Finite Group Databasebeta

← #223171

Subgroups — #223171

Order 512.

883 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic
3D4×C4⋊C8C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9128C212yesmaximal, complemented
7#9127C212yesmaximal, complemented
8#8727C212yesmaximal, complemented
9#8587C211yesmaximal, characteristic, complemented
10C4×#333C211yesmaximal, characteristic, complemented
11#8039C212yesmaximal, complemented
12C2×C8×D4—21no—
13C2×#431C2212yes—
14C22×C4⋊C8C2212yes—
15#2868C2211yescharacteristic
16#2857C2212yes—
17#2571—21no—
18#2563—21no—
19#2538C412yes—
20#2497—22no—
21#2492C414yes—
22#2439—22no—
23#2369—21no—
24#2355C412yes—
25#2354—21no—
26C4×C4⋊Q8C2211yescharacteristic
27#2285C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1939—21no—
31#1923—22no—
32#1850—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1797C2212yes—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#333C2211yescharacteristic
41C2×#333C2211yescharacteristic
42C2×#333C2212yes—
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50C23×C8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2×C412yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C22⋊Q8C2×C414yes—
56C2×C4×Q8C2×C412yes—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×#117—22no—
61C22×C42C2312yes—
62C8×D4—44no—
63#431—24no—
64#431—42no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#400—24no—
85#396—24no—
86#393—24no—
87#391—24no—
88#391C2×C412yes—
89#391C2×C414yes—
90#390—22no—
91#389—22no—
92#389—21no—
93#389—21no—
94#389—22no—
95#389—22no—
96#388—24no—
97#387—24no—
98#384—21no—
99#384—21no—
100#384C2×C412yes—
101#384C2311yescharacteristic
102#384C2312yes—
103#382—22no—
104#382—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380C2×C414yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C414yes—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4D412yes—
119C4×C22⋊C4D412yes—
120C43C2311yescharacteristic
121#336—22no—
122#336—22no—
123#336—21no—
124#336—21no—
125#333—22no—
126#333—22no—
127#333—42no—
128C22×Q8—22no—
129C22×Q8C22×C412yes—
130C22×D4C22⋊C411yescharacteristic
131C23×C4C22×C412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—44no—
141C22×C8—44no—
142C22×C8—82no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—81no—
150C4⋊Q8—81no—
151C4⋊Q8—82no—
152C22⋊Q8—84no—
153C4×Q8—44no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—42no—
157C4×D4SD1614yes—
158C4×D4Q1614yes—
159#117—82no—
160#117—82no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C414yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22⋊C412yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—24no—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2411yescharacteristic
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—44no—
209C4⋊C8—42no—
210C4⋊C8—42no—
211C4⋊C8—81no—
212C4⋊C8—81no—
213C4⋊C8#4112yes—
214C4⋊C8#4112yes—
215C22⋊C8—44no—
216C22⋊C8—44no—
217C22⋊C8—82no—
218C4×C8—44no—
219C4×C8—42no—
220C4×C8—42no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95—22no—
225#95—24no—
226#95—24no—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230C24C2×C22⋊C412yes—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4#10314yes—
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—22no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×SD1614yes—
266C22×C4C2×Q1614yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—88no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—44no—
307C4⋊C4—84no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4#10314yes—
311C22⋊C4—24no—
312C22⋊C4—42no—
313C22⋊C4—84no—
314C22⋊C4—84no—
315C22⋊C4#10318yes—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—22no—
327C42—41no—
328C42—44no—
329C42—84no—
330C42C2×SD1612yes—
331C42C2×Q1612yes—
332C42#13512yes—
333C42#13512yes—
334C42C2×#4111yescharacteristic
335C42C2×#4111yescharacteristic
336C42H(5,𝔽2)11yescharacteristic
337C422−1+411yescharacteristic
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23C2×#10314yes—
345C23C22×C22⋊C411yescharacteristic
346Q8—82no—
347Q8—82no—
348Q8—164no—
349D4—28no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4#41312yes—
387C2×C4C2×#10312yes—
388C2×C4C2×#10314yes—
389C2×C4C2×#10312yes—
390C2×C4#41612yes—
391C2×C4#41612yes—
392C2×C4C4×#4111yescharacteristic
393C2×C4#52011yescharacteristic
394C2×C4#54611yescharacteristic
395C2×C4#54611yescharacteristic
396C2×C4#54711yescharacteristic
397C2×C4#54711yescharacteristic
398C2×C4C22×SD1611yescharacteristic
399C2×C4C22×Q1611yescharacteristic
400C2×C4C2×#13511yescharacteristic
401C2×C4C2×#13511yescharacteristic
402C2×C4#57611yescharacteristic
403C2×C4#57711yescharacteristic
404C2×C4#57711yescharacteristic
405C2×C4#57811yescharacteristic
406C8—42no—
407C8—42no—
408C8—81no—
409C8—81no—
410C8—82no—
411C8—82no—
412C22—24no—
413C22—24no—
414C22—24no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22#233911yescharacteristic
420C22C2×#41311yescharacteristic
421C22#288811yescharacteristic
422C22C22×#10311yescharacteristic
423C22C2×#41611yescharacteristic
424C22#289711yescharacteristic
425C22#289811yescharacteristic
426C4—22no—
427C4—22no—
428C4—21no—
429C4—21no—
430C4—24no—
431C4—41no—
432C4—42no—
433C4—42no—
434C4—84no—
435C4—82no—
436C4—82no—
437C4#154214yes—
438C4#329511yescharacteristic
439C4#329911yescharacteristic
440C4#330211yescharacteristic
441C4#330411yescharacteristic
442C4#333511yescharacteristic
443C4#333811yescharacteristic
444C4#334011yescharacteristic
445C4#334211yescharacteristic
446C2—44no—
447C2C2×#154211yescharacteristic
448C2#1772211yescharacteristic
449C2#1772411yescharacteristic
450C2#1777211yescharacteristic
451C2#2248311yescharacteristic
452C2#2248711yescharacteristic
453C2#2249011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9898C2semidirect
5. #9896C2semidirect
6. #9128C2semidirect
7. #9127C2semidirect
8. #8727C2semidirect
9. #8587C2semidirect
10. C4×#333C2semidirect
11. #8039C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#227952
D4×C4⋊C82
#98982
#98962
#91282
#91272
#87272
#85872
C4×#3332
#80392