Finite Group Databasebeta

← #127804

Subgroups — #127804

Order 512.

618 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21374C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#20723C211yesmaximal, characteristic
4#7282C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7C2×#431C411yescharacteristic, complemented
8C2×C4×M16C411yescharacteristic, complemented
9#2283—21no—
10#1985—21no—
11#1955—21no—
12#1870C412yescomplemented
13#1870C2212yes—
14#1822C412yescomplemented
15#1820C412yes—
16#1801C412yescomplemented
17#1800—21no—
18#1799—21no—
19#1799C412yes—
20#1785—21no—
21#1782—21no—
22C4×#101—21no—
23C4×#100C2212yes—
24C4×#99—21no—
25C4×#99C2212yes—
26C4×C22⋊C8—21no—
27C4×C22⋊C8—21no—
28C22×#41D411yescharacteristic
29C22×M16C2×C412yes—
30C4×#41—41no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8D411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4D411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4Q811yescharacteristic
41C2×C4×D4C2311yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×#117C2×C411yescharacteristic
44C2×#117Q811yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#432—41no—
49#432—41no—
50#431—41no—
51#431—41no—
52#431—41no—
53#431—41no—
54#423—22no—
55C2×C4⋊C8—21no—
56C2×#101—22no—
57C2×#100—22no—
58C2×#99—22no—
59C2×#99—22no—
60#407—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8C2×C412yes—
64C4×M16—22no—
65C4×M16—41no—
66C4×M16—41no—
67C2×#97—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70#383—41no—
71#382—41no—
72#381—22no—
73#380—22no—
74C4×C4⋊C4—41no—
75C4×C22⋊C4—22no—
76#376—41no—
77#343—22no—
78#343C2×C414yes—
79#342—22no—
80#342C2×C414yes—
81#336—22no—
82#336—22no—
83C2×#41—41no—
84C2×#41—41no—
85C2×#41—42no—
86C2×#41—42no—
87C22×Q8C22⋊C411yescharacteristic
88C22×D4C22⋊C411yescharacteristic
89C22×D4C4⋊C411yescharacteristic
90C22×D4C2×D411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C4⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—24no—
101C2×M16—44no—
102C2×M16—42no—
103C22×C8—22no—
104C22×C8—22no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—41no—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—41no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121#117—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C4211yescharacteristic
129C2×C4⋊C4C22⋊C411yescharacteristic
130C2×C4⋊C4C4⋊C411yescharacteristic
131C2×C4⋊C4C4⋊C411yescharacteristic
132C2×C4⋊C4C4⋊C411yescharacteristic
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C22⋊C4—24no—
135C2×C22⋊C4C4211yescharacteristic
136C2×C22⋊C4C22⋊C411yescharacteristic
137C2×C22⋊C4C4⋊C411yescharacteristic
138C2×C22⋊C4C4⋊C411yescharacteristic
139C2×C22⋊C4C22×C411yescharacteristic
140C2×C22⋊C4C2×Q811yescharacteristic
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—42no—
154C2×C42C4211yescharacteristic
155C2×C42C4211yescharacteristic
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C4⋊C411yescharacteristic
161C2×C42C4⋊C411yescharacteristic
162C2×C42C4⋊C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C4⋊C8—42no—
165C4⋊C8—42no—
166C4⋊C8—42no—
167#101—44no—
168#100—44no—
169#99—44no—
170#99—44no—
171C22⋊C8—44no—
172C22⋊C8—44no—
173C22⋊C8—42no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176#97—42no—
177#97—42no—
178C4×C8—42no—
179C4×C8—41no—
180C4×C8—41no—
181C4×C8—41no—
182C4×C8—41no—
183#95—42no—
184#95—42no—
185#95—42no—
186#95—42no—
187C24#9511yescharacteristic
188C24C2×C22⋊C411yescharacteristic
189C24C2×C4⋊C411yescharacteristic
190#41—48no—
191#41—48no—
192C2×Q8—22no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4#9511yescharacteristic
236C22×C4#9511yescharacteristic
237C22×C4#9511yescharacteristic
238C22×C4#9511yescharacteristic
239C22×C4#9511yescharacteristic
240C22×C4#9511yescharacteristic
241C22×C4#9511yescharacteristic
242C22×C4C2×C4211yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4C2×C4⋊C411yescharacteristic
247M16—44no—
248M16—44no—
249M16—44no—
250M16—44no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—44no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—84no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—84no—
292C42#9911yescharacteristic
293C42#9911yescharacteristic
294C42#9911yescharacteristic
295C42#9911yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C2×#9511yescharacteristic
311C23#42112yes—
312Q8—44no—
313D4—44no—
314D4—44no—
315D4—44no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4#34211yescharacteristic
362C2×C4#34211yescharacteristic
363C2×C4#34211yescharacteristic
364C2×C4#34211yescharacteristic
365C2×C4C2×#9911yescharacteristic
366C2×C4C2×#9911yescharacteristic
367C2×C4#42112yes—
368C8—44no—
369C8—82no—
370C8—84no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22#173311yescharacteristic
389C22#173311yescharacteristic
390C22C2×#34211yescharacteristic
391C4—21no—
392C4—21no—
393C4—22no—
394C4—21no—
395C4—21no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—164no—
405C4#139512yes—
406C4#212012yes—
407C2—21no—
408C2—21no—
409C2—42no—
410C2—81no—
411C2—81no—
412C2#804911yescharacteristic
413C2#1257311yescharacteristic
414C2C2×#139511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7282C2semidirect
7. C2×#431C4semidirect
8. C2×C4×M16C4semidirect
12. #1870C4semidirect
14. #1822C4semidirect
16. #1801C4semidirect

Maximal subgroups

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

subgroupindex
#213742
#207282
#207232
#72822