Finite Group Databasebeta

← #223178

Subgroups — #223178

Order 512.

837 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic
3#22795C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9132C212yesmaximal, complemented
7#9130C212yesmaximal, complemented
8#8729C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10C4×#332C211yesmaximal, characteristic, complemented
11#8038C212yesmaximal, complemented
12C2×#431C2212yes—
13C2×#431C2212yes—
14#2868C2211yescharacteristic
15#2857C2212yes—
16#2571C412yes—
17#2563—21no—
18#2538—21no—
19#2497—22no—
20#2492—22no—
21#2439C414yes—
22#2369—21no—
23#2355—21no—
24#2354C412yes—
25C4×C4⋊Q8C2211yescharacteristic
26#2285C2212yes—
27C42×D4C2211yescharacteristic
28#1954—21no—
29#1954—21no—
30#1923—22no—
31#1923—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1797C2212yes—
35#1797C2212yes—
36C8×C4⋊C4—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42C2×#332C2212yes—
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2×C412yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C22⋊Q8—22no—
55C2×C4×Q8—21no—
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×#117C2×C414yes—
60C22×C42C2312yes—
61#431—24no—
62#431—24no—
63#431—42no—
64#431—42no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2311yescharacteristic
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83#400—24no—
84#396—24no—
85#393—24no—
86#391—21no—
87#391—22no—
88#391—24no—
89#390—22no—
90#389—22no—
91#389—21no—
92#389—22no—
93#389C2×C412yes—
94#389C2×C414yes—
95#388—24no—
96#387—24no—
97#384—21no—
98#384—21no—
99#384C2×C412yes—
100#384C2311yescharacteristic
101#384C2312yes—
102#382—22no—
103#382C2×C414yes—
104#380—22no—
105#380—22no—
106#380—22no—
107#380C2×C414yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C411yescharacteristic
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C22⋊C4C2×C412yes—
117C4×C22⋊C4D412yes—
118C4×C22⋊C4D412yes—
119C43C2311yescharacteristic
120#336—22no—
121#336—22no—
122#336—21no—
123#336—21no—
124#332—22no—
125#332—22no—
126#332—42no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×D4C22⋊C411yescharacteristic
130C23×C4C22×C412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—81no—
146C4⋊Q8—81no—
147C4⋊Q8—82no—
148C22⋊Q8—82no—
149C22⋊Q8—82no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×D4—24no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157#117—84no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C414yes—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2411yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—42no—
206C4⋊C8—42no—
207C4⋊C8—81no—
208C4⋊C8—81no—
209C4⋊C8#4112yes—
210C4⋊C8#4112yes—
211C22⋊C8—44no—
212C22⋊C8—44no—
213C22⋊C8—82no—
214C22⋊C8—82no—
215C4×C8—44no—
216C4×C8—44no—
217C4×C8—42no—
218C4×C8—42no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—24no—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228C24C2×C22⋊C412yes—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—44no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—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—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—42no—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
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—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—42no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—84no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4#10114yes—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—84no—
314C22⋊C4—84no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—22no—
326C42—41no—
327C42—44no—
328C42—84no—
329C42#13512yes—
330C42#13512yes—
331C42#13512yes—
332C42#13512yes—
333C42C2×#4111yescharacteristic
334C42C2×#4111yescharacteristic
335C42H(5,𝔽2)11yescharacteristic
336C422−1+411yescharacteristic
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23C22×C22⋊C411yescharacteristic
345Q8—81no—
346Q8—81no—
347Q8—81no—
348Q8—81no—
349Q8—164no—
350D4—44no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
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—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4C2×#10114yes—
388C2×C4C2×#10112yes—
389C2×C4#41612yes—
390C2×C4#41612yes—
391C2×C4#41612yes—
392C2×C4#41612yes—
393C2×C4C4×#4111yescharacteristic
394C2×C4#52011yescharacteristic
395C2×C4#54611yescharacteristic
396C2×C4#54611yescharacteristic
397C2×C4#54711yescharacteristic
398C2×C4#54711yescharacteristic
399C2×C4C2×#13511yescharacteristic
400C2×C4C2×#13511yescharacteristic
401C2×C4C2×#13511yescharacteristic
402C2×C4C2×#13511yescharacteristic
403C2×C4#57611yescharacteristic
404C2×C4#57711yescharacteristic
405C2×C4#57711yescharacteristic
406C2×C4#57811yescharacteristic
407C8—42no—
408C8—42no—
409C8—81no—
410C8—81no—
411C8—82no—
412C8—82no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22#233911yescharacteristic
421C22C22×#10111yescharacteristic
422C22#288811yescharacteristic
423C22C2×#41611yescharacteristic
424C22C2×#41611yescharacteristic
425C22#289711yescharacteristic
426C22#289811yescharacteristic
427C4—22no—
428C4—22no—
429C4—21no—
430C4—21no—
431C4—24no—
432C4—41no—
433C4—42no—
434C4—42no—
435C4—84no—
436C4—82no—
437C4—82no—
438C4#154414yes—
439C4#330211yescharacteristic
440C4#330311yescharacteristic
441C4#330411yescharacteristic
442C4#330511yescharacteristic
443C4#334011yescharacteristic
444C4#334011yescharacteristic
445C4#334111yescharacteristic
446C4#334211yescharacteristic
447C2—82no—
448C2C2×#154411yescharacteristic
449C2#1772411yescharacteristic
450C2#1772611yescharacteristic
451C2#1777711yescharacteristic
452C2#2248111yescharacteristic
453C2#2248911yescharacteristic
454C2#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. #9132C2semidirect
7. #9130C2semidirect
8. #8729C2semidirect
9. #8586C2semidirect
10. C4×#332C2semidirect
11. #8038C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#227952
#98982
#98962
#91322
#91302
#87292
#85862
C4×#3322
#80382