Finite Group Databasebeta

← #127190

Subgroups — #127190

Order 512.

677 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20688C211yesmaximal, characteristic
2#20377C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#7214C214yesmaximal, complemented
5C22×#117C2211yescharacteristic
6#2920—21no—
7C2×#407—21no—
8C2×C4×M16C411yescharacteristic, complemented
9C2×C4×M16C411yescharacteristic, complemented
10#2280—21no—
11#1820—21no—
12#1820C2212yes—
13#1819—21no—
14#1819C412yes—
15#1750—21no—
16#1750—21no—
17#1750C412yescomplemented
18#1750C2212yes—
19#1740—21no—
20#1740C412yes—
21#1728—21no—
22#1464—21no—
23#1464C2212yes—
24#1463C412yescomplemented
25#1462—21no—
26#1458—21no—
27#1458C412yescomplemented
28C24×C4D411yescharacteristic
29C22×M16—21no—
30C22×M16—21no—
31C2×#117—22no—
32C2×#117—41no—
33C2×#117—41no—
34C2×#117—41no—
35C2×#117—41no—
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×#117C2×C411yescharacteristic
39C2×#117D411yescharacteristic
40C2×#117D411yescharacteristic
41C2×#117D411yescharacteristic
42C2×#117Q811yescharacteristic
43C2×#117C2311yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4Q811yescharacteristic
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C22⋊C4D411yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42D411yescharacteristic
50#432—22no—
51#432—41no—
52#431—22no—
53#431—41no—
54#423—41no—
55#407—28no—
56#407—22no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C4×M16—41no—
60C4×M16—41no—
61C4×M16—41no—
62C4×M16—41no—
63C4×M16—41no—
64C2×#97—21no—
65C2×#97C2×C412yes—
66C2×C4×C8—21no—
67C2×C4×C8C2×C412yes—
68#381—22no—
69#380—41no—
70#380—41no—
71#379—22no—
72C4×C22⋊C4—41no—
73C4×C22⋊C4—41no—
74#342—22no—
75#342—22no—
76#342—22no—
77#342C2×C414yes—
78#336—22no—
79#336—22no—
80#324—22no—
81#324C2×C414yes—
82#323—22no—
83#323—22no—
84C25C22⋊C411yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—24no—
90C23×C4—42no—
91C23×C4C4211yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—24no—
102C2×M16—24no—
103C2×M16—42no—
104C2×M16—42no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—42no—
110C22×C8—22no—
111C22×C8—22no—
112#117—41no—
113#117—41no—
114#117—41no—
115#117—41no—
116#117—41no—
117#117—41no—
118#117—41no—
119#117—41no—
120#117—41no—
121#117—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128#117—44no—
129#117—44no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C4211yescharacteristic
136C2×C4⋊C4C22⋊C411yescharacteristic
137C2×C4⋊C4C4⋊C411yescharacteristic
138C2×C4⋊C4C4⋊C411yescharacteristic
139C2×C4⋊C4C4⋊C411yescharacteristic
140C2×C4⋊C4C4⋊C411yescharacteristic
141C2×C4⋊C4C22×C411yescharacteristic
142C2×C4⋊C4C2×Q811yescharacteristic
143C2×C22⋊C4—24no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4C4211yescharacteristic
151C2×C22⋊C4C22⋊C411yescharacteristic
152C2×C22⋊C4C22⋊C411yescharacteristic
153C2×C22⋊C4C22⋊C411yescharacteristic
154C2×C22⋊C4C4⋊C411yescharacteristic
155C2×C22⋊C4C4⋊C411yescharacteristic
156C2×C22⋊C4C22×C411yescharacteristic
157C2×C22⋊C4C2×D411yescharacteristic
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—42no—
167C2×C42C4211yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C4⋊C411yescharacteristic
172C2×C42C4⋊C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C4⋊C8—42no—
176C4⋊C8—42no—
177C22⋊C8—28no—
178C22⋊C8—28no—
179C22⋊C8—24no—
180C22⋊C8—42no—
181#97—42no—
182#97—42no—
183#97—42no—
184C4×C8—42no—
185C4×C8—42no—
186C4×C8—42no—
187#95—42no—
188#95—42no—
189C24—21no—
190C24—21no—
191C24—22no—
192C24—41no—
193C24—81no—
194C24—81no—
195C24#9511yescharacteristic
196C24#9511yescharacteristic
197C24C2×C22⋊C411yescharacteristic
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4#9511yescharacteristic
237C22×C4#9511yescharacteristic
238C22×C4#9511yescharacteristic
239C22×C4#9511yescharacteristic
240C22×C4#9511yescharacteristic
241C22×C4#9511yescharacteristic
242C22×C4#9911yescharacteristic
243C22×C4#9911yescharacteristic
244C22×C4#9911yescharacteristic
245C22×C4#9911yescharacteristic
246C22×C4#10414yes—
247C22×C4C2×C4211yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253M16—44no—
254M16—44no—
255M16—44no—
256M16—44no—
257M16—84no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—44no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—84no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—84no—
285C22⋊C4—84no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—84no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—42no—
313C23—42no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23—81no—
321C23—81no—
322C23—84no—
323C23—84no—
324C23#34211yescharacteristic
325C23#34211yescharacteristic
326C23C2×#9511yescharacteristic
327C23C2×#10412yes—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4H(3,ℤ/4)12yes—
377C2×C4#33912yes—
378C2×C4#33912yes—
379C2×C4#34211yescharacteristic
380C2×C4#34211yescharacteristic
381C2×C4C2×#9911yescharacteristic
382C2×C4C2×#9911yescharacteristic
383C8—44no—
384C8—44no—
385C8—84no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—81no—
406C22—84no—
407C22—84no—
408C22C2×H(3,ℤ/4)11yescharacteristic
409C22C2×#33911yescharacteristic
410C22C2×#34211yescharacteristic
411C4—21no—
412C4—21no—
413C4—22no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—164no—
424C4#128412yes—
425C4#128412yes—
426C2—21no—
427C2—21no—
428C2—42no—
429C2—41no—
430C2—41no—
431C2—81no—
432C2C2×#128411yescharacteristic
433C2#1257711yescharacteristic
434C2#1258911yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7214C2semidirect
8. C2×C4×M16C4semidirect
9. C2×C4×M16C4semidirect
17. #1750C4semidirect
24. #1463C4semidirect
27. #1458C4semidirect

Maximal subgroups

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

subgroupindex
#206882
#203772
#173572
#72142