Finite Group Databasebeta

← #214861

Subgroups — #214861

Order 512.

897 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36040C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic
3#17370C211yesmaximal, characteristic
4#9487C212yesmaximal, complemented
5#9399C212yesmaximal, complemented
6#9191C212yesmaximal, complemented
7#9033C212yesmaximal, complemented
8#8766C212yesmaximal, complemented
9#8392C212yesmaximal, complemented
10C2×#557C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2975C2212yes—
13#2905C2212yes—
14C2×#413—21no—
15#2628C412yes—
16#2628C2211yescharacteristic
17#2580C412yes—
18#2580C2211yescharacteristic
19#2563C412yes—
20#2563C2211yescharacteristic
21#2454—22no—
22#2454—21no—
23#2442—22no—
24#2442—21no—
25#2396C412yes—
26#2396C2211yescharacteristic
27#2351—21no—
28#2351C2211yescharacteristic
29#2341—21no—
30#1801—21no—
31#1801C2212yes—
32#1800—21no—
33#1800C2212yes—
34C2×#343—21no—
35#1548C2214yes—
36#1487C2214yes—
37#1465—22no—
38#1463—22no—
39#1461C2212yes—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44C22×M16—22no—
45#557—44no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—22no—
49C2×#125—21no—
50#520—44no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57C2×C4×D4C2×C412yes—
58C2×C4×D4D412yes—
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2311yescharacteristic
61C2×#117C2×C412yes—
62C2×#117C2311yescharacteristic
63C22×C4⋊C4D412yes—
64C22×C4⋊C4D411yescharacteristic
65#436—44no—
66#431C2314yes—
67#424—44no—
68#423—22no—
69#423C2314yes—
70#413—44no—
71C2×#101—22no—
72C2×#100—22no—
73#407—24no—
74#400—22no—
75#400—21no—
76#400C2×C411yescharacteristic
77#399—22no—
78#399—21no—
79#399C2×C411yescharacteristic
80#399C2×C412yes—
81#398—22no—
82#398—21no—
83#396—22no—
84#396—21no—
85#395C2×C411yescharacteristic
86#395C2×C412yes—
87#393C2×C411yescharacteristic
88#393C2×C412yes—
89#393C2×C412yes—
90#393C2×C412yes—
91#393C2×C412yes—
92#390—22no—
93#390—22no—
94#390—21no—
95#390—21no—
96#389—22no—
97#389—21no—
98#389—21no—
99#389—22no—
100#389—22no—
101#389—21no—
102#385—22no—
103#385—22no—
104#385—22no—
105#382—22no—
106#382—21no—
107#382C2×C412yes—
108#382C2311yescharacteristic
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2311yescharacteristic
118C4×C22⋊C4—22no—
119C4×C22⋊C4—21no—
120#343—22no—
121#343—22no—
122#333—44no—
123Q8⋊C8—44no—
124#324—22no—
125#324—24no—
126#323—22no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—24no—
130C2×#41—42no—
131C22×Q8C22×C412yes—
132C22×Q8C22×C411yescharacteristic
133C22×Q8C2×D411yescharacteristic
134C22×D4C22⋊C412yes—
135C22×D4C2×D411yescharacteristic
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—44no—
144C2×M16—44no—
145C2×M16—44no—
146C22×C8—42no—
147C4⋊Q8—24no—
148C4⋊Q8—44no—
149#125—48no—
150#125—44no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—44no—
154C4×Q8—42no—
155C4×Q8—44no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×D4—24no—
159C4×D4—44no—
160C4×D4—42no—
161C4×D4—44no—
162#117—24no—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22×C412yes—
197C2×C22⋊C4C22×C411yescharacteristic
198C2×C22⋊C4C2×D411yescharacteristic
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22⋊C412yes—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207C4⋊C8—24no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8—44no—
211#101—44no—
212#100—44no—
213C22⋊C8—24no—
214C22⋊C8—44no—
215C4×C8—24no—
216#95—21no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95C22×C411yescharacteristic
228#95C22×C412yes—
229#95C22×C411yescharacteristic
230C24C2×C22⋊C412yes—
231C24C2×C22⋊C411yescharacteristic
232#41—48no—
233#41—48no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—24no—
238C2×Q8—81no—
239C2×Q8—82no—
240C2×Q8AGL(1,ℤ/8)12yes—
241C2×Q8#13712yes—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—24no—
248C2×D4—42no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4#13714yes—
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277M16—44no—
278M16—44no—
279M16—84no—
280M16—164no—
281C2×C8—44no—
282C2×C8—44no—
283C2×C8—84no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—44no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C22⋊C4—22no—
308C22⋊C4—24no—
309C22⋊C4—22no—
310C22⋊C4—42no—
311C22⋊C4—81no—
312C22⋊C4—82no—
313C22⋊C4—81no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—24no—
319C42—22no—
320C42—42no—
321C42—44no—
322C42—42no—
323C42—41no—
324C42—41no—
325C42—44no—
326C42H(5,𝔽2)14yes—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—41no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23#41912yes—
337C23C22×C22⋊C411yescharacteristic
338Q8—24no—
339Q8—24no—
340Q8—44no—
341Q8—84no—
342D4—48no—
343D4—42no—
344D4—42no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—48no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—162no—
382C2×C4—162no—
383C2×C4#41714yes—
384C2×C4#41912yes—
385C2×C4#51812yes—
386C2×C4#53412yes—
387C2×C4#53412yes—
388C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
389C2×C4C2×#13712yes—
390C2×C4C2×#13711yescharacteristic
391C8—84no—
392C8—84no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—42no—
397C22—44no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—81no—
404C22—82no—
405C22#230811yescharacteristic
406C22C2×#41911yescharacteristic
407C22C2×#41711yescharacteristic
408C4—22no—
409C4—24no—
410C4—22no—
411C4—22no—
412C4—41no—
413C4—42no—
414C4—41no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—82no—
419C4—81no—
420C4—82no—
421C4—81no—
422C4#319312yes—
423C4#319512yes—
424C2—21no—
425C2—21no—
426C2—42no—
427C2—82no—
428C2#2176511yescharacteristic
429C2#2178111yescharacteristic
430C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9487C2semidirect
5. #9399C2semidirect
6. #9191C2semidirect
7. #9033C2semidirect
8. #8766C2semidirect
9. #8392C2semidirect

Maximal subgroups

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

subgroupindex
#360402
#206402
#173702
#94872
#93992
#91912
#90332
#87662
#83922