Finite Group Databasebeta

← #190275

Subgroups — #190275

Order 512.

1074 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#8771C214yesmaximal, complemented
5#8770C214yesmaximal, complemented
6#8098C212yesmaximal, complemented
7#7959C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2357—21no—
12#2357—21no—
13#2357—22no—
14#2357C414yes—
15#2357C2212yes—
16#2357C2212yes—
17C2×#390—21no—
18C2×#388—22no—
19C2×#382C2211yescharacteristic
20C2×#381C412yes—
21C2×C4×C4⋊C4—21no—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C412yes—
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—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4C2×C412yes—
43C2×C4×D4D411yescharacteristic
44C2×C4×D4D411yescharacteristic
45C2×C4×D4D412yes—
46C2×C4×D4C2311yescharacteristic
47C2×C4×D4C2311yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D412yes—
54C22×C42—21no—
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57#432—22no—
58#432—22no—
59#432—22no—
60#432—22no—
61#431—22no—
62#431—22no—
63#431—22no—
64#431C2314yes—
65#423—22no—
66#423C2314yes—
67C2×#100—22no—
68C2×#100—22no—
69#407—22no—
70#407—22no—
71C4×M16—22no—
72C4×M16—22no—
73#390—24no—
74#390—22no—
75#390—22no—
76#388—24no—
77#388—24no—
78#388—24no—
79#388—24no—
80#382—21no—
81#382—21no—
82#382C2×C412yes—
83#382C2312yes—
84#381—22no—
85#381—22no—
86#381C2×C414yes—
87#381C2×C414yes—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4C2312yes—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4C2×C414yes—
98C4×C22⋊C4C2×C414yes—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C412yes—
103C2×#95C2×C411yescharacteristic
104#343—22no—
105#343—22no—
106#323—24no—
107#323—24no—
108C25C22⋊C412yes—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—42no—
112C22×D4—42no—
113C22×D4C22⋊C412yes—
114C22×D4C2×D411yescharacteristic
115C22×D4C2×D411yescharacteristic
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
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—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—48no—
164C2×C22⋊C4—84no—
165C2×C22⋊C4C22⋊C412yes—
166C2×C22⋊C4C22⋊C412yes—
167C2×C22⋊C4C2×D412yes—
168C2×C22⋊C4C2×D412yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—44no—
174C2×C42—84no—
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42#4112yes—
179C2×C42#4112yes—
180C2×C42#4112yes—
181C2×C42#4112yes—
182C2×C42C2411yescharacteristic
183C4⋊C8—24no—
184C4⋊C8—24no—
185C4⋊C8—24no—
186C4⋊C8—42no—
187#100—44no—
188#100—44no—
189C22⋊C8—24no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193#97—42no—
194#97—42no—
195C4×C8—24no—
196C4×C8—42no—
197#95—24no—
198#95—24no—
199#95—22no—
200#95—21no—
201#95—22no—
202#95—21no—
203#95—22no—
204#95C22×C412yes—
205#95C22×C414yes—
206#95C22×C412yes—
207C24—22no—
208C24—22no—
209C24—42no—
210C24—42no—
211C24—44no—
212C24—44no—
213C24C2×C22⋊C411yescharacteristic
214C24C2×C22⋊C412yes—
215C24C2×C22⋊C412yes—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—48no—
226C2×D4—48no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
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—24no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4#11711yescharacteristic
268C22×C4#11711yescharacteristic
269C22×C4#11711yescharacteristic
270C22×C4#11711yescharacteristic
271C22×C4#12312yes—
272C22×C4#12312yes—
273C22×C4#12312yes—
274C22×C4#12312yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278M16—44no—
279M16—44no—
280M16—84no—
281C2×C8—44no—
282C2×C8—44no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—48no—
295C22⋊C4—48no—
296C22⋊C4—84no—
297C22⋊C4—84no—
298C42—44no—
299C42—44no—
300C42—84no—
301C42—84no—
302C42#11712yes—
303C42#11712yes—
304C42#12312yes—
305C42#12312yes—
306C42#12312yes—
307C42#12312yes—
308C42C2×#4112yes—
309C42C2×#4112yes—
310C23—22no—
311C23—22no—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—44no—
329C23—44no—
330C23—88no—
331C23—88no—
332C23#38111yescharacteristic
333C23#38111yescharacteristic
334C23C22×C22⋊C411yescharacteristic
335D4—48no—
336D4—44no—
337D4—44no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—84no—
377C2×C4—88no—
378C2×C4#38112yes—
379C2×C4#38112yes—
380C2×C4#38111yescharacteristic
381C2×C4#38111yescharacteristic
382C2×C4C2×#11711yescharacteristic
383C2×C4C2×#11711yescharacteristic
384C2×C4C2×#12312yes—
385C2×C4C2×#12312yes—
386C2×C4#57512yes—
387C2×C4#57512yes—
388C8—84no—
389C8—84no—
390C22—24no—
391C22—24no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—88no—
409C22—88no—
410C22#182712yes—
411C22#182912yes—
412C22C2×#38111yescharacteristic
413C22#289611yescharacteristic
414C22#289611yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—44no—
423C4—42no—
424C4—42no—
425C4—82no—
426C4—84no—
427C4#309314yes—
428C2—21no—
429C2—21no—
430C2—22no—
431C2—22no—
432C2—82no—
433C2—82no—
434C2#2070111yescharacteristic
435C2C2×#182711yescharacteristic
436C2C2×#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. #8771C2semidirect
5. #8770C2semidirect
6. #8098C2semidirect
7. #7959C2semidirect
34. #1303C22semidirect
35. #1286C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
#207262
#173642
#87712
#87702
#80982
#79592