Finite Group Databasebeta

← #480464

Subgroups — #480464

Order 512.

884 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35679C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#11546C214yesmaximal, complemented
5#11542C214yesmaximal, complemented
6#11452C212yesmaximal, complemented
7#11427C212yesmaximal
8C22×#124C411yescharacteristic
9C2×#519C2211yescharacteristic
10#2983—21no—
11#2982—21no—
12#2482C412yes—
13#2482C412yes—
14#2482C2212yes—
15#2482C2212yes—
16#2448—22no—
17#2448—21no—
18#2448—21no—
19#2448—22no—
20#2448—21no—
21#2448—21no—
22#2310C411yescharacteristic
23#2309C411yescharacteristic
24#2309C2211yescharacteristic
25#2296C411yescharacteristic
26#2296C2211yescharacteristic
27#1467—22no—
28#1464C2214yes—
29#1461—22no—
30#1461—21no—
31#1461—21no—
32#1458C2214yes—
33#1458C2212yes—
34#1458C2212yes—
35#1326C2214yes—
36#1326C2214yes—
37#1316C2218yes—
38C23×Q8C2×C411yescharacteristic
39C23×D4C2×C411yescharacteristic
40C2×#124—21no—
41C2×#124—22no—
42C2×#124—21no—
43C2×#124—44no—
44#519—44no—
45C2×C4×Q8D412yes—
46C2×C4×Q8D411yescharacteristic
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D412yes—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8D411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C412yes—
55C2×#117C2311yescharacteristic
56C2×#117C2312yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C42C2×C411yescharacteristic
60#446—44no—
61C8×Q8—44no—
62#432—24no—
63#431—22no—
64#431—24no—
65#431—22no—
66#399—22no—
67#399—22no—
68#399—22no—
69#399—22no—
70#397—22no—
71#397—22no—
72#397—22no—
73#397—22no—
74#391—21no—
75#391—22no—
76#391—21no—
77#390—21no—
78#390—21no—
79#390—21no—
80#390—21no—
81#390—22no—
82#390—21no—
83#390—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#385C2×C411yescharacteristic
97#385C2×C411yescharacteristic
98#385C2×C412yes—
99#385C2×C412yes—
100#385C2311yescharacteristic
101#385C2311yescharacteristic
102#381C2×C412yes—
103#381C2×C412yes—
104#379C2×C412yes—
105#379C2×C412yes—
106#379C2×C412yes—
107#379C2312yes—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111#324—22no—
112#324—24no—
113#324—22no—
114#324—24no—
115#323C2314yes—
116#323C2314yes—
117C25C22×C412yes—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—42no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22⋊C412yes—
123C22×Q8C22⋊C411yescharacteristic
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—24no—
127C22×D4—42no—
128C22×D4—42no—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133#124—48no—
134#124—44no—
135#124—44no—
136C4×Q8—24no—
137C4×Q8—24no—
138C4×Q8—44no—
139C4×Q8—42no—
140C4×Q8—42no—
141C4×Q8—44no—
142#117—24no—
143#117—42no—
144#117—44no—
145#117—42no—
146#117—44no—
147#117—42no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D412yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4C22×C411yescharacteristic
179C2×C22⋊C4C22×C411yescharacteristic
180C2×C22⋊C4C22×C412yes—
181C2×C22⋊C4C22×C412yes—
182C2×C22⋊C4C22×C411yescharacteristic
183C2×C22⋊C4C2411yescharacteristic
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—42no—
189C2×C42C22⋊C412yes—
190C2×C42C22⋊C412yes—
191C2×C42C22⋊C412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C4⋊C8—44no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—44no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202#97—44no—
203C4×C8—44no—
204C4×C8—42no—
205C4×C8—42no—
206#95—22no—
207#95—22no—
208#95—21no—
209#95—21no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—21no—
215#95—21no—
216C24—22no—
217C24—24no—
218C24—42no—
219C24—42no—
220C24—82no—
221C24—82no—
222C24C23×C411yescharacteristic
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—24no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—416no—
232C2×Q8#9914yes—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—88no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4#13114yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C2×C8—44no—
266C2×C8—44no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—41no—
286C22⋊C4—42no—
287C22⋊C4—41no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—42no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C42—24no—
302C42—22no—
303C42—22no—
304C42—24no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—42no—
312C23—21no—
313C23—21no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—84no—
324C23—84no—
325C23—84no—
326C23—164no—
327C23—164no—
328C23C22×C22⋊C411yescharacteristic
329C23C2×#13112yes—
330Q8—28no—
331Q8—48no—
332Q8—44no—
333D4—84no—
334D4—84no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—28no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—48no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4#40812yes—
374C2×C4#40812yes—
375C2×C4C2×#9914yes—
376C2×C4C2×#9912yes—
377C2×C4#41312yes—
378C8—84no—
379C8—84no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—168no—
395C22C2×#40811yescharacteristic
396C22C22×#9911yescharacteristic
397C22C2×#41311yescharacteristic
398C4—22no—
399C4—24no—
400C4—41no—
401C4—41no—
402C4—42no—
403C4—41no—
404C4—41no—
405C4—42no—
406C4—42no—
407C4—81no—
408C4—81no—
409C4—82no—
410C4—81no—
411C4—81no—
412C4#146814yes—
413C2—21no—
414C2—21no—
415C2—42no—
416C2—162no—
417C2C2×#146811yescharacteristic
418C2#3335111yescharacteristic
419C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #17352C2semidirect
4. #11546C2semidirect
5. #11542C2semidirect
6. #11452C2semidirect

Maximal subgroups

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

subgroupindex
#356792
#173712
#173522
#115462
#115422
#114522
#114272