Finite Group Databasebeta

← #412665

Subgroups — #412665

Order 512.

856 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23670C212yesmaximal
2C2×#1951C211yesmaximal, characteristic
3C2×#1941C211yesmaximal, characteristic
4C2×#1848C211yesmaximal, characteristic
5#20435C211yesmaximal, characteristic
6#11249C212yesmaximal
7#11242C212yesmaximal
8#11204C212yesmaximal
9#10869C211yesmaximal, characteristic
10#10869C211yesmaximal, characteristic
11#10869C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C22×#108C2212yes—
14C22×#108C2211yescharacteristic
15C2×#428C2212yes—
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2212yes—
20C2×#405C2211yescharacteristic
21C2×#405C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2296C2211yescharacteristic
24C2×#389C2212yes—
25#2088—22no—
26#2088—22no—
27#2087—22no—
28#2086—22no—
29#1951—24no—
30#1941—21no—
31#1941—21no—
32#1941—21no—
33#1941—21no—
34#1848—24no—
35#1771—21no—
36#1771—21no—
37C8×C4⋊C4—21no—
38C8×C4⋊C4—21no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45#1736C2212yes—
46C2×#340C2212yes—
47#1729C2211yescharacteristic
48#1728C2211yescharacteristic
49C22×M16C2312yes—
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C2×#117C2311yescharacteristic
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42C2311yescharacteristic
61C2×#108—22no—
62C2×#108—22no—
63C2×#108—24no—
64C2×#108—22no—
65C2×#108—24no—
66C2×#108—22no—
67C2×#108—22no—
68C2×#108—42no—
69C2×#108D414yes—
70#428—24no—
71#428—42no—
72C2×#107—22no—
73C2×#106—21no—
74C2×#106—21no—
75C2×#106—22no—
76C2×#106—21no—
77C2×#106—21no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106Q814yes—
82#424—22no—
83#424Q814yes—
84#423—24no—
85#423—42no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90#405—22no—
91#405—22no—
92#405—22no—
93#405Q814yes—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8—22no—
97C2×C4×C8—21no—
98C2×C4×C8—22no—
99C2×C4×C8—21no—
100C2×C4×C8D411yescharacteristic
101C2×C4×C8D411yescharacteristic
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389D412yes—
109#389D412yes—
110#385—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C2×#95—22no—
114#343—24no—
115#343—42no—
116#343—42no—
117#343—42no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#336—21no—
127#336—21no—
128#336—21no—
129#336—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4#4112yes—
133C23×C4C2411yescharacteristic
134C2×M16—21no—
135C2×M16—21no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—22no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16C2×D414yes—
156C2×M16C2×Q812yes—
157C2×M16C2×Q812yes—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D412yes—
171C22×C8C2×Q812yes—
172C22×C8C2×Q812yes—
173C22×C8C2×Q812yes—
174C22×C8#4112yes—
175C22×C8#4112yes—
176C22×C8#4112yes—
177C22×C8#4112yes—
178#117—21no—
179#117—21no—
180#117—21no—
181#117—21no—
182#117—21no—
183#117—21no—
184#117C2×Q812yes—
185#117C2×Q812yes—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—84no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×Q814yes—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217#108—28no—
218#108—48no—
219#108—44no—
220#107—44no—
221#106—24no—
222#106—24no—
223#106—44no—
224#106—42no—
225#106—42no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—44no—
229C4⋊C8—44no—
230#97—24no—
231#97—42no—
232C4×C8—22no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—24no—
237C4×C8—42no—
238#95—22no—
239#95—22no—
240#95—42no—
241#95—42no—
