Finite Group Databasebeta

← #413557

Subgroups — #413557

Order 512.

822 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23669C212yesmaximal
2C2×#1946C211yesmaximal, characteristic
3#21158C211yesmaximal, characteristic
4#21157C211yesmaximal, characteristic
5C2×#1773C211yesmaximal, characteristic
6#11296C212yesmaximal
7#11296C212yesmaximal
8#11242C212yesmaximal
9#10870C211yesmaximal, characteristic
10#10869C211yesmaximal, characteristic
11#10863C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#427C2211yescharacteristic
15C2×#427C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#424C2212yes—
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21C22×#97C2211yescharacteristic
22#2316C2212yes—
23#2316C2212yes—
24#2088—22no—
25#2088—22no—
26#2087—22no—
27#2086—22no—
28#1946—24no—
29#1944—22no—
30#1943—21no—
31#1943—21no—
32#1942—22no—
33#1941—21no—
34#1941—21no—
35#1773—24no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42C2×#340C2212yes—
43C2×#339C2212yes—
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C4⋊#95C2212yes—
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42C2311yescharacteristic
58C2×#108—22no—
59C2×#108—22no—
60C2×#108—22no—
61C2×#108—22no—
62C2×#108—22no—
63C2×#108—22no—
64#427—22no—
65#427—22no—
66#427—22no—
67#427—22no—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—22no—
71C2×#107D412yes—
72C2×#107D412yes—
73C2×#107Q814yes—
74C2×#106—21no—
75C2×#106—21no—
76C2×#106—22no—
77C2×#106D412yes—
78C2×#106D411yescharacteristic
79C2×#106D411yescharacteristic
80C2×#106Q814yes—
81#424—24no—
82#424—24no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85#405—22no—
86#405—22no—
87#405—22no—
88#405—22no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97—21no—
92C2×#97—21no—
93C2×#97D414yes—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#380—42no—
105#380—42no—
106C2×#95—42no—
107#343—24no—
108#343—24no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#339—24no—
120#339—22no—
121#339—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2411yescharacteristic
126C2×M16—21no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—21no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16#4112yes—
143C2×M16#4112yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D412yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D412yes—
157C22×C8C2×Q812yes—
158C22×C8C2×Q812yes—
159C22×C8#4112yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162C22×C8#4112yes—
163#117—21no—
164#117—21no—
165#117—22no—
166#117—21no—
167#117—21no—
168#117—22no—
169#117—22no—
170#117—21no—
171#117—21no—
172#117—21no—
173#117—21no—
174#117—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—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×Q814yes—
194C2×C4⋊C4C2×Q814yes—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42C2×D412yes—
207#108—28no—
208#108—44no—
209#107—24no—
210#107—24no—
211#107—24no—
212#107—24no—
213#106—24no—
214#106—22no—
215#106—22no—
216#106—24no—
217#106—24no—
218C4⋊C8—44no—
219C4⋊C8—44no—
220#97—28no—
221#97—42no—
222#97—42no—
223C4×C8—42no—
224C4×C8—42no—
225#95—42no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—84no—
230C24H(5,𝔽2)11yescharacteristic
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×Q811yescharacteristic
254C22×C4C22×Q811yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C42−1+411yescharacteristic
262C22×C42−1+411yescharacteristic
263C22×C42−1+411yescharacteristic
264M16—22no—
265M16—22no—
266M16—22no—
267M16—22no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—41no—
272M16—42no—
273M16—42no—
274M16—42no—
275M16—42no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8#12112yes—
294C2×C8#12112yes—
295C2×C8#12112yes—
296C2×C8#12112yes—
297C2×C8C22⋊Q812yes—
298C2×C8C22⋊Q812yes—
299C2×C8C22⋊Q812yes—
300C2×C8C22⋊Q812yes—
301C2×C8#12314yes—
302C2×C8#12514yes—
303C2×C8#12514yes—
304C2×C8C4⋊Q814yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—162no—
322C4⋊C4—162no—
323C4⋊C4—162no—
324C4⋊C4—162no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—42no—
331C42—22no—
332C42—22no—
333C42—41no—
334C42—41no—
335C42—42no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—42no—
341C23—22no—
342C23—22no—
343C23#53711yescharacteristic
344C23#54211yescharacteristic
345C23#54211yescharacteristic
346C23D4211yescharacteristic
347C23#55411yescharacteristic
348C23#55411yescharacteristic
349C23#55511yescharacteristic
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—162no—
376C2×C4—162no—
377C2×C4C2×#12111yescharacteristic
378C2×C4C2×#12111yescharacteristic
379C2×C4C2×C22⋊Q811yescharacteristic
380C2×C4C2×C22⋊Q811yescharacteristic
381C2×C4C2×#12311yescharacteristic
382C2×C4#52511yescharacteristic
383C2×C4C2×#12511yescharacteristic
384C2×C4C2×#12511yescharacteristic
385C2×C4#52911yescharacteristic
386C2×C4#52911yescharacteristic
387C2×C4C2×C4⋊Q811yescharacteristic
388C2×C4#53211yescharacteristic
389C2×C4#53211yescharacteristic
390C2×C4#53611yescharacteristic
391C2×C4#54111yescharacteristic
392C2×C4#54111yescharacteristic
393C2×C4#54711yescharacteristic
394C2×C4#54711yescharacteristic
395C2×C4#54811yescharacteristic
396C2×C4#54811yescharacteristic
397C2×C4D4×Q811yescharacteristic
398C2×C4D4×Q811yescharacteristic
399C2×C4#55211yescharacteristic
400C2×C4#55211yescharacteristic
401C2×C4#55711yescharacteristic
402C2×C4#55711yescharacteristic
403C2×C4#55711yescharacteristic
404C2×C4#55711yescharacteristic
405C8—22no—
406C8—22no—
407C8—42no—
408C8—42no—
409C8—82no—
410C8#39814yes—
411C8#39814yes—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22#245211yescharacteristic
421C22#245211yescharacteristic
422C22#250811yescharacteristic
423C22#250811yescharacteristic
424C22#253711yescharacteristic
425C22#258411yescharacteristic
426C22#262511yescharacteristic
427C22#328812yes—
428C22#328912yes—
429C4—82no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—82no—
434C4—84no—
435C4—84no—
436C4C2×#39811yescharacteristic
437C4#245311yescharacteristic
438C4#245311yescharacteristic
439C4#250611yescharacteristic
440C4#250611yescharacteristic
441C4#253611yescharacteristic
442C4#254611yescharacteristic
443C4#285411yescharacteristic
444C2—21no—
445C2—21no—
446C2—21no—
447C2—21no—
448C2#1514911yescharacteristic
449C2#2650312yes—
450C2#2650411yescharacteristic
451C2#2650511yescharacteristic
452C2#2824812yes—
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ 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
#236692
C2×#19462
#211582
#211572
C2×#17732
#112962
#112422
#108702
#108692
#108632