Finite Group Databasebeta

← #161844

Subgroups — #161844

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3C2×#1831C211yesmaximal, characteristic
4#7723C212yesmaximal, complemented
5#7722C212yesmaximal, complemented
6#7709C214yesmaximal, complemented
7#7589C214yesmaximal
8C22×C4⋊C8—21no—
9#2864—21no—
10#2863—21no—
11#2863C2212yes—
12#2862C2211yescharacteristic
13#2859C2211yescharacteristic
14#2857C411yescharacteristic
15#2857C411yescharacteristic
16#2855—21no—
17#2855C412yes—
18#2851—21no—
19C4×C4⋊Q8C411yescharacteristic
20C4×C4⋊Q8C411yescharacteristic
21C4×#125—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8C2212yes—
24#2295—21no—
25#2295C412yes—
26#2283—21no—
27C2×C43C2211yescharacteristic
28#1839C2212yes—
29#1831—21no—
30#1831—21no—
31#1831—21no—
32#1831C2212yes—
33#1831C2212yes—
34#1817—21no—
35#1797C2212yes—
36#1767—21no—
37#1749—21no—
38#1745C2212yes—
39C2×#336—21no—
40C2×#336C2212yes—
41#1272C2218yes—
42#1271C2218yes—
43C23×C8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C412yes—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—44no—
62C2×C22⋊C8—22no—
63C2×#97—22no—
64#391—21no—
65#391—21no—
66#391—21no—
67#391—21no—
68#391C2×C412yes—
69#391C2×C412yes—
70#391C2×C412yes—
71#391C2312yes—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2312yes—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#385—22no—
93#385—22no—
94#384C2×C411yescharacteristic
95#384C2×C411yescharacteristic
96#383—22no—
97#382—21no—
98#382—21no—
99#382—21no—
100#381—22no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380C2×C412yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111C4×C22⋊C4—22no—
112#376—22no—
113C43C2×C411yescharacteristic
114C43C2×C411yescharacteristic
115C43D411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43C2311yescharacteristic
120C43C2311yescharacteristic
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336C2314yes—
128#336C2314yes—
129C22×Q8—21no—
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—24no—
140C22×C8—44no—
141C22×C8—44no—
142C22×C8—44no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145#125—42no—
146#125—42no—
147C22⋊Q8—28no—
148C22⋊Q8—44no—
149C4×Q8—44no—
150C4×Q8—44no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207C4⋊C8—44no—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C22⋊C8—44no—
211#97—44no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226C24C2×C22⋊C411yescharacteristic
227C2×Q8—24no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4#13114yes—
269C22×C4#13714yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—88no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C42—22no—
300C42—22no—
301C42—21no—
302C42—22no—
303C42—21no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—44no—
328C42—44no—
329C42AGL(1,ℤ/8)12yes—
330C42#13712yes—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23#40812yes—
336C23#41912yes—
337C23#42112yes—
338C23C22×C22⋊C411yescharacteristic
339Q8—84no—
340Q8—84no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—24no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
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—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4#40812yes—
394C2×C4#41012yes—
395C2×C4#41312yes—
396C2×C4#41712yes—
397C2×C4#41712yes—
398C2×C4#41912yes—
399C2×C4#42112yes—
400C2×C4C2×#13112yes—
401C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
402C2×C4C2×#13712yes—
403C2×C4C2×#13711yescharacteristic
404C8—84no—
405C8—84no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22#149714yes—
413C22C2×#40811yescharacteristic
414C22C2×#41011yescharacteristic
415C22C2×#41311yescharacteristic
416C22C2×#41911yescharacteristic
417C22C2×#41711yescharacteristic
418C22C2×#42111yescharacteristic
419C22C2×#42111yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—21no—
425C4—21no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—82no—
436C4—82no—
437C4—82no—
438C4—82no—
439C2—24no—
440C2#1738211yescharacteristic
441C2#1746611yescharacteristic
442C2C2×#149711yescharacteristic
443C2#1758811yescharacteristic
444C2#1759611yescharacteristic
445C2#1766911yescharacteristic
446C2#1774911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7723C2semidirect
5. #7722C2semidirect
6. #7709C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#207782
C2×#18312
#77232
#77222
#77092
#75892