Finite Group Databasebeta

← #362965

Subgroups — #362965

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2961C211yesmaximal, characteristic
2#21323C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4C2×#1847C211yesmaximal, characteristic
5C2×#1778C211yesmaximal, characteristic
6#11239C212yesmaximal
7#11238C212yesmaximal
8#10870C212yesmaximal
9#10869C212yesmaximal
10#10858C211yesmaximal, characteristic
11#10857C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13#2961—24no—
14#2961C418yescomplemented
15C2×#446—24no—
16C22×#108C2212yes—
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C2×#424C2212yes—
22C2×#423C2211yescharacteristic
23C2×#405C2212yes—
24C2×C4×M16C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29#1993—22no—
30#1992—22no—
31#1991—22no—
32#1990—22no—
33#1847—24no—
34#1778—24no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C22×M16C2312yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49#533—22no—
50#533—22no—
51#533C2×C414yes—
52#533Q814yes—
53C2×C4⋊Q8D412yes—
54C2×#125D412yes—
55C2×C22⋊Q8—22no—
56C2×C4×Q8—22no—
57C2×#117C2312yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#446—232no—
65C2×#108—24no—
66C2×#108—24no—
67C2×#108Q818yescomplemented
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—24no—
71C2×#107—42no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—24no—
75C2×#106—42no—
76#424—24no—
77#424C2×C418yes—
78#423—22no—
79#423C2×C414yes—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82#405—24no—
83#405C2×C418yes—
84C4×M16—22no—
85C4×M16—22no—
86C4×M16—22no—
87C4×M16—22no—
88C2×#97—22no—
89C2×#97—21no—
90C2×#97—21no—
91C2×#97C2×C414yes—
92C2×#97D414yes—
93C2×#97D414yes—
94C2×C4×C8—22no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97#384—22no—
98#384—21no—
99#384—21no—
100#382—22no—
101#382—21no—
102#382—21no—
103C2×#95—22no—
104#343—42no—
105#343—42no—
106#343—42no—
107#343—42no—
108#341—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113C22×Q8—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4#4112yes—
119C23×C4C2411yescharacteristic
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16C22×C414yes—
136C2×M16C2×Q814yes—
137C2×M16C2×Q814yes—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8C22×C412yes—
144C22×C8C22×C412yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C22×C8C2×Q814yes—
150C22×C8#4114yes—
151C4⋊Q8—24no—
152C4⋊Q8C4⋊C418yes—
153#125—24no—
154#125C4⋊C418yes—
155C22⋊Q8—28no—
156C22⋊Q8—28no—
157C4×Q8—28no—
158C4×Q8—28no—
159#117—22no—
160#117—22no—
161#117C22×C414yes—
162#117C2×Q814yes—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4C2×D414yes—
185C2×C22⋊C4—22no—
186C2×C42—22no—
187C2×C42C22×C412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×Q812yes—
193C2×C42#4112yes—
194#108—216no—
195#107—44no—
196#107—44no—
197#106—44no—
198#106—44no—
199C4⋊C8—216no—
200C4⋊C8—28no—
201#97—28no—
202#97—42no—
203#97—42no—
204#97C22⋊C4116yes—
205C4×C8—28no—
206C4×C8—42no—
207C4×C8—42no—
208#95—42no—
209#95—42no—
210C24C2×#4111yescharacteristic
211C2×Q8—28no—
212C2×Q8—82no—
213C2×Q8—82no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4#11714yes—
230C22×C4#12112yes—
231C22×C4#12112yes—
232C22×C4C22⋊Q814yes—
233C22×C4C23×C411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C22×Q811yescharacteristic
238C22×C4C22×Q812yes—
239C22×C4C2×#4112yes—
240C22×C4C2×#4111yescharacteristic
241C22×C4H(5,𝔽2)12yes—
242C22×C42−1+412yes—
243M16—24no—
244M16—24no—
245M16—24no—
246M16—24no—
247M16—24no—
248M16—24no—
249M16—42no—
250M16—42no—
251M16—42no—
252M16—42no—
253M16—42no—
254M16—42no—
255C2×C8—24no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8C2×C22⋊C414yes—
269C2×C8C2×C22⋊C414yes—
270C2×C8C4×Q818yes—
271C2×C8C22⋊Q818yes—
272C2×C8#12414yes—
273C2×C8C4⋊Q814yes—
274C4⋊C4—24no—
275C4⋊C4—216no—
276C4⋊C4—28no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—84no—
290C4⋊C4—84no—
291C4⋊C4C2×C4⋊C418yes—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C42—24no—
295C42—24no—
296C42C2×C22⋊C414yes—
297C42C2×C4⋊C412yes—
298C42C2×C4⋊C412yes—
299C42#12112yes—
300C42#12112yes—
301C42C22⋊Q814yes—
302C23—22no—
303C23—22no—
304C23C2×#11711yescharacteristic
305C23C2×#12111yescharacteristic
306C23C2×C22⋊Q811yescharacteristic
307C23#55712yes—
308C23#55912yes—
309Q8—88no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—28no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—162no—
328C2×C4—162no—
329C2×C4—162no—
330C2×C4—162no—
331C2×C4#38014yes—
332C2×C4#38014yes—
333C2×C4C22×C22⋊C411yescharacteristic
334C2×C4C22×C4⋊C411yescharacteristic
335C2×C4C2×C4×Q812yes—
336C2×C4#52012yes—
337C2×C4C2×#12111yescharacteristic
338C2×C4C2×C22⋊Q811yescharacteristic
339C2×C4C2×C22⋊Q812yes—
340C2×C4C2×#12411yescharacteristic
341C2×C4C2×C4⋊Q811yescharacteristic
342C2×C4#53511yescharacteristic
343C2×C4#53611yescharacteristic
344C2×C4#53712yes—
345C2×C4D4×Q812yes—
346C2×C4#55012yes—
347C2×C4#55412yes—
348C2×C4#55612yes—
349C2×C4#55712yes—
350C2×C4Q8212yes—
351C8—24no—
352C8—44no—
353C8—44no—
354C8—44no—
355C8#39118yes—
356C22—24no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22C2×#38011yescharacteristic
362C22#235512yes—
363C22#254611yescharacteristic
364C22#254811yescharacteristic
365C22#258212yes—
366C22#306812yes—
367C22#306912yes—
368C4—24no—
369C4—24no—
370C4—48no—
371C4—161no—
372C4—161no—
373C4—161no—
374C4—161no—
375C4C2×#39111yescharacteristic
376C4#231111yescharacteristic
377C4Q8×C4⋊C412yes—
378C4#253812yes—
379C4#256711yescharacteristic
380C4#256811yescharacteristic
381C2—22no—
382C2—22no—
383C2#1392611yescharacteristic
384C2#2067912yes—
385C2#2870312yes—
386C2#2876711yescharacteristic
387C2#2876811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2961C4semidirect
67. C2×#108Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#29612
#213232
#213222
C2×#18472
C2×#17782
#112392
#112382
#108702
#108692
#108582
#108572