Finite Group Databasebeta

← #214857

Subgroups — #214857

Order 512.

939 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35748C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic
3#17370C211yesmaximal, characteristic
4#9487C212yesmaximal, complemented
5#9403C212yesmaximal, complemented
6#9193C212yesmaximal, complemented
7#9033C212yesmaximal, complemented
8#8766C212yesmaximal, complemented
9#8396C212yesmaximal, complemented
10C2×Q82C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2975C2212yes—
13#2905C2212yes—
14C2×#413—21no—
15#2628C412yes—
16#2628C2211yescharacteristic
17#2580C412yes—
18#2580C2211yescharacteristic
19#2542C412yes—
20#2542C2211yescharacteristic
21#2452—22no—
22#2452—21no—
23#2440—22no—
24#2440—21no—
25#2396C412yes—
26#2396C2211yescharacteristic
27#2351C2211yescharacteristic
28Q8×C4⋊C4—21no—
29#2341—21no—
30#1801—21no—
31#1801C2212yes—
32#1800—21no—
33#1800C2212yes—
34C2×#343—21no—
35#1549C2214yes—
36#1487C2214yes—
37#1465—22no—
38#1463—22no—
39#1461C2212yes—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44C22×M16—22no—
45Q82—44no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53#520—44no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×D4C2×C412yes—
61C2×C4×D4D412yes—
62C2×C4×D4D411yescharacteristic
63C2×C4×D4C2311yescharacteristic
64C2×#117C2×C412yes—
65C2×#117C2311yescharacteristic
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D411yescharacteristic
68#436—44no—
69#431C2314yes—
70#424—44no—
71#423—22no—
72#423C2314yes—
73#413—44no—
74C2×#101—22no—
75C2×#100—22no—
76#407—24no—
77#400C2×C411yescharacteristic
78#400C2×C412yes—
79#399C2×C412yes—
80#399C2×C412yes—
81#399C2×C412yes—
82#399C2×C412yes—
83#399C2×C411yescharacteristic
84#398—22no—
85#398—21no—
86#396—22no—
87#396—21no—
88#395—22no—
89#395—21no—
90#395C2×C411yescharacteristic
91#393—22no—
92#393—21no—
93#393C2×C411yescharacteristic
94#393C2×C412yes—
95#390—22no—
96#390—22no—
97#390—21no—
98#390—21no—
99#389—22no—
100#389—21no—
101#389—21no—
102#389—22no—
103#389—22no—
104#389—21no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384—22no—
109#384—21no—
110#382C2×C412yes—
111#382C2311yescharacteristic
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#332—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×Q8—22no—
132C22×Q8—21no—
133C22×Q8C22×C411yescharacteristic
134C22×Q8C2×D411yescharacteristic
135C22×D4C22⋊C412yes—
136C22×D4C2×D411yescharacteristic
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C2×M16—24no—
143C2×M16—42no—
144C2×M16—44no—
145C2×M16—44no—
146C2×M16—44no—
147C22×C8—42no—
148C4⋊Q8—24no—
149C4⋊Q8—44no—
150C4⋊Q8—48no—
151C4⋊Q8—44no—
152C22⋊Q8—84no—
153C22⋊Q8—84no—
154C22⋊Q8—84no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—44no—
158C4×Q8—42no—
159C4×Q8—44no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×D4—24no—
163C4×D4—44no—
164C4×D4—42no—
165C4×D4—44no—
166#117—24no—
167#117—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22×C412yes—
202C2×C22⋊C4C22×C411yescharacteristic
203C2×C22⋊C4C2×D411yescharacteristic
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22⋊C412yes—
209C2×C42C22×C412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2411yescharacteristic
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—44no—
216#101—44no—
217#100—44no—
218C22⋊C8—24no—
219C22⋊C8—44no—
220C4×C8—24no—
221#95—21no—
222#95—22no—
223#95—21no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—21no—
228#95—22no—
229#95—21no—
230#95C22×C411yescharacteristic
231#95C22×C412yes—
232#95C22×C411yescharacteristic
233C24C2×C22⋊C412yes—
234C24C2×C22⋊C411yescharacteristic
235#41—48no—
236#41—48no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8AGL(1,ℤ/8)12yes—
247C2×Q8#13712yes—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—24no—
254C2×D4—42no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4#13714yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283M16—44no—
284M16—44no—
285M16—84no—
286M16—164no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—84no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—44no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C22⋊C4—22no—
315C22⋊C4—24no—
316C22⋊C4—22no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C22⋊C4—44no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—24no—
326C42—22no—
327C42—42no—
328C42—44no—
329C42—42no—
330C42—41no—
331C42—41no—
332C42—44no—
333C42H(5,𝔽2)14yes—
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—41no—
340C23—42no—
341C23—41no—
342C23—42no—
343C23#41912yes—
344C23C22×C22⋊C411yescharacteristic
345Q8—24no—
346Q8—24no—
347Q8—24no—
348Q8—42no—
349Q8—162no—
350Q8—164no—
351D4—48no—
352D4—42no—
353D4—42no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—48no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—84no—
390C2×C4—162no—
391C2×C4—162no—
392C2×C4#41714yes—
393C2×C4#41912yes—
394C2×C4#51812yes—
395C2×C4#53412yes—
396C2×C4#53412yes—
397C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
398C2×C4C2×#13712yes—
399C2×C4C2×#13711yescharacteristic
400C8—84no—
401C8—84no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—42no—
406C22—44no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—81no—
413C22—82no—
414C22#230811yescharacteristic
415C22C2×#41911yescharacteristic
416C22C2×#41711yescharacteristic
417C4—22no—
418C4—24no—
419C4—22no—
420C4—22no—
421C4—42no—
422C4—41no—
423C4—42no—
424C4—42no—
425C4—41no—
426C4—42no—
427C4—81no—
428C4—82no—
429C4—82no—
430C4—81no—
431C4#319312yes—
432C4#319512yes—
433C2—21no—
434C2—21no—
435C2—42no—
436C2—82no—
437C2#2176511yescharacteristic
438C2#2178011yescharacteristic
439C2#2178211yescharacteristic
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. #9403C2semidirect
6. #9193C2semidirect
7. #9033C2semidirect
8. #8766C2semidirect
9. #8396C2semidirect

Maximal subgroups

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

subgroupindex
#357482
#206402
#173702
#94872
#94032
#91932
#90332
#87662
#83962