Finite Group Databasebeta

← #160609

Subgroups — #160609

Order 512.

954 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3C2×#1839C211yesmaximal, characteristic
4#7604C212yesmaximal, complemented
5#7604C212yesmaximal, complemented
6#7585C214yesmaximal, complemented
7#7579C214yesmaximal
8C2×#533C411yescharacteristic
9C4×C4⋊Q8C411yescharacteristic
10C4×C4⋊Q8C411yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13C4×#125C412yes—
14C4×#125C2212yes—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19#2296C2212yes—
20#2295—21no—
21#2295—21no—
22#2285C411yescharacteristic
23C42×Q8—21no—
24C42×Q8—21no—
25C4×#117C412yes—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1839—21no—
29#1839C2212yes—
30#1839C2212yes—
31#1839C2212yes—
32#1831—21no—
33#1817C2212yes—
34#1797—21no—
35C4×C4⋊C8C2212yes—
36C4×C22⋊C8—21no—
37#1744—21no—
38C2×#336—21no—
39C2×#336—21no—
40C2×#336C2212yes—
41#1271C2218yes—
42#1271C2218yes—
43C23×C8—22no—
44#533—24no—
45#533C818yescomplemented
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#125C2×C412yes—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×#117C2×C412yes—
58C2×#117C2×C412yes—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C4×C8—22no—
69#391—21no—
70#391—21no—
71#391—21no—
72#391—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389C2×C412yes—
78#389C2×C412yes—
79#389C2312yes—
80#389C2312yes—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#385—22no—
86#385—22no—
87#384C2×C411yescharacteristic
88#384C2×C411yescharacteristic
89#382C2×C412yes—
90#380—21no—
91#380—21no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2×C412yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110#376—22no—
111C43—22no—
112C43C2×C411yescharacteristic
113C43C2×C411yescharacteristic
114C43D411yescharacteristic
115C43D411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43C2311yescharacteristic
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336C2314yes—
129#336C2314yes—
130C22×Q8—21no—
131C22×Q8—21no—
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—44no—
142C22×C8—44no—
143C22×C8—44no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8C2×C814yescomplemented
147C4⋊Q8C2×C814yescomplemented
148#125—24no—
149#125C2×C818yescomplemented
150C22⋊Q8—28no—
151C22⋊Q8—28no—
152C4×Q8—28no—
153C4×Q8—28no—
154C4×Q8—44no—
155C4×Q8—44no—
156C4×Q8—44no—
157C4×Q8—44no—
158#117—24no—
159#117—44no—
160#117C2×C818yes—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C2×C814yes—
204C2×C42M1614yes—
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2411yescharacteristic
221C4⋊C8—44no—
222C22⋊C8—44no—
223C4×C8—44no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234C24C2×C22⋊C411yescharacteristic
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
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—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C22⋊C814yes—
263C22×C4C22⋊C814yes—
264C22×C4#10114yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C22×C812yes—
277C22×C4C2×M1612yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—88no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4C22×C814yes—
305C4⋊C4C22×C814yes—
306C22⋊C4—28no—
307C22⋊C4—44no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—28no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—44no—
334C42—44no—
335C42—44no—
336C42—44no—
337C42—44no—
338C42C22⋊C814yes—
339C42C22⋊C814yes—
340C42C22×C812yes—
341C42C22×C812yes—
342C42C2×M1612yes—
343C42C2×M1612yes—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23C2×C22⋊C812yes—
349C23#41012yes—
350C23C2×#10112yes—
351C23C22×C22⋊C411yescharacteristic
352Q8—48no—
353Q8—48no—
354C2×C4—28no—
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—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4C2×C22⋊C812yes—
405C2×C4C2×C22⋊C812yes—
406C2×C4C2×C22⋊C812yes—
407C2×C4C2×C22⋊C812yes—
408C2×C4C2×C22⋊C812yes—
409C2×C4C2×#10112yes—
410C2×C4C2×#10112yes—
411C2×C4#42112yes—
412C2×C4#42112yes—
413C2×C4C23×C811yescharacteristic
414C2×C4C22×M1611yescharacteristic
415C2×C4#57512yes—
416C2×C4#57512yes—
417C8—84no—
418C8—84no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22#146114yes—
426C22C22×C22⋊C811yescharacteristic
427C22C2×#41011yescharacteristic
428C22C22×#10111yescharacteristic
429C22#289611yescharacteristic
430C22#289611yescharacteristic
431C22C2×#42111yescharacteristic
432C22C2×#42111yescharacteristic
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—21no—
444C4—21no—
445C4—22no—
446C4—41no—
447C4—44no—
448C4—44no—
449C4—41no—
450C4—82no—
451C4—82no—
452C2—24no—
453C2C2×#146111yescharacteristic
454C2#1743711yescharacteristic
455C2#1743711yescharacteristic
456C2#1758611yescharacteristic
457C2#1759311yescharacteristic
458C2#1767211yescharacteristic
459C2#1775511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×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. #7604C2semidirect
5. #7604C2semidirect
6. #7585C2semidirect
45. #533C8semidirect
146. C4⋊Q8C2×C8semidirect
147. C4⋊Q8C2×C8semidirect
149. #125C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
#207772
C2×#18392
#76042
#75852
#75792