Finite Group Databasebeta

← #190272

Subgroups — #190272

Order 512.

1097 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8767C214yesmaximal, complemented
5#8764C214yesmaximal, complemented
6#8098C212yesmaximal, complemented
7#7961C212yesmaximal, complemented
8C2×C4×#41—21no—
9C2×C4×#41C2211yescharacteristic
10#2920C2212yes—
11#2920C2212yes—
12#2351C2212yes—
13Q8×C4⋊C4—21no—
14#2349C414yes—
15#2349C2212yes—
16D4×C4⋊C4—21no—
17D4×C4⋊C4—22no—
18#2296—22no—
19#2296—21no—
20#2285C412yes—
21#2285C2211yescharacteristic
22C2×C4×C4⋊C4C412yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24#1822—21no—
25#1801C2212yes—
26#1800C2212yes—
27#1785—21no—
28C4×#101—21no—
29C4×#100—21no—
30#1465—22no—
31#1464—22no—
32#1461C2212yes—
33#1460C2212yes—
34#1303C2218yes—
35#1286C2218yes—
36C22×#41D411yescharacteristic
37C4×#41—22no—
38C4×#41—22no—
39C4×#41—22no—
40C4×#41—22no—
41C4×#41—48no—
42C2×C4×Q8—22no—
43C2×C4×Q8D411yescharacteristic
44C2×C4×Q8C2311yescharacteristic
45C2×C4×D4—24no—
46C2×C4×D4—22no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D411yescharacteristic
50C2×C4×D4C2311yescharacteristic
51C2×#117—22no—
52C2×#117—21no—
53C2×#117D411yescharacteristic
54C2×#117D412yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C42—22no—
58C22×C42—21no—
59C22×C42C2×C412yes—
60C22×C42C2311yescharacteristic
61#432—22no—
62#432—22no—
63#432—22no—
64#432—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431C2314yes—
69#423—22no—
70#423C2314yes—
71C2×#101—22no—
72C2×#100—22no—
73#407—22no—
74#407—22no—
75C4×M16—22no—
76C4×M16—22no—
77#389—24no—
78#389—22no—
79#389—24no—
80#389—22no—
81#385—24no—
82#385—24no—
83#385—24no—
84#384—22no—
85#384—21no—
86#382C2×C414yes—
87#382C2312yes—
88#380—22no—
89#380—21no—
90#380C2×C412yes—
91#380C2×C414yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2×C414yes—
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4C2×C411yescharacteristic
102C4×C4⋊C4C2312yes—
103C2×#95C2×C412yes—
104C2×#95C2×C414yes—
105#343—22no—
106#343—22no—
107#324—24no—
108#323—24no—
109C2×#41—24no—
110C2×#41—24no—
111C2×#41—22no—
112C2×#41—22no—
113C22×Q8C2×D411yescharacteristic
114C22×D4C22⋊C412yes—
115C22×D4C2×D411yescharacteristic
116C23×C4—22no—
117C23×C4—24no—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C412yes—
120C23×C4C22×C412yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—24no—
125C2×M16—42no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—48no—
131C4×D4—24no—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—22no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4—416no—
140C4×D4—48no—
141#117—22no—
142#117—22no—
143#117—24no—
144#117—24no—
145#117—44no—
146#117—48no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4C22⋊C412yes—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C2×D412yes—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—48no—
177C2×C42—44no—
178C2×C42C22⋊C412yes—
179C2×C42C22⋊C412yes—
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C2×C42C2411yescharacteristic
189C4⋊C8—24no—
190C4⋊C8—24no—
191C4⋊C8—24no—
192C4⋊C8—42no—
193#101—44no—
194#100—44no—
195C22⋊C8—24no—
196C22⋊C8—42no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199#97—42no—
200#97—42no—
201C4×C8—24no—
202C4×C8—42no—
203#95—24no—
204#95—22no—
205#95—24no—
206#95—22no—
207#95C22×C412yes—
208#95C22×C414yes—
209#95C22×C412yes—
210C24C2×C22⋊C411yescharacteristic
211C24C2×C22⋊C412yes—
212#41—216no—
213#41—216no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8#13714yes—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—24no—
222C2×D4AGL(1,ℤ/8)14yes—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—48no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4—81no—
243C22×C4—82no—
244C22×C4—84no—
245C22×C4—82no—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4#11712yes—
253C22×C4#11712yes—
254C22×C4#12312yes—
255C22×C4#12312yes—
256C22×C4#12312yes—
257C22×C4#12312yes—
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261M16—44no—
262M16—44no—
263M16—84no—
264C2×C8—44no—
265C2×C8—44no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—44no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C4⋊C4—84no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—44no—
283C22⋊C4—48no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—42no—
291C42—44no—
292C42—42no—
293C42—48no—
294C42—44no—
295C42#11711yescharacteristic
296C42#11711yescharacteristic
297C42#11711yescharacteristic
298C42#11711yescharacteristic
299C42#12312yes—
300C42#12312yes—
301C42#12312yes—
302C42#12312yes—
303C42C2×#4112yes—
304C42C2×#4112yes—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—22no—
309C23—41no—
310C23—41no—
311C23—42no—
312C23—42no—
313C23#38112yes—
314C23#41814yes—
315C23C22×C22⋊C411yescharacteristic
316Q8—28no—
317D4—216no—
318D4—28no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—816no—
353C2×C4—88no—
354C2×C4#38111yescharacteristic
355C2×C4#38111yescharacteristic
356C2×C4#38111yescharacteristic
357C2×C4#38112yes—
358C2×C4#38111yescharacteristic
359C2×C4#41914yes—
360C2×C4C2×#11711yescharacteristic
361C2×C4C2×#11711yescharacteristic
362C2×C4C2×#12312yes—
363C2×C4C2×#12312yes—
364C2×C4C2×AGL(1,ℤ/8)12yes—
365C2×C4C2×#13712yes—
366C8—84no—
367C8—84no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—24no—
372C22—22no—
373C22—22no—
374C22—44no—
375C22—42no—
376C22—42no—
377C22—44no—
378C22—42no—
379C22—42no—
380C22C2×#38111yescharacteristic
381C22C2×#41811yescharacteristic
382C22C2×#41911yescharacteristic
383C4—21no—
384C4—21no—
385C4—24no—
386C4—22no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—22no—
391C4—44no—
392C4—42no—
393C4—42no—
394C4—84no—
395C4—82no—
396C4#182712yes—
397C4#309214yes—
398C2—21no—
399C2—21no—
400C2—42no—
401C2—44no—
402C2#2070011yescharacteristic
403C2C2×#182711yescharacteristic
404C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8767C2semidirect
5. #8764C2semidirect
6. #8098C2semidirect
7. #7961C2semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
#207282
#173662
#87672
#87642
#80982
#79612