Finite Group Databasebeta

← #497657

Subgroups — #497657

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36538C211yesmaximal, characteristic
2#22517C212yesmaximal, complemented
3#13228C214yesmaximal, complemented
4#13226C212yesmaximal
5#12802C211yesmaximal, characteristic, complemented
6#12801C212yesmaximal, complemented
7C4×#356C211yesmaximal, characteristic, complemented
8C4×#355C212yesmaximal, complemented
9C2×#559C411yescharacteristic
10C2×#557C2211yescharacteristic
11Q8×M16—22no—
12#2861C412yes—
13#2858C411yescharacteristic
14#2568C411yescharacteristic
15#2566C412yes—
16#2558—24no—
17#2548C2211yescharacteristic
18#2545—22no—
19#2545C2212yes—
20#2367—22no—
21C4×C4⋊Q8C2211yescharacteristic
22C4×#124C2212yes—
23C42×Q8C411yescharacteristic
24C2×#356C2212yes—
25C2×#356C2212yes—
26C2×#355C2214yes—
27C2×#355C2214yes—
28#1998—24no—
29#1997—22no—
30#1801C2214yes—
31#1801C2212yes—
32C4×#101C2214yes—
33C4×#101C2212yes—
34#1394C2211yescharacteristic
35#1394C2212yes—
36#1394C2211yescharacteristic
37#559—42no—
38#559—41no—
39#559—41no—
40#557—41no—
41#557—42no—
42#557—41no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8D412yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×#127C2×C411yescharacteristic
47C2×#125C2×C412yes—
48C2×#125D414yes—
49C2×#124—22no—
50C2×#124—21no—
51C2×#124C2×C412yes—
52C2×#124C2312yes—
53C2×#123—22no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2312yes—
60C2×C4×D4—21no—
61C2×C4×D4C2×C412yes—
62#446—44no—
63#446—42no—
64C8×Q8—42no—
65#423—22no—
66#423—24no—
67C2×#101C2314yes—
68C2×#101C2312yes—
69C2×#101C2312yes—
70C4×M16—22no—
71C4×M16—24no—
72#400—22no—
73#396—24no—
74#396—24no—
75#394—22no—
76#391—22no—
77#391C2×C411yescharacteristic
78#391C2×C412yes—
79#391C2311yescharacteristic
80#390—22no—
81#390—22no—
82#390C2×C412yes—
83#390C2×C412yes—
84#390C2×C412yes—
85#389—22no—
86#388—24no—
87#388—24no—
88#388—24no—
89#388—24no—
90#385—24no—
91#384C2×C411yescharacteristic
92#382C2×C412yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C22⋊C4—24no—
99C43C2×C411yescharacteristic
100C43C2×C412yes—
101#356—24no—
102#356—42no—
103#355—28no—
104#355—44no—
105#343—24no—
106#343—22no—
107#343—24no—
108#343—22no—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C411yescharacteristic
112C22×Q8C2411yescharacteristic
113C22×D4—21no—
114C22×D4—21no—
115C22×D4C22×C412yes—
116C23×C4—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—24no—
120C4⋊Q8—22no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—42no—
126C4⋊Q8—42no—
127#127—41no—
128#127—41no—
129#125—42no—
130#125—42no—
131#125—44no—
132#125—44no—
133#124—24no—
134#124—22no—
135#124—22no—
136#124—48no—
137#124—42no—
138#124—42no—
139#124—42no—
140#124—42no—
141#123—44no—
142#123—44no—
143C4×Q8—22no—
144C4×Q8—21no—
145C4×Q8—21no—
146C4×Q8—22no—
147C4×Q8—24no—
148C4×Q8—22no—
149C4×Q8—42no—
150C4×Q8—41no—
151C4×Q8—42no—
152C4×Q8—41no—
153C4×Q8—44no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×D4—42no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—41no—
160C4×D4—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C414yes—
168C2×C4⋊C4C22⋊C414yes—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D414yes—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C2×D414yes—
194C2×C42C2×D412yes—
195C4⋊C8—44no—
196C4⋊C8—42no—
197C4⋊C8—44no—
198C4⋊C8—42no—
199#101—22no—
200#101—24no—
201#101—22no—
202#101—22no—
203#101—24no—
204#101#4114yes—
205#97—44no—
206#97—42no—
207C4×C8—44no—
208C4×C8—42no—
209#95—24no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95—24no—
214#95—22no—
215C24—22no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8C2×#4112yes—
229C2×Q8H(5,𝔽2)11yescharacteristic
230C2×Q82−1+411yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—44no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—82no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—84no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C414yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D412yes—
256M16—24no—
257M16—42no—
258M16—42no—
259M16—44no—
260M16—44no—
261M16—42no—
262C2×C8—42no—
263C2×C8—44no—
264C2×C8—42no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—82no—
282C4⋊C4#9912yes—
283C4⋊C4#9911yescharacteristic
284C4⋊C4#9911yescharacteristic
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—48no—
288C22⋊C4—48no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—84no—
294C42—24no—
295C42—24no—
296C42—22no—
297C42—24no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—21no—
305C42—22no—
306C42—21no—
307C42—41no—
308C42—44no—
309C42—41no—
310C42—41no—
311C42—82no—
312C42—81no—
313C23—44no—
314C23—44no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23C22×C22⋊C411yescharacteristic
319Q8—42no—
320Q8—41no—
321Q8—41no—
322Q8—42no—
323Q8—42no—
324Q8—44no—
325Q8—82no—
326Q8—81no—
327Q8—81no—
328D4—82no—
329D4—82no—
330D4—82no—
331D4—82no—
332D4—162no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—162no—
362C2×C4—162no—
363C2×C4C2×#9911yescharacteristic
364C2×C4C2×#9912yes—
365C2×C4C2×#9912yes—
366C2×C4C2×#9911yescharacteristic
367C2×C4C4×#4111yescharacteristic
368C2×C4#52011yescharacteristic
369C2×C4#54812yes—
370C2×C4#55012yes—
371C8—44no—
372C8—84no—
373C8—82no—
374C22—21no—
375C22—21no—
376C22—84no—
377C22—84no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22#234111yescharacteristic
382C22C22×#9911yescharacteristic
383C22#288411yescharacteristic
384C4—22no—
385C4—22no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—44no—
395C4—42no—
396C4—41no—
397C4—82no—
398C4—81no—
399C4—81no—
400C4#213212yes—
401C2—21no—
402C2—21no—
403C2—162no—
404C2#2253411yescharacteristic
405C2C2×#213211yescharacteristic
406C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#557 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22517C2semidirect
3. #13228C2semidirect
5. #12802C2semidirect
6. #12801C2semidirect
7. C4×#356C2semidirect
8. C4×#355C2semidirect

Maximal subgroups

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

subgroupindex
#365382
#225172
#132282
#132262
#128022
#128012
C4×#3562
C4×#3552