Finite Group Databasebeta

← #190284

Subgroups — #190284

Order 512.

1179 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#8765C214yesmaximal, complemented
5#8764C214yesmaximal, complemented
6#8094C212yesmaximal, complemented
7#7961C212yesmaximal, complemented
8C2×#518—21no—
9C22×C4×D4C2211yescharacteristic
10#2920C2212yes—
11#2920C2212yes—
12#2349C414yes—
13#2349C2212yes—
14#2349C2212yes—
15D4×C4⋊C4—22no—
16D4×C4⋊C4—21no—
17D4×C4⋊C4—21no—
18#2316—21no—
19#2315—22no—
20#2285C2211yescharacteristic
21#2285C2211yescharacteristic
22C2×#380C412yes—
23C2×#380C412yes—
24#1821—21no—
25#1800C2212yes—
26#1800C2212yes—
27#1783—21no—
28C4×#100—21no—
29C4×#100—21no—
30#1466—22no—
31#1462—22no—
32#1460C2212yes—
33#1460C2212yes—
34#1303C2218yescomplemented
35#1286C2218yescomplemented
36C23×D4D411yescharacteristic
37C24×C4C2×C412yes—
38#518—48no—
39C2×C4×D4—24no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4D412yes—
50C2×C4×D4C2311yescharacteristic
51C2×C4×D4C2311yescharacteristic
52C2×#117—21no—
53C2×#117—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4D411yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D412yes—
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×#100—22no—
72C2×#100—22no—
73#407—22no—
74#407—22no—
75C4×M16—22no—
76C4×M16—22no—
77#389—24no—
78#389—22no—
79#389—22no—
80#385—24no—
81#385—24no—
82#385—24no—
83#385—24no—
84#384—21no—
85#384—21no—
86#382C2×C412yes—
87#382C2312yes—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—21no—
93#380—21no—
94#380C2×C414yes—
95#380C2×C414yes—
96#380C2×C412yes—
97#380C2×C411yescharacteristic
98#380C2×C411yescharacteristic
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2312yes—
103C2×#95C2×C414yes—
104C2×#95C2×C414yes—
105#343—22no—
106#343—22no—
107#323—24no—
108#323—24no—
109C25C22⋊C412yes—
110C22×D4—24no—
111C22×D4—24no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4C22⋊C412yes—
115C22×D4C2×D411yescharacteristic
116C22×D4C2×D411yescharacteristic
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—24no—
123C23×C4—84no—
124C23×C4C22⋊C412yes—
125C23×C4C22⋊C412yes—
126C23×C4C22⋊C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—24no—
133C2×M16—42no—
134C4×D4—24no—
135C4×D4—24no—
136C4×D4—22no—
137C4×D4—22no—
138C4×D4—24no—
139C4×D4—22no—
140C4×D4—22no—
141C4×D4—24no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—22no—
145C4×D4—22no—
146C4×D4—416no—
147C4×D4—48no—
148C4×D4—48no—
149#117—44no—
150#117—44no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—84no—
163C2×C4⋊C4—84no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C2×D412yes—
179C2×C22⋊C4C2×D412yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4112yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189C2×C42C2411yescharacteristic
190C4⋊C8—24no—
191C4⋊C8—24no—
192C4⋊C8—24no—
193C4⋊C8—42no—
194#100—44no—
195#100—44no—
196C22⋊C8—24no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200#97—42no—
201#97—42no—
202C4×C8—24no—
203C4×C8—42no—
204#95—24no—
205#95—24no—
206#95—22no—
207#95—22no—
208#95C22×C414yes—
209#95C22×C412yes—
210#95C22×C412yes—
211C24—22no—
212C24—22no—
213C24—24no—
214C24—24no—
215C24—82no—
216C24—82no—
217C24C2×C22⋊C411yescharacteristic
218C24C2×C22⋊C412yes—
219C24C2×C22⋊C412yes—
220C2×D4—216no—
221C2×D4—216no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4AGL(1,ℤ/8)14yes—
230C2×D4#13714yes—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—48no—
250C22×C4—48no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4—81no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4#11711yescharacteristic
269C22×C4#11711yescharacteristic
270C22×C4#11711yescharacteristic
271C22×C4#11711yescharacteristic
272C22×C4#12312yes—
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4#12312yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—44no—
280M16—44no—
281M16—84no—
282C2×C8—44no—
283C2×C8—44no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—44no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—48no—
304C22⋊C4—48no—
305C42—44no—
306C42—44no—
307C42#11712yes—
308C42#11712yes—
309C42#12312yes—
310C42#12312yes—
311C42#12312yes—
312C42#12312yes—
313C42C2×#4112yes—
314C42C2×#4112yes—
315C23—28no—
316C23—28no—
317C23—21no—
318C23—22no—
319C23—21no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—82no—
333C23—88no—
334C23—88no—
335C23#38111yescharacteristic
336C23#38111yescharacteristic
337C23#41814yes—
338C23#41914yes—
339C23C22×C22⋊C411yescharacteristic
340D4—216no—
341D4—28no—
342D4—28no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—168no—
381C2×C4—168no—
382C2×C4#38112yes—
383C2×C4#38112yes—
384C2×C4#38111yescharacteristic
385C2×C4#38111yescharacteristic
386C2×C4C2×#11711yescharacteristic
387C2×C4C2×#11711yescharacteristic
388C2×C4C2×#12312yes—
389C2×C4C2×#12312yes—
390C2×C4C2×AGL(1,ℤ/8)12yes—
391C2×C4C2×#13712yes—
392C8—84no—
393C8—84no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—24no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—88no—
415C22—88no—
416C22C2×#38111yescharacteristic
417C22C2×#41811yescharacteristic
418C22C2×#41911yescharacteristic
419C4—22no—
420C4—22no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—42no—
426C4—42no—
427C4—84no—
428C4—82no—
429C4—82no—
430C4#182912yes—
431C4#309214yes—
432C2—21no—
433C2—21no—
434C2—41no—
435C2—41no—
436C2—44no—
437C2—44no—
438C2#2070011yescharacteristic
439C2#2070811yescharacteristic
440C2C2×#182911yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #8765C2semidirect
5. #8764C2semidirect
6. #8094C2semidirect
7. #7961C2semidirect
34. #1303C22semidirect
35. #1286C22semidirect

Maximal subgroups

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

subgroupindex
#341812
#207262
#173642
#87652
#87642
#80942
#79612