Finite Group Databasebeta

← #214813

Subgroups — #214813

Order 512.

882 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35845C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic
3#17352C211yesmaximal, characteristic
4#9484C212yesmaximal, complemented
5#9392C212yesmaximal, complemented
6#9191C212yesmaximal
7#9033C212yesmaximal
8#8767C212yesmaximal, complemented
9#8392C212yesmaximal, complemented
10C2×#557C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2982C2212yes—
13#2905C2212yes—
14C22×#101—21no—
15#2628C412yes—
16#2628C2211yescharacteristic
17#2581C412yes—
18#2581C2211yescharacteristic
19#2567C412yes—
20#2567C2211yescharacteristic
21#2453—21no—
22#2453—22no—
23#2441—21no—
24#2441—22no—
25#2396C412yes—
26#2396C2211yescharacteristic
27#2351C2211yescharacteristic
28Q8×C4⋊C4—21no—
29Q8×C22⋊C4—21no—
30#1801—21no—
31#1801—21no—
32#1801C2212yes—
33#1801C2212yes—
34#1742—21no—
35#1548C2214yes—
36#1487C2214yes—
37#1461—22no—
38#1461C2212yes—
39#1458—22no—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C23×Q8D411yescharacteristic
44C22×M16—22no—
45#557—44no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—22no—
49C2×#125—21no—
50#519—44no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×#117C2×C412yes—
62C2×#117D411yescharacteristic
63C2×#117D412yes—
64C2×#117C2311yescharacteristic
65C8×Q8—44no—
66#431—24no—
67#431C2314yes—
68#424—44no—
69#423—22no—
70#423C2314yes—
71C2×#101—22no—
72C2×#101—22no—
73C2×#101—44no—
74#400—22no—
75#400—21no—
76#400C2×C411yescharacteristic
77#400C2×C412yes—
78#399—22no—
79#399—21no—
80#399C2×C411yescharacteristic
81#398—22no—
82#398—21no—
83#396—22no—
84#396—21no—
85#395C2×C411yescharacteristic
86#395C2×C412yes—
87#395C2×C412yes—
88#395C2×C412yes—
89#395C2×C412yes—
90#393C2×C411yescharacteristic
91#393C2×C412yes—
92#391—22no—
93#391—22no—
94#391—21no—
95#391—21no—
96#389—22no—
97#389—21no—
98#389—21no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—21no—
105#387—22no—
106#387—22no—
107#387—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#333—44no—
123Q8⋊C8—44no—
124#324—24no—
125#324—22no—
126#323—22no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8C22⋊C412yes—
130C22×Q8SD1614yes—
131C22×Q8Q1614yes—
132C22×Q8C22×C412yes—
133C22×Q8C22×C411yescharacteristic
134C22×Q8C2×D411yescharacteristic
135C22×Q8C2×D411yescharacteristic
136C23×C4C22⋊C412yes—
137C23×C4C2×D411yescharacteristic
138C2×M16—24no—
139C2×M16—42no—
140C2×M16—44no—
141C2×M16—44no—
142C22×C8—42no—
143C4⋊Q8—24no—
144C4⋊Q8—44no—
145#125—48no—
146#125—44no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—44no—
151C4×Q8—44no—
152C4×Q8—44no—
153C4×Q8—42no—
154C4×Q8—42no—
155C4×Q8—44no—
156C4×Q8—44no—
157#117—24no—
158#117—44no—
159#117—44no—
160#117—42no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
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—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22×C411yescharacteristic
191C2×C22⋊C4C2×D411yescharacteristic
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C412yes—
199C2×C42C22×C412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203C4⋊C8—24no—
204C4⋊C8—24no—
205C4⋊C8—24no—
206C4⋊C8—44no—
207C4⋊C8—44no—
208#101—44no—
209#101—44no—
210C22⋊C8—24no—
211C4×C8—24no—
212C4×C8—44no—
213#95—21no—
214#95—22no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—22no—
220#95—21no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95C22×C411yescharacteristic
227#95C22×C412yes—
228#95C22×C411yescharacteristic
229C24C2×C22⋊C411yescharacteristic
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—48no—
239C2×Q8—48no—
240C2×Q8—81no—
241C2×Q8—82no—
242C2×Q8#10318yes—
243C2×Q8C2×SD1612yes—
244C2×Q8C2×SD1612yes—
245C2×Q8C2×Q1612yes—
246C2×Q8C2×Q1612yes—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4#10318yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×SD1612yes—
266C22×C4C2×Q1612yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270M16—44no—
271M16—44no—
272M16—84no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—84no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—81no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—81no—
306C22⋊C4—82no—
307C42—24no—
308C42—24no—
309C42—22no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—44no—
314C42—44no—
315C42—42no—
316C42—42no—
317C42H(5,𝔽2)14yes—
318C23—21no—
319C23—21no—
320C23—41no—
321C23—41no—
322C23C2×#10312yes—
323C23C22×C22⋊C411yescharacteristic
324Q8—24no—
325Q8—24no—
326Q8—24no—
327Q8—24no—
328Q8—48no—
329Q8—44no—
330Q8—84no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—48no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4C2×#10314yes—
369C2×C4C2×#10314yes—
370C2×C4C2×#10312yes—
371C2×C4#51812yes—
372C2×C4#53412yes—
373C2×C4#53412yes—
374C2×C4C22×SD1611yescharacteristic
375C2×C4C22×Q1611yescharacteristic
376C2×C4#57512yes—
377C8—84no—
378C8—84no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—42no—
383C22—41no—
384C22—41no—
385C22—81no—
386C22#230811yescharacteristic
387C22C22×#10311yescharacteristic
388C22#289611yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—24no—
393C4—24no—
394C4—42no—
395C4—41no—
396C4—42no—
397C4—41no—
398C4—42no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—82no—
403C4—81no—
404C4#318712yes—
405C4#318912yes—
406C2—21no—
407C2—21no—
408C2—42no—
409C2#2175911yescharacteristic
410C2#2178111yescharacteristic
411C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
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. #9484C2semidirect
5. #9392C2semidirect
8. #8767C2semidirect
9. #8392C2semidirect

Maximal subgroups

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

subgroupindex
#358452
#206392
#173522
#94842
#93922
#91912
#90332
#87672
#83922