Finite Group Databasebeta

← #497655

Subgroups — #497655

Order 512.

850 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36508C211yesmaximal, characteristic
2#22517C212yesmaximal, complemented
3#13228C214yesmaximal
4#13227C212yesmaximal, complemented
5#12802C212yesmaximal, complemented
6#12801C211yesmaximal, characteristic
7C4×#356C212yesmaximal, complemented
8C4×#355C211yesmaximal, characteristic
9C2×#557C2211yescharacteristic
10C2×#556C411yescharacteristic
11Q8×M16—22no—
12#2862C411yescharacteristic
13#2861C412yes—
14#2565C412yes—
15#2546C2211yescharacteristic
16#2545C2212yes—
17#2542C411yescharacteristic
18#2519—24no—
19#2429—22no—
20#2362—22no—
21C4×C4⋊Q8C2212yes—
22C4×#124C2211yescharacteristic
23C42×Q8C411yescharacteristic
24C2×#356C2214yes—
25C2×#356C2214yes—
26C2×#355C2212yes—
27C2×#355C2212yes—
28#1998—24no—
29#1997—22no—
30#1801C2212yes—
31#1801C2214yes—
32C4×#101C2212yes—
33C4×#101C2214yes—
34#1394C2212yes—
35#1394C2211yescharacteristic
36#1394C2211yescharacteristic
37#557—41no—
38#557—42no—
39#557—41no—
40#556—42no—
41#556—41no—
42#556—41no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8D412yes—
46C2×C4⋊Q8C2312yes—
47C2×#126—22no—
48C2×#125C2×C412yes—
49C2×#125D414yes—
50C2×#124C2×C412yes—
51C2×#124C2311yescharacteristic
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8C2312yes—
59C2×C4×Q8C2311yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×#117—21no—
62#446—44no—
63#446—42no—
64C8×Q8—42no—
65#423—22no—
66#423—24no—
67C2×#101C2312yes—
68C2×#101C2314yes—
69C2×#101C2312yes—
70C4×M16—22no—
71C4×M16—24no—
72#399—22no—
73#398—24no—
74#398—22no—
75#396—24no—
76#391—22no—
77#391C2×C412yes—
78#391C2×C412yes—
79#391C2×C412yes—
80#391C2311yescharacteristic
81#390C2×C411yescharacteristic
82#390C2×C412yes—
83#389—24no—
84#389—22no—
85#388—24no—
86#388—24no—
87#388—22no—
88#388—24no—
89#387—24no—
90#384C2×C412yes—
91#382—22no—
92#382C2×C411yescharacteristic
93C4×C4⋊C4—22no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C22⋊C4—22no—
99C43C2×C412yes—
100C43C2×C411yescharacteristic
101#356—28no—
102#356—44no—
103#355—24no—
104#355—42no—
105#343—24no—
106#343—22no—
107#343—24no—
108#343—22no—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C412yes—
112C22×Q8C2411yescharacteristic
113C22×D4C22×C411yescharacteristic
114C23×C4—21no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—24no—
118C4⋊Q8—24no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125C4⋊Q8—41no—
126C4⋊Q8—41no—
127#126—44no—
128#126—44no—
129#125—42no—
130#125—42no—
131#125—44no—
132#125—44no—
133#124—22no—
134#124—21no—
135#124—21no—
136#124—42no—
137#124—42no—
138C22⋊Q8—48no—
139C22⋊Q8—42no—
140C22⋊Q8—42no—
141C4×Q8—22no—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—22no—
145C4×Q8—24no—
146C4×Q8—22no—
147C4×Q8—42no—
148C4×Q8—41no—
149C4×Q8—42no—
150C4×Q8—41no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×Q8—44no—
154C4×D4—41no—
155C4×D4—41no—
156#117—42no—
157#117—41no—
158#117—41no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
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—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D414yes—
193C2×C42C2×D412yes—
194C4⋊C8—44no—
195C4⋊C8—42no—
196C4⋊C8—44no—
197C4⋊C8—42no—
198#101—22no—
199#101—22no—
200#101—24no—
201#101—24no—
202#101—22no—
203#101#4114yes—
204#97—44no—
205#97—42no—
206C4×C8—44no—
207C4×C8—42no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—22no—
215C24—21no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8C2×#4112yes—
229C2×Q8H(5,𝔽2)11yescharacteristic
230C2×Q82−1+411yescharacteristic
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—44no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D412yes—
251M16—24no—
252M16—42no—
253M16—42no—
254M16—44no—
255M16—42no—
256M16—44no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—42no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—41no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4#9912yes—
281C4⋊C4#9911yescharacteristic
282C4⋊C4#9911yescharacteristic
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—48no—
286C22⋊C4—44no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C42—24no—
292C42—22no—
293C42—24no—
294C42—22no—
295C42—21no—
296C42—21no—
297C42—22no—
298C42—22no—
299C42—21no—
300C42—22no—
301C42—21no—
302C42—22no—
303C42—41no—
304C42—44no—
305C42—42no—
306C42—42no—
307C42—44no—
308C42—42no—
309C42—42no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23C22×C22⋊C411yescharacteristic
316Q8—42no—
317Q8—41no—
318Q8—41no—
319Q8—42no—
320Q8—42no—
321Q8—44no—
322Q8—82no—
323Q8—82no—
324Q8—82no—
325D4—81no—
326D4—81no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4C2×#9912yes—
358C2×C4C2×#9912yes—
359C2×C4C2×#9911yescharacteristic
360C2×C4C2×#9911yescharacteristic
361C2×C4C4×#4111yescharacteristic
362C2×C4#52011yescharacteristic
363C2×C4#54812yes—
364C2×C4#55012yes—
365C8—44no—
366C8—82no—
367C8—84no—
368C22—21no—
369C22—21no—
370C22—82no—
371C22—82no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22#234111yescharacteristic
376C22C22×#9911yescharacteristic
377C22#288411yescharacteristic
378C4—22no—
379C4—22no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—22no—
385C4—44no—
386C4—42no—
387C4—42no—
388C4—44no—
389C4—41no—
390C4—42no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4#213212yes—
395C2—21no—
396C2—21no—
397C2—161no—
398C2#2253411yescharacteristic
399C2C2×#213211yescharacteristic
400C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×D4C22×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
4. #13227C2semidirect
5. #12802C2semidirect
7. C4×#356C2semidirect

Maximal subgroups

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

subgroupindex
#365082
#225172
#132282
#132272
#128022
#128012
C4×#3562
C4×#3552