Finite Group Databasebeta

← #162591

Subgroups — #162591

Order 512.

898 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3C4×#423C211yesmaximal, characteristic
4#7689C214yesmaximal, complemented
5#7673C214yesmaximal, complemented
6#7609C212yesmaximal
7#7607C212yesmaximal
8C2×#423—21no—
9C2×C4×M16—21no—
10#2863—21no—
11#2863—21no—
12#2863—21no—
13#2863—21no—
14#2863C412yes—
15#2863C412yes—
16#2863C2212yes—
17#2863C2212yes—
18C2×#391C411yescharacteristic
19C2×#391C411yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#389C411yescharacteristic
22C2×#389C411yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387—21no—
25C2×#387—21no—
26C2×#387—21no—
27C2×#383—21no—
28C2×C43C2211yescharacteristic
29#1839C2212yes—
30#1831—21no—
31#1817C2212yes—
32#1797—21no—
33#1767—21no—
34#1767C2212yes—
35C4×C4⋊C8—21no—
36C4×C4⋊C8C2212yes—
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8—21no—
39#1744—21no—
40#1728C2212yes—
41#1272C2218yes—
42#1271C2218yes—
43C23×Q8C2×C411yescharacteristic
44C22×M16—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58#423—28no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—22no—
63C4×M16—44no—
64C2×#97—22no—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67#391—21no—
68#391—21no—
69#391—21no—
70#391—21no—
71#391—21no—
72#391—21no—
73#391C2×C412yes—
74#391C2×C412yes—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C412yes—
78#391C2312yes—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2312yes—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#383—22no—
104#383—22no—
105#383—22no—
106#383—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336C2314yes—
129#336C2314yes—
130C22×Q8—21no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8C22×C412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—28no—
147C2×M16—44no—
148C2×M16—44no—
149C22×C8—24no—
150C22×C8—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C42—24no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
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×C42SD1614yes—
188C2×C42Q1614yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
205C4⋊C8—28no—
206C4⋊C8—28no—
207C4⋊C8—44no—
208C22⋊C8—44no—
209#97—44no—
210C4×C8—44no—
211C4×C8—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—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230C24C2×C22⋊C411yescharacteristic
231C2×Q8—24no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—88no—
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×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4#10314yes—
272C22×C4#10314yes—
273C22×C4#10414yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×SD1612yes—
286C22×C4C2×Q1612yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290M16—416no—
291C2×C8—28no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
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#10314yes—
334C42#10314yes—
335C42C2×SD1612yes—
336C42C2×SD1612yes—
337C42C2×Q1612yes—
338C42C2×Q1612yes—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23#41012yes—
344C23C2×#10312yes—
345C23C2×#10412yes—
346C23C22×C22⋊C411yescharacteristic
347Q8—84no—
348Q8—84no—
349C2×C4—28no—
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—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4#41312yes—
404C2×C4C2×#10312yes—
405C2×C4C2×#10312yes—
406C2×C4C2×#10312yes—
407C2×C4C2×#10312yes—
408C2×C4C2×#10312yes—
409C2×C4C2×#10412yes—
410C2×C4C2×#10412yes—
411C2×C4#42112yes—
412C2×C4#56812yes—
413C2×C4C22×SD1611yescharacteristic
414C2×C4C22×Q1611yescharacteristic
415C2×C4#57512yes—
416C8—48no—
417C8—84no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22#150314yes—
425C22#287911yescharacteristic
426C22C2×#41011yescharacteristic
427C22C2×#41311yescharacteristic
428C22C22×#10311yescharacteristic
429C22#289611yescharacteristic
430C22C22×#10411yescharacteristic
431C22C2×#42111yescharacteristic
432C4—22no—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—21no—
439C4—21no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—41no—
446C4—41no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4—82no—
451C2—24no—
452C2#1739811yescharacteristic
453C2#1748611yescharacteristic
454C2#1750211yescharacteristic
455C2C2×#150311yescharacteristic
456C2#1758011yescharacteristic
457C2#1764211yescharacteristic
458C2#1770811yescharacteristic
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. #7689C2semidirect
5. #7673C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28632
#207772
C4×#4232
#76892
#76732
#76092
#76072