Finite Group Databasebeta

← #190205

Subgroups — #190205

Order 512.

997 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#34121C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17349C211yesmaximal, characteristic
4#8763C218yesmaximal, complemented
5#8014C212yesmaximal, complemented
6#7981C212yesmaximal, complemented
7C2×C4×#41C2211yescharacteristic
8C2×#431C2212yes—
9C2×#410—21no—
10#2357C412yescomplemented
11#2356C412yescomplemented
12#2355—21no—
13#2354—21no—
14#2353C2214yes—
15#2352—22no—
16#2298—22no—
17#2297—21no—
18#2285C2212yes—
19#2285C2211yescharacteristic
20#2283C412yescomplemented
21C4×#117—21no—
22C4×#117C412yescomplemented
23C2×C4×C4⋊C4C2211yescharacteristic
24#1799—22no—
25#1799C2214yes—
26#1740—21no—
27#1461—21no—
28#1460—21no—
29#1459C2214yes—
30#1458—22no—
31#1294C2218yes—
32#1289C2218yes—
33C22×#41D411yescharacteristic
34C4×#41—22no—
35C4×#41—22no—
36C4×#41—22no—
37C4×#41—22no—
38C2×C4×Q8C2×C411yescharacteristic
39C2×C4×Q8D411yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×C4×D4D412yes—
42C2×C4×D4D411yescharacteristic
43C2×C4×D4C2312yes—
44C2×#117—22no—
45C2×#117—21no—
46C2×#117—22no—
47C2×#117D411yescharacteristic
48C2×#117D412yes—
49C22×C4⋊C4—21no—
50C22×C42—21no—
51C22×C42C2×C412yes—
52C22×C42C2311yescharacteristic
53#431—22no—
54#431—22no—
55#431—22no—
56#431—22no—
57C2×C4⋊C8—22no—
58#410—44no—
59C2×#99—24no—
60C2×C22⋊C8C2314yes—
61C2×C4×C8—22no—
62#391—22no—
63#390—22no—
64#388—24no—
65#388—24no—
66#387—24no—
67#387—24no—
68#386—24no—
69#384—21no—
70#384—22no—
71#383C2×C414yes—
72#382—22no—
73#382—22no—
74#382C2×C412yes—
75#381C2×C414yes—
76#380—22no—
77#380—21no—
78#380—22no—
79#380C2314yes—
80#380C2312yes—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4C2×C412yes—
90C4×C4⋊C4C2×C411yescharacteristic
91C4×C4⋊C4C2×C411yescharacteristic
92C4×C4⋊C4C2×C412yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4C2×C414yes—
97#376—22no—
98#376C2×C414yes—
99C2×#95C2312yes—
100C43—22no—
101#342—24no—
102#324—24no—
103#323—24no—
104C2×#41—22no—
105C2×#41—22no—
106C2×#41—24no—
107C2×#41—22no—
108C2×#41—22no—
109C22×Q8C22⋊C411yescharacteristic
110C22×D4C22⋊C411yescharacteristic
111C22×D4C2×D412yes—
112C23×C4—22no—
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C412yes—
115C23×C4C22×C412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—24no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×D4—24no—
125C4×D4—24no—
126C4×D4—24no—
127C4×D4—24no—
128C4×D4—22no—
129C4×D4—22no—
130C4×D4—22no—
131C4×D4—22no—
132#117—22no—
133#117—22no—
134#117—24no—
135#117—24no—
136#117—48no—
137#117—42no—
138#117—42no—
139#117—44no—
140#117—44no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C2×D412yes—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4C22⋊C412yes—
160C2×C22⋊C4C2×D412yes—
161C2×C22⋊C4C2×D412yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—24no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—44no—
173C2×C42—42no—
174C2×C42C22⋊C412yes—
175C2×C42C22⋊C412yes—
176C2×C42C22×C412yes—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42#4112yes—
182C2×C42#4112yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185C4⋊C8—42no—
186C4⋊C8—42no—
187#99—48no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190C4×C8—42no—
191C4×C8—42no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—22no—
197#95—22no—
198#95—24no—
199#95—42no—
200#95C22×C412yes—
201#95C22×C412yes—
202#95C22×C414yes—
203C24C2×C22⋊C412yes—
204C24C22×D411yescharacteristic
205#41—44no—
206#41—44no—
207#41—44no—
208#41—44no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—81no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4C2×C22⋊C412yes—
239C22×C4C2×C22⋊C411yescharacteristic
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4#12312yes—
246C22×C4#12312yes—
247C22×C4#12312yes—
248C22×C4#12312yes—
249C22×C4#13514yes—
250C22×C4AGL(1,ℤ/8)12yes—
251C22×C4#13712yes—
252C22×C4C23×C411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C2×#4112yes—
255C22×C4C2×#4112yes—
256C2×C8—44no—
257C2×C8—84no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—24no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—84no—
268C4⋊C4—82no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—48no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—84no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—48no—
287C42—42no—
288C42—44no—
289C42—42no—
290C42—44no—
291C42—84no—
292C42—84no—
293C42#11711yescharacteristic
294C42#11711yescharacteristic
295C42#11712yes—
296C42#11712yes—
297C42#11711yescharacteristic
298C42#11711yescharacteristic
299C42#12312yes—
300C42#12312yes—
301C42#12312yes—
302C42#12312yes—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—41no—
307C23—41no—
308C23—42no—
309C23—42no—
310C23C22×C22⋊C411yescharacteristic
311C23C2×#12312yes—
312C23C2×#13512yes—
313C23C2×AGL(1,ℤ/8)11yescharacteristic
314C23C2×#13711yescharacteristic
315Q8—44no—
316D4—44no—
317D4—44no—
318D4—44no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—88no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4#38111yescharacteristic
354C2×C4#38112yes—
355C2×C4#38111yescharacteristic
356C2×C4#38112yes—
357C2×C4#38111yescharacteristic
358C2×C4#38111yescharacteristic
359C2×C4#41612yes—
360C2×C4#41612yes—
361C2×C4#41712yes—
362C2×C4#41712yes—
363C2×C4#48214yes—
364C2×C4#48214yes—
365C2×C4#48514yes—
366C2×C4#48514yes—
367C2×C4C2×#11711yescharacteristic
368C2×C4C2×#11711yescharacteristic
369C2×C4C2×#12312yes—
370C8—84no—
371C22—21no—
372C22—22no—
373C22—21no—
374C22—22no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—82no—
382C22—82no—
383C22C2×#38111yescharacteristic
384C22C2×#41611yescharacteristic
385C22C2×#41711yescharacteristic
386C22C2×#48212yes—
387C22C2×#48512yes—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—44no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—84no—
398C4—82no—
399C4—164no—
400C4#182512yes—
401C4#182512yes—
402C4#182514yes—
403C4#182712yes—
404C2—21no—
405C2—21no—
406C2—22no—
407C2—22no—
408C2—82no—
409C2C2×#182511yescharacteristic
410C2C2×#182711yescharacteristic
411C2#2071011yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8763C2semidirect
5. #8014C2semidirect
6. #7981C2semidirect
10. #2357C4semidirect
11. #2356C4semidirect
20. #2283C4semidirect
22. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
#341212
#206322
#173492
#87632
#80142
#79812