Finite Group Databasebeta

← #413361

Subgroups — #413361

Order 512.

898 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2090C212yesmaximal
2C2×#2089C212yesmaximal
3#21144C211yesmaximal, characteristic
4#20801C211yesmaximal, characteristic
5#20800C211yesmaximal, characteristic
6C2×#1847C211yesmaximal, characteristic
7#11331C212yesmaximal
8#11330C212yesmaximal
9#10858C211yesmaximal, characteristic
10#10857C211yesmaximal, characteristic
11#10705C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C2×#428C2212yes—
15C2×#428C2212yes—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C2×#424C2212yes—
21C2×#423C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C22×#97C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28#2090—28no—
29#2089—28no—
30#1940—24no—
31#1847—24no—
32#1847—21no—
33#1847—21no—
34#1847—22no—
35#1846—22no—
36#1845—21no—
37#1844—21no—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#341C2212yes—
43C2×#341C2212yes—
44#1729C2212yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47C22×M16C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#117C2312yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#428—24no—
58#428—24no—
59#428—24no—
60#428—24no—
61#428—42no—
62#428—42no—
63#428—42no—
64#428—42no—
65C2×#107—24no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—42no—
73C2×#106—24no—
74C2×#106—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—42no—
81#424—24no—
82#424—24no—
83#423—22no—
84#423—22no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—22no—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16Q814yes—
90C4×M16Q814yes—
91C2×#97—21no—
92C2×#97—21no—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97D414yes—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98#384—21no—
99#384—21no—
100#384—22no—
101#384D411yescharacteristic
102#384D411yescharacteristic
103#382—21no—
104#382—21no—
105#382—22no—
106#382D412yes—
107#380—22no—
108C4×C4⋊C4—22no—
109C2×#95—22no—
110#343—24no—
111#343—24no—
112#343—24no—
113#343—24no—
114#341D418yes—
115#341Q818yes—
116#336—24no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4#4112yes—
122C23×C4C2411yescharacteristic
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×M16C2×D414yes—
136C2×M16C2×D414yes—
137C2×M16C2×Q814yes—
138C2×M16C2×Q814yes—
139C2×M16C2×Q814yes—
140C2×M16C2×Q814yes—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×D414yes—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C22×C8C2×Q814yes—
154#117—22no—
155#117—22no—
156#117—22no—
157#117—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—84no—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2×D412yes—
181C2×C22⋊C4—22no—
182C2×C42—22no—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×Q812yes—
187C2×C42C2×Q812yes—
188C2×C42#4112yes—
189C2×C42#4112yes—
190#107—44no—
191#107—44no—
192#107—44no—
193#107—44no—
194#107—44no—
195#107—44no—
196#106—44no—
197#106—44no—
198#106—44no—
199#106—44no—
200#106—44no—
201#106—44no—
202C4⋊C8—48no—
203C4⋊C8—44no—
204#97—24no—
205#97—24no—
206#97—28no—
207C4×C8—24no—
208C4×C8—24no—
209#95—24no—
210#95—42no—
211#95—42no—
212C24C2×#4111yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
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#12112yes—
230C22×C4#12112yes—
231C22×C4#12514yes—
232C22×C4#12514yes—
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D412yes—
236C22×C4C22×Q811yescharacteristic
237C22×C4C22×Q811yescharacteristic
238C22×C4C22×Q812yes—
239C22×C4C2×#4111yescharacteristic
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—
253M16C4⋊Q818yes—
254M16C4⋊Q818yes—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8#12714yes—
273C2×C8C4⋊Q814yes—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C42—42no—
297C42—42no—
298C42C22⋊Q812yes—
299C42C22⋊Q812yes—
300C42C22⋊Q812yes—
301C42C22⋊Q812yes—
302C42#12314yes—
303C42C4⋊Q814yes—
304C23—22no—
305C23—22no—
306C23C2×#12111yescharacteristic
307C23C2×#12511yescharacteristic
308C23C2×#12511yescharacteristic
309C23#55012yes—
310C23#55712yes—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—162no—
330C2×C4—162no—
331C2×C4—162no—
332C2×C4—162no—
333C2×C4#39814yes—
334C2×C4#39814yes—
335C2×C4C2×C22⋊Q811yescharacteristic
336C2×C4C2×C22⋊Q811yescharacteristic
337C2×C4C2×#12311yescharacteristic
338C2×C4C2×#12711yescharacteristic
339C2×C4C2×C4⋊Q811yescharacteristic
340C2×C4C2×C4⋊Q812yes—
341C2×C4C2×C4⋊Q812yes—
342C2×C4C2×C4⋊Q811yescharacteristic
343C2×C4#53511yescharacteristic
344C2×C4#53611yescharacteristic
345C2×C4#54412yes—
346C2×C4#54412yes—
347C2×C4D4212yes—
348C2×C4#54712yes—
349C2×C4#54712yes—
350C2×C4D4×Q812yes—
351C2×C4#55412yes—
352C2×C4#55412yes—
353C8—28no—
354C8—28no—
355C8—24no—
356C8—24no—
357C8—82no—
358C22—24no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22C2×#39811yescharacteristic
364C22#254111yescharacteristic
365C22#254611yescharacteristic
366C22#255212yes—
367C22#255212yes—
368C22#306812yes—
369C22#306912yes—
370C4—24no—
371C4—24no—
372C4—82no—
373C4—82no—
374C4—161no—
375C4—161no—
376C4—161no—
377C4—161no—
378C4#244012yes—
379C4#244012yes—
380C4#250111yescharacteristic
381C4#250211yescharacteristic
382C4#266011yescharacteristic
383C4#286811yescharacteristic
384C2—22no—
385C2—22no—
386C2#1514511yescharacteristic
387C2#2367512yes—
388C2#2849911yescharacteristic
389C2#2850011yescharacteristic
390C2#2870312yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20902
C2×#20892
#211442
#208012
#208002
C2×#18472
#113312
#113302
#108582
#108572
#107052