Finite Group Databasebeta

← #373989

Subgroups — #373989

Order 512.

842 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C211yesmaximal, characteristic
2#21158C211yesmaximal, characteristic
3#21036C212yesmaximal
4#20455C211yesmaximal, characteristic
5#11303C212yesmaximal
6#11296C212yesmaximal
7#10899C212yesmaximal
8#10870C211yesmaximal, characteristic
9#10862C211yesmaximal, characteristic
10#10859C211yesmaximal, characteristic
11#10706C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#427C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2316C2212yes—
21#2274C2212yes—
22#1952C418yescomplemented
23#1943—22no—
24#1942—21no—
25#1942—21no—
26#1921—22no—
27#1920—22no—
28#1919—22no—
29#1918—22no—
30#1776—22no—
31C4×#107—21no—
32C4×#107—21no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#341C2211yescharacteristic
36#1736C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39#1735C2211yescharacteristic
40C2×#340C2212yes—
41C2×#339C2211yescharacteristic
42C2×#339C2212yes—
43C2×#339C2211yescharacteristic
44#1729C2212yes—
45C4⋊#95C2212yes—
46C22×M16C2311yescharacteristic
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C2×#117C2312yes—
53C2×#117C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42C2311yescharacteristic
57C2×#108—22no—
58C2×#108—22no—
59C2×#108C2×C414yes—
60#428—24no—
61#428C2×C418yes—
62#427—22no—
63#427—22no—
64C2×#107—21no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#107—21no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#107C2×C414yes—
72C2×#107D411yescharacteristic
73C2×#107D411yescharacteristic
74C2×#107Q814yes—
75C2×#106—22no—
76C2×#106D412yes—
77#405—22no—
78#405—22no—
79#405—22no—
80#405C2×C414yes—
81C2×#97—21no—
82C2×#97—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#380—42no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C22⋊C4—42no—
98C2×#95—42no—
99#343—24no—
100#343C2×C418yes—
101#341—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340D411yescharacteristic
111#340D411yescharacteristic
112#340D411yescharacteristic
113#340D411yescharacteristic
114#339—22no—
115#339—22no—
116#339—24no—
117#336—22no—
118#336—22no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4C2411yescharacteristic
123C2×M16—21no—
124C2×M16—22no—
125C2×M16—21no—
126C2×M16—22no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—22no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16C22×C412yes—
139C2×M16C22×C414yes—
140C2×M16#4112yes—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8C22×C412yes—
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C22×C8C2×D411yescharacteristic
155C22×C8C2×Q812yes—
156C22×C8#4112yes—
157C22×C8#4112yes—
158C22×C8#4112yes—
159C22×C8#4112yes—
160#117—21no—
161#117—21no—
162#117—22no—
163#117—21no—
164#117—21no—
165#117—21no—
166#117—22no—
167#117—22no—
168#117—21no—
169#117—22no—
170#117C22×C412yes—
171#117#4112yes—
172C2×C4⋊C4—22no—
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—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C22×C414yes—
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×Q814yes—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42#4112yes—
206#108—28no—
207#107—24no—
208#107—24no—
209#107—24no—
210#107—24no—
211#107—22no—
212#107—22no—
213#107—28no—
214#106—24no—
215#106—28no—
216#97—24no—
217#97—42no—
218C4×C8—24no—
219C4×C8—42no—
220C4×C8—44no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—42no—
225#95—84no—
226C24H(5,𝔽2)11yescharacteristic
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—41no—
239C22×C4—41no—
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×C4C23×C411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×Q811yescharacteristic
254C22×C4C2×#4111yescharacteristic
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—24no—
265M16—22no—
266M16—24no—
267M16—22no—
268M16—22no—
269M16—22no—
270M16—42no—
271M16—42no—
272M16—41no—
273M16—41no—
274M16—41no—
275M16—41no—
276C2×C8—22no—
277C2×C8—24no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8#11714yes—
296C2×C8C4×D412yes—
297C2×C8C4×D412yes—
298C2×C8C4×Q814yes—
299C2×C8C22⋊Q812yes—
300C2×C8C22⋊Q812yes—
301C2×C8#12312yes—
302C2×C8#12312yes—
303C2×C8#12514yes—
304C2×C8#12614yes—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—162no—
324C4⋊C4—162no—
325C4⋊C4C4×D412yes—
326C4⋊C4C4×D412yes—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—41no—
338C42—41no—
339C42—42no—
340C42—41no—
341C42—42no—
342C42—41no—
343C42—162no—
344C42—162no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23#52011yescharacteristic
350C23#54211yescharacteristic
351C23#54211yescharacteristic
352C23D4211yescharacteristic
353C23#55311yescharacteristic
354C23#55311yescharacteristic
355C23#55411yescharacteristic
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—162no—
390C2×C4—162no—
391C2×C4C2×#11711yescharacteristic
392C2×C4C2×C4×D411yescharacteristic
393C2×C4C2×C4×D411yescharacteristic
394C2×C4C2×C4×Q811yescharacteristic
395C2×C4C4×#4111yescharacteristic
396C2×C4#51911yescharacteristic
397C2×C4C2×C22⋊Q811yescharacteristic
398C2×C4C2×#12311yescharacteristic
399C2×C4C2×#12511yescharacteristic
400C2×C4C2×#12611yescharacteristic
401C2×C4#52911yescharacteristic
402C2×C4#52911yescharacteristic
403C2×C4#53211yescharacteristic
404C2×C4#53211yescharacteristic
405C2×C4#54111yescharacteristic
406C2×C4#54111yescharacteristic
407C2×C4#54711yescharacteristic
408C2×C4#54711yescharacteristic
409C2×C4#54711yescharacteristic
410C2×C4#54811yescharacteristic
411C2×C4#54811yescharacteristic
412C2×C4D4×Q811yescharacteristic
413C2×C4#55211yescharacteristic
414C2×C4#55211yescharacteristic
415C2×C4#55611yescharacteristic
416C2×C4#55611yescharacteristic
417C2×C4#55711yescharacteristic
418C2×C4#55711yescharacteristic
419C8—42no—
420C8—42no—
421C8—41no—
422C8—41no—
423C8—44no—
424C8#38714yes—
425C8#38714yes—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22#232011yescharacteristic
439C22#236111yescharacteristic
440C22#237011yescharacteristic
441C22#242211yescharacteristic
442C22#245211yescharacteristic
443C22#249411yescharacteristic
444C22#250911yescharacteristic
445C22#329011yescharacteristic
446C22#329011yescharacteristic
447C22#329211yescharacteristic
448C22#329311yescharacteristic
449C4—44no—
450C4—42no—
451C4—42no—
452C4—84no—
453C4—84no—
454C4—82no—
455C4—81no—
456C4—82no—
457C4—81no—
458C4C2×#38711yescharacteristic
459C4#229311yescharacteristic
460C4#236211yescharacteristic
461C4#236511yescharacteristic
462C4#242111yescharacteristic
463C4#245311yescharacteristic
464C4#249611yescharacteristic
465C4#250611yescharacteristic
466C2—21no—
467C2—21no—
468C2—21no—
469C2—21no—
470C2#1413311yescharacteristic
471C2#2304511yescharacteristic
472C2#2304611yescharacteristic
473C2#2505211yescharacteristic
474C2#2505311yescharacteristic
475C2#2650111yescharacteristic
476C2#2650211yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1952C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
#211582
#210362
#204552
#113032
#112962
#108992
#108702
#108622
#108592
#107062