Finite Group Databasebeta

← #416966

Subgroups — #416966

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 448 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#20755C211yesmaximal, characteristic
6C4×#428C211yesmaximal, characteristic
7#11303C212yesmaximal
8#11303C212yesmaximal
9#11249C212yesmaximal
10#11249C212yesmaximal
11#10847C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C4×#117C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1998—24no—
22#1989—22no—
23#1989—22no—
24#1989—21no—
25#1989—21no—
26#1989—22no—
27#1950—24no—
28#1832—22no—
29#1831—22no—
30#1776—22no—
31C4×#107—21no—
32C4×#106—21no—
33C2×#343C2211yescharacteristic
34C2×#343C2211yescharacteristic
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39C2×#341C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1729C2212yes—
46#1729C2212yes—
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
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⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C2×#108—24no—
70C2×#108—24no—
71C2×#108—24no—
72#428—24no—
73#428—24no—
74C2×#107—22no—
75C2×#106—22no—
76#423—22no—
77#423—22no—
78#423—22no—
79#423—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D412yes—
87C2×C4⋊C8D412yes—
88C4×M16—22no—
89C4×M16D414yes—
90C2×#97—21no—
91C2×C4×C8—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D411yescharacteristic
97C4×C4⋊C4D411yescharacteristic
98C4×C4⋊C4D412yes—
99C4×C22⋊C4—21no—
100C4×C22⋊C4—21no—
101#376—21no—
102C2×#95—21no—
103C43—21no—
104#343—22no—
105#343—22no—
106#343—22no—
107#343—22no—
108#343—24no—
109#343—24no—
110#341—24no—
111#340—24no—
112#340—24no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#336—24no—
120#336—24no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4#4111yescharacteristic
125C23×C4#4111yescharacteristic
126C23×C4C2411yescharacteristic
127C2×M16—21no—
128C2×M16—22no—
129C2×M16—21no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16C2×D412yes—
147C2×M16C2×D412yes—
148C2×M16#4114yes—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8C2×D414yes—
161#125—24no—
162#125—24no—
163#125—24no—
164#125—24no—
165#125—42no—
166#125—42no—
167#125—42no—
168#125—42no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×Q812yes—
199C2×C4⋊C4C2×Q812yes—
200C2×C4⋊C4#4114yes—
201C2×C22⋊C4—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—82no—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217#108—48no—
218#107—28no—
219#106—28no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224C4⋊C8—42no—
225C4⋊C8—42no—
226#97—24no—
227C4×C8—24no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—41no—
232#95—41no—
233C24C2×#4111yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4C22⋊Q812yes—
258C22×C4C22⋊Q812yes—
259C22×C4#12612yes—
260C22×C4#12612yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×Q811yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C22×C42−1+412yes—
270C22×C42−1+412yes—
271M16—24no—
272M16—22no—
273M16—22no—
274M16—22no—
275M16—42no—
276M16—42no—
277M16—42no—
278M16—42no—
279M16—42no—
280M16—41no—
281M16—41no—
282M16—41no—
283M16—41no—
284M16—42no—
285M16—42no—
286M16#12414yes—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C42—44no—
328C42—42no—
329C42—42no—
330C42—81no—
331C42—81no—
332C42—82no—
333C42—82no—
334C42#12112yes—
335C42#12112yes—
336C42#12112yes—
337C42#12112yes—
338C42C22⋊Q812yes—
339C42C22⋊Q812yes—
340C42C22⋊Q812yes—
341C42C22⋊Q812yes—
342C23—22no—
343C23—21no—
344C23—21no—
345C23C2×#12111yescharacteristic
346C23C2×C22⋊Q811yescharacteristic
347C23C2×#12611yescharacteristic
348C23#55212yes—
349C23#55612yes—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
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—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—84no—
380C2×C4#39612yes—
381C2×C4#39612yes—
382C2×C4#39612yes—
383C2×C4#39612yes—
384C2×C4C2×#12111yescharacteristic
385C2×C4C2×#12111yescharacteristic
386C2×C4C2×C22⋊Q811yescharacteristic
387C2×C4C2×C22⋊Q811yescharacteristic
388C2×C4#52512yes—
389C2×C4C2×#12411yescharacteristic
390C2×C4#52912yes—
391C2×C4#53211yescharacteristic
392C2×C4#53612yes—
393C2×C4#53611yescharacteristic
394C2×C4#53611yescharacteristic
395C2×C4#54112yes—
396C2×C4#54712yes—
397C2×C4#54812yes—
398C2×C4#55612yes—
399C2×C4#55612yes—
400C2×C4#55712yes—
401C2×C4#55712yes—
402C8—24no—
403C8—42no—
404C8—44no—
405C8—82no—
406C8—81no—
407C8—81no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22C2×#39611yescharacteristic
417C22#241512yes—
418C22#251412yes—
419C22#256511yescharacteristic
420C22#256811yescharacteristic
421C22#306712yes—
422C22#306811yescharacteristic
423C22#306911yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4—84no—
432C4—82no—
433C4#245312yes—
434C4#254511yescharacteristic
435C4#254711yescharacteristic
436C4#258112yes—
437C4#262111yescharacteristic
438C4#285011yescharacteristic
439C2—21no—
440C2—21no—
441C2—22no—
442C2#1523711yescharacteristic
443C2#2360111yescharacteristic
444C2#2360411yescharacteristic
445C2#2472412yes—
446C2#2472512yes—
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
#207552
C4×#4282
#113032
#112492
#108472