Finite Group Databasebeta

← #410629

Subgroups — #410629

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21033C212yesmaximal
2#21020C211yesmaximal, characteristic
3C2×C4×#107C211yesmaximal, characteristic
4#11365C211yesmaximal, characteristic
5#11365C211yesmaximal, characteristic
6#11331C211yesmaximal, characteristic
7#11270C211yesmaximal, characteristic
8#11249C212yesmaximal
9#11242C212yesmaximal
10#10864C211yesmaximal, characteristic
11#10802C212yesmaximal
12C22×#108C2212yes—
13C2×#428C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2296C2211yescharacteristic
19C2×C4×C4⋊C4C2212yes—
20#1921—22no—
21#1920—22no—
22#1919—22no—
23#1918—22no—
24#1916—22no—
25#1916—22no—
26C4×#107—22no—
27C4×#107—22no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#341C2211yescharacteristic
33C2×#341C2211yescharacteristic
34#1736C2212yes—
35#1736C2212yes—
36#1736C2212yes—
37C2×#340C2212yes—
38#1729C2211yescharacteristic
39#1729C2211yescharacteristic
40#1729C2211yescharacteristic
41#1728C2211yescharacteristic
42#1728C2212yes—
43C22×M16C2311yescharacteristic
44C22×M16C2312yes—
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C42—22no—
54C22×C42C2311yescharacteristic
55C2×#108—24no—
56C2×#108—24no—
57C2×#108—42no—
58#428—24no—
59#428—24no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—24no—
65C2×#107Q814yes—
66C2×#106—22no—
67#424—24no—
68C2×C4⋊C8—22no—
69C2×C4×C8—24no—
70C2×C4×C8D412yes—
71C2×C4×C8D412yes—
72#389D411yescharacteristic
73#389D411yescharacteristic
74#389D411yescharacteristic
75#389D411yescharacteristic
76#385—21no—
77#385—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C2×#95—22no—
85#343—24no—
86#343—24no—
87#343—42no—
88#343—42no—
89#341—22no—
90#341—22no—
91#340—24no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#336—22no—
99#336—22no—
100#336—22no—
101#336—21no—
102#336—21no—
103#336—22no—
104#336—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4#4111yescharacteristic
108C23×C4#4111yescharacteristic
109C23×C4C2411yescharacteristic
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—21no—
127C2×M16—21no—
128C2×M16—22no—
129C2×M16#4114yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C22×C8C2×Q812yes—
142C22×C8#4112yes—
143C22×C8#4114yes—
144#117—22no—
145#117—22no—
146#117—22no—
147#117—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4C2×Q814yes—
167C2×C4⋊C4#4114yes—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—84no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181#108—48no—
182#107—28no—
183#107—28no—
184#107—28no—
185#106—28no—
186C4⋊C8—44no—
187C4×C8—24no—
188C4×C8—24no—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—42no—
194#95—42no—
195C24C2×#4111yescharacteristic
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C22×Q811yescharacteristic
223C22×C4C2×#4111yescharacteristic
224C22×C4C2×#4112yes—
225C22×C4C2×#4112yes—
226C22×C4C2×#4112yes—
227C22×C4H(5,𝔽2)11yescharacteristic
228C22×C42−1+412yes—
229C22×C42−1+411yescharacteristic
230M16—24no—
231M16—24no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—42no—
236M16—42no—
237M16—42no—
238M16—42no—
239M16—42no—
240M16—42no—
241M16—42no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—24no—
245C2×C8—22no—
246C2×C8—24no—
247C2×C8—22no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8C22⋊Q814yes—
263C2×C8#12314yes—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—22no—
287C42—22no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—41no—
292C42—84no—
293C42—84no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C2×#12111yescharacteristic
299C23#52912yes—
300C23#55212yes—
301C23#55411yescharacteristic
302C23#55611yescharacteristic
303C2×C4—28no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—44no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—44no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4#47212yes—
338C2×C4#47312yes—
339C2×C4C2×C22⋊Q811yescharacteristic
340C2×C4C2×#12311yescharacteristic
341C2×C4#52511yescharacteristic
342C2×C4#52512yes—
343C2×C4#52912yes—
344C2×C4#53211yescharacteristic
345C2×C4#53312yes—
346C2×C4#53611yescharacteristic
347C2×C4#53711yescharacteristic
348C2×C4#54112yes—
349C2×C4#54212yes—
350C2×C4#54712yes—
351C2×C4D4×Q811yescharacteristic
352C2×C4#55011yescharacteristic
353C2×C4#55112yes—
354C2×C4#55212yes—
355C2×C4#55212yes—
356C2×C4#55712yes—
357C8—24no—
358C8—24no—
359C8—44no—
360C8—81no—
361C8—82no—
362C8—81no—
363C22—22no—
364C22—22no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22#239211yescharacteristic
372C22#248611yescharacteristic
373C22#252012yes—
374C22#253312yes—
375C22#253811yescharacteristic
376C22C2×#47211yescharacteristic
377C22C2×#47311yescharacteristic
378C22#306711yescharacteristic
379C22#306711yescharacteristic
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—81no—
385C4—81no—
386C4—84no—
387C4—81no—
388C4—84no—
389C4—81no—
390C4#246212yes—
391C4#247912yes—
392C4#249811yescharacteristic
393C4#256511yescharacteristic
394C4#263711yescharacteristic
395C4#285511yescharacteristic
396C2—22no—
397C2—22no—
398C2#1508311yescharacteristic
399C2#2371611yescharacteristic
400C2#2371711yescharacteristic
401C2#2767311yescharacteristic
402C2#2767511yescharacteristic
403C2#2829511yescharacteristic
404C2#2829711yescharacteristic
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
#210332
#210202
C2×C4×#1072
#113652
#113312
#112702
#112492
#112422
#108642
#108022