242C24C2×#4111yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4#12112yes—
261C22×C4#12112yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×Q811yescharacteristic
266C22×C4C22×Q811yescharacteristic
267C22×C4C22×Q811yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C4H(5,𝔽2)11yescharacteristic
274C22×C42−1+411yescharacteristic
275C22×C42−1+411yescharacteristic
276M16—24no—
277M16—22no—
278M16—22no—
279M16—24no—
280M16—22no—
281M16—22no—
282M16—41no—
283M16—42no—
284M16—41no—
285M16—41no—
286M16—41no—
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291M16—42no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8#12112yes—
313C2×C8#12112yes—
314C2×C8C22⋊Q812yes—
315C2×C8C22⋊Q812yes—
316C2×C8C22⋊Q812yes—
317C2×C8C22⋊Q812yes—
318C2×C8C22⋊Q812yes—
319C2×C8C22⋊Q812yes—
320C2×C8#12314yes—
321C2×C8#12514yes—
322C2×C8#12514yes—
323C2×C8C4⋊Q814yes—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—84no—
339C4⋊C4—84no—
340C4⋊C4C4⋊Q812yes—
341C4⋊C4C4⋊Q812yes—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C42—21no—
347C42—21no—
348C42—22no—
349C42—22no—
350C42—21no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C23—22no—
357C23—22no—
358C23C2×#12111yescharacteristic
359C23#53311yescharacteristic
360C23#53311yescharacteristic
361C23#54811yescharacteristic
362C23#55311yescharacteristic
363C23#55411yescharacteristic
364C23#55611yescharacteristic
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—162no—
388C2×C4—162no—
389C2×C4#47114yes—
390C2×C4C2×#12111yescharacteristic
391C2×C4C2×C22⋊Q811yescharacteristic
392C2×C4C2×C22⋊Q811yescharacteristic
393C2×C4C2×C22⋊Q811yescharacteristic
394C2×C4C2×#12311yescharacteristic
395C2×C4C2×#12511yescharacteristic
396C2×C4C2×#12511yescharacteristic
397C2×C4C2×C4⋊Q811yescharacteristic
398C2×C4C2×C4⋊Q811yescharacteristic
399C2×C4#53211yescharacteristic
400C2×C4#53711yescharacteristic
401C2×C4#53711yescharacteristic
402C2×C4#54011yescharacteristic
403C2×C4#54111yescharacteristic
404C2×C4#54411yescharacteristic
405C2×C4#54411yescharacteristic
406C2×C4D4211yescharacteristic
407C2×C4#54611yescharacteristic
408C2×C4#54711yescharacteristic
409C2×C4D4×Q811yescharacteristic
410C2×C4D4×Q811yescharacteristic
411C2×C4#55011yescharacteristic
412C2×C4#55211yescharacteristic
413C2×C4#55411yescharacteristic
414C2×C4#55711yescharacteristic
415C2×C4#55711yescharacteristic
416C2×C4Q8211yescharacteristic
417C2×C4#55911yescharacteristic
418C8—22no—
419C8—22no—
420C8—42no—
421C8—42no—
422C8—44no—
423C8—82no—
424C8#39814yes—
425C8#39814yes—
426C22—22no—
427C22—22no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—22no—
433C22—22no—
434C22#241611yescharacteristic
435C22#244611yescharacteristic
436C22#250711yescharacteristic
437C22#253811yescharacteristic
438C22#255011yescharacteristic
439C22#257011yescharacteristic
440C22#285711yescharacteristic
441C22C2×#47112yes—
442C22#306712yes—
443C4—24no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—84no—
448C4—82no—
449C4—162no—
450C4C2×#39811yescharacteristic
451C4#241711yescharacteristic
452C4#244511yescharacteristic
453C4#250611yescharacteristic
454C4#255111yescharacteristic
455C4#255211yescharacteristic
456C4#258211yescharacteristic
457C4#262911yescharacteristic
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2#1513311yescharacteristic
463C2#2480212yes—
464C2#2829612yes—
465C2#3322011yescharacteristic
466C2#3322111yescharacteristic
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×#19512
C2×#19412
C2×#18482
#204352
#112492
#112422
#112042
#108692