Finite Group Databasebeta

← #472429

Subgroups — #472429

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23670C212yesmaximal
2C2×#1942C211yesmaximal, characteristic
3#21020C211yesmaximal, characteristic
4#11365C211yesmaximal, characteristic
5#11365C211yesmaximal, characteristic
6#11331C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11270C211yesmaximal, characteristic
9#11216C212yesmaximal
10#11204C212yesmaximal
11#10844C211yesmaximal, characteristic
12C2×#428C2212yes—
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C2×#424C2211yescharacteristic
16C2×#423C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2296C2211yescharacteristic
19C2×#389C2212yes—
20#2088—22no—
21#2088—22no—
22#2087—22no—
23#2086—22no—
24#1942—22no—
25#1942—22no—
26#1916—22no—
27#1916—22no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#341C2211yescharacteristic
33C2×#341C2211yescharacteristic
34C2×#341C2212yes—
35#1736C2212yes—
36#1736C2212yes—
37#1736C2212yes—
38C2×#340C2212yes—
39#1729C2211yescharacteristic
40#1729C2211yescharacteristic
41#1729C2211yescharacteristic
42#1728C2211yescharacteristic
43C22×M16C2312yes—
44C22×M16C2311yescharacteristic
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55#428—24no—
56#428—42no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—42no—
62C2×#107Q814yes—
63C2×#106—22no—
64#424—42no—
65#423—24no—
66#423—42no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4×C8—42no—
71C2×C4×C8D412yes—
72C2×C4×C8D412yes—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389D411yescharacteristic
78#389D411yescharacteristic
79#389D411yescharacteristic
80#389D411yescharacteristic
81#385—21no—
82#385—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C2×#95—22no—
86#343—24no—
87#343—42no—
88#343—42no—
89#343—42no—
90#341—22no—
91#341—22no—
92#341—42no—
93#340—24no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#336—22no—
101#336—22no—
102#336—22no—
103#336—21no—
104#336—21no—
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—21no—
120C2×M16—21no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16#4114yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×Q812yes—
143C22×C8#4112yes—
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—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×Q814yes—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181#107—28no—
182#107—44no—
183#107—44no—
184#106—44no—
185C4⋊C8—44no—
186C4⋊C8—44no—
187C4⋊C8—44no—
188C4×C8—42no—
189C4×C8—42no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—42no—
195#95—42no—
196C24C2×#4111yescharacteristic
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—24no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4—82no—
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×Q811yescharacteristic
224C22×C4C2×#4111yescharacteristic
225C22×C4C2×#4112yes—
226C22×C4H(5,𝔽2)12yes—
227C22×C4H(5,𝔽2)11yescharacteristic
228C22×C42−1+412yes—
229C22×C42−1+412yes—
230C22×C42−1+411yescharacteristic
231M16—24no—
232M16—24no—
233M16—42no—
234M16—42no—
235M16—42no—
236M16—42no—
237M16—42no—
238M16—42no—
239M16—42no—
240M16—42no—
241M16—42no—
242M16—42no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8C22⋊Q814yes—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—84no—
282C4⋊C4—84no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—22no—
290C42—22no—
291C42—41no—
292C42—41no—
293C42—41no—
294C42—41no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23C2×#12111yescharacteristic
300C23#54112yes—
301C23#55312yes—
302C23#55411yescharacteristic
303C23#55611yescharacteristic
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—44no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
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—82no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—162no—
337C2×C4—162no—
338C2×C4#47212yes—
339C2×C4#47312yes—
340C2×C4C2×C22⋊Q811yescharacteristic
341C2×C4#53211yescharacteristic
342C2×C4#53411yescharacteristic
343C2×C4#53611yescharacteristic
344C2×C4#53611yescharacteristic
345C2×C4#53612yes—
346C2×C4#53711yescharacteristic
347C2×C4#53912yes—
348C2×C4#53912yes—
349C2×C4#54112yes—
350C2×C4#54412yes—
351C2×C4#54612yes—
352C2×C4D4×Q811yescharacteristic
353C2×C4#55011yescharacteristic
354C2×C4#55212yes—
355C2×C4#55712yes—
356C2×C4#56212yes—
357C2×C4#56312yes—
358C8—24no—
359C8—44no—
360C8—42no—
361C8—81no—
362C8—82no—
363C8—81no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22#253811yescharacteristic
373C22#254311yescharacteristic
374C22#262111yescharacteristic
375C22#277612yes—
376C22#280212yes—
377C22C2×#47211yescharacteristic
378C22C2×#47311yescharacteristic
379C22#306711yescharacteristic
380C22#306711yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—84no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—162no—
391C4#256511yescharacteristic
392C4#262711yescharacteristic
393C4#266411yescharacteristic
394C4#269412yes—
395C4#271312yes—
396C4#281411yescharacteristic
397C2—22no—
398C2—22no—
399C2#1682011yescharacteristic
400C2#2829511yescharacteristic
401C2#2829711yescharacteristic
402C2#2859111yescharacteristic
403C2#2859211yescharacteristic
404C2#2988611yescharacteristic
405C2#2988811yescharacteristic
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
#236702
C2×#19422
#210202
#113652
#113312
#113032
#112702
#112162
#112042
#108442