Finite Group Databasebeta

← #415144

Subgroups — #415144

Order 512.

894 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
1C2×#1998C211yesmaximal, characteristic
2#21314C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1950C211yesmaximal, characteristic
5#20793C211yesmaximal, characteristic
6C4×#428C211yesmaximal, characteristic
7#11242C212yesmaximal
8#11242C212yesmaximal
9#11216C212yesmaximal
10#11216C212yesmaximal
11#10847C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#424C2212yes—
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C4×#117C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1998—24no—
24#1989—22no—
25#1989—22no—
26#1989—21no—
27#1989—21no—
28#1989—22no—
29#1950—24no—
30#1843—22no—
31#1842—22no—
32#1776—22no—
33C4×#107—21no—
34C4×#106—21no—
35C2×#343C2211yescharacteristic
36C2×#343C2211yescharacteristic
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#341C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C22×M16C2311yescharacteristic
48C22×M16C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C23×C8C2312yes—
53C2×#125—22no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125—21no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125Q814yes—
61C2×#117C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C2×#108—24no—
70C2×#108—42no—
71C2×#108—42no—
72#428—24no—
73#428Q818yes—
74C2×#107—22no—
75C2×#106—22no—
76#424—24no—
77#424—24no—
78#424—42no—
79#424—42no—
80#423—22no—
81#423—22no—
82#423—22no—
83#423—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8D412yes—
93C2×C4⋊C8D412yes—
94C4×M16—22no—
95C4×M16D414yes—
96C2×#97—21no—
97C2×C4×C8—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4D411yescharacteristic
103C4×C4⋊C4D411yescharacteristic
104C4×C4⋊C4D412yes—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107#376—21no—
108C2×#95—21no—
109C43—21no—
110#343—22no—
111#343—22no—
112#343—22no—
113#343—22no—
114#343—42no—
115#343—42no—
116#341—24no—
117#340—24no—
118#340—24no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4#4111yescharacteristic
129C23×C4#4111yescharacteristic
130C23×C4C2411yescharacteristic
131C2×M16—21no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—22no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16C2×D412yes—
151C2×M16C2×D412yes—
152C2×M16C2×Q814yes—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D414yes—
165#125—24no—
166#125—24no—
167#125—24no—
168#125—24no—
169#125—42no—
170#125—42no—
171#125—42no—
172#125—42no—
173#117—22no—
174#117—22no—
175#117C2×Q814yes—
176#117#4114yes—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×Q812yes—
203C2×C4⋊C4C2×Q812yes—
204C2×C4⋊C4C2×Q814yes—
205C2×C22⋊C4—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42—82no—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221#108—48no—
222#107—28no—
223#106—28no—
224C4⋊C8—24no—
225C4⋊C8—24no—
226C4⋊C8—48no—
227C4⋊C8—48no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232#97—24no—
233C4×C8—24no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95—41no—
238#95—41no—
239C24C2×#4111yescharacteristic
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4C22⋊Q812yes—
264C22×C4C22⋊Q812yes—
265C22×C4#12612yes—
266C22×C4#12612yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C22×Q812yes—
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4H(5,𝔽2)12yes—
276C22×C42−1+412yes—
277M16—22no—
278M16—24no—
279M16—22no—
280M16—22no—
281M16—42no—
282M16—41no—
283M16—41no—
284M16—42no—
285M16—41no—
286M16—41no—
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291M16—42no—
292M16C4⋊Q814yes—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4C22⋊Q814yes—
330C4⋊C4C22⋊Q814yes—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C42—24no—
334C42—24no—
335C42—44no—
336C42—81no—
337C42—81no—
338C42—82no—
339C42—82no—
340C42#12112yes—
341C42#12112yes—
342C42#12112yes—
343C42#12112yes—
344C42C22⋊Q812yes—
345C42C22⋊Q812yes—
346C42C22⋊Q812yes—
347C42C22⋊Q812yes—
348C23—22no—
349C23—21no—
350C23—21no—
351C23C2×#12111yescharacteristic
352C23C2×C22⋊Q811yescharacteristic
353C23C2×#12611yescharacteristic
354C23#55212yes—
355C23#55412yes—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—84no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4#39612yes—
387C2×C4#39612yes—
388C2×C4#39612yes—
389C2×C4#39612yes—
390C2×C4C2×#12111yescharacteristic
391C2×C4C2×#12111yescharacteristic
392C2×C4C2×C22⋊Q811yescharacteristic
393C2×C4C2×C22⋊Q812yes—
394C2×C4C2×C22⋊Q811yescharacteristic
395C2×C4C2×C4⋊Q811yescharacteristic
396C2×C4#53211yescharacteristic
397C2×C4#53312yes—
398C2×C4#53511yescharacteristic
399C2×C4#53611yescharacteristic
400C2×C4#53712yes—
401C2×C4#54212yes—
402C2×C4#54612yes—
403C2×C4D4×Q812yes—
404C2×C4#55712yes—
405C2×C4#55712yes—
406C2×C4Q8212yes—
407C2×C4#55912yes—
408C8—24no—
409C8—42no—
410C8—44no—
411C8—82no—
412C8—81no—
413C8—81no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22C2×#39611yescharacteristic
423C22#241712yes—
424C22#251112yes—
425C22#256511yescharacteristic
426C22#256711yescharacteristic
427C22#306712yes—
428C22#306811yescharacteristic
429C22#306911yescharacteristic
430C4—22no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—44no—
435C4—44no—
436C4—84no—
437C4—82no—
438C4—82no—
439C4#245112yes—
440C4#254411yescharacteristic
441C4#254611yescharacteristic
442C4#258212yes—
443C4#262011yescharacteristic
444C4#285711yescharacteristic
445C2—21no—
446C2—21no—
447C2—22no—
448C2#1518411yescharacteristic
449C2#2360111yescharacteristic
450C2#2360411yescharacteristic
451C2#2485912yes—
452C2#2486012yes—
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
C2×#19982
#213142
C2×#19892
C2×#19502
#207932
C4×#4282
#112422
#112162
#108472