yes—
436C4—22no—
437C4—22no—
438C4—41no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—44no—
444C4—42no—
445C4—42no—
446C4—41no—
447C4—42no—
448C4—42no—
449C4—82no—
450C4#292211yescharacteristic
451C4#292311yescharacteristic
452C4#292512yes—
453C4#311712yes—
454C4#311712yes—
455C2—42no—
456C2#1751312yes—
457C2#1751412yes—
458C2#2072011yescharacteristic
459C2#2074111yescharacteristic
460C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8570C2semidirect
6. #8566C2semidirect
8. #8180C2semidirect
9. #8178C2semidirect
10. #8173C2semidirect
11. #8171C2semidirect

Maximal subgroups

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

subgroupindex
#341452
#214312
#207862
#87072
#85702
#85662
#83142
#81802
#81782
#81732
#81712