Finite Group Databasebeta

← #416992

Subgroups — #416992

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1997C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4#21171C211yesmaximal, characteristic
5C2×#1837C211yesmaximal, characteristic
6C4×#424C211yesmaximal, characteristic
7#11303C214yesmaximal
8#11216C214yesmaximal
9#10857C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#428C2212yes—
12C2×#424C2212yes—
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C4×#117C2211yescharacteristic
19#1997—24no—
20#1990—22no—
21#1990—22no—
22#1988—21no—
23#1988—21no—
24#1987—21no—
25#1987—21no—
26#1948—21no—
27#1948—21no—
28#1947—21no—
29#1947—21no—
30#1837—24no—
31#1768—22no—
32C4×C4⋊C8—22no—
33C2×#343C2214yes—
34C2×#343C2214yes—
35C2×#341C2214yes—
36#1736C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39C2×#340C2214yes—
40#1729C2212yes—
41C23×Q8—21no—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8Q814yes—
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C42C2311yescharacteristic
62#428—28no—
63C2×#107—22no—
64C2×#106—22no—
65#424—24no—
66#424—24no—
67#423—22no—
68#423—22no—
69#423—22no—
70#423D414yes—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C4×M16—22no—
77C4×M16—22no—
78C2×#97—21no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8—21no—
82C2×C4×C8D412yes—
83C2×C4×C8D412yes—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88C4×C4⋊C4D411yescharacteristic
89C4×C4⋊C4D411yescharacteristic
90C4×C4⋊C4D411yescharacteristic
91C4×C4⋊C4D411yescharacteristic
92C4×C22⋊C4—21no—
93C4×C22⋊C4—21no—
94#376—21no—
95C43—21no—
96#343—28no—
97#343—28no—
98#341—28no—
99#340—28no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#336—24no—
107C22×Q8—24no—
108C22×Q8—41no—
109C22×Q8—41no—
110C22×Q8—42no—
111C22×Q8—82no—
112C22×Q8—82no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4C2411yescharacteristic
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—24no—
128C2×M16—22no—
129C2×M16—24no—
130C2×M16C2×D414yes—
131C2×M16#4114yes—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C2×D414yes—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117#4114yes—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C4⋊C4C2×Q814yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42C2×D412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×Q812yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194#107—44no—
195#106—44no—
196C4⋊C8—216no—
197C4⋊C8—24no—
198C4⋊C8—24no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201#97—42no—
202C4×C8—24no—
203C4×C8—24no—
204C4×C8—42no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209C24C2×#4111yescharacteristic
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—88no—
213C2×Q8—82no—
214C2×Q8—82no—
215C2×Q8—88no—
216C2×Q8—82no—
217C2×Q8—82no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4C22⋊Q812yes—
240C22×C4C22⋊Q812yes—
241C22×C4#12612yes—
242C22×C4#12612yes—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C22×C4C22×Q811yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4C2×#4112yes—
250C22×C4C2×#4112yes—
251C22×C42−1+412yes—
252C22×C42−1+412yes—
253M16—28no—
254M16—24no—
255M16—24no—
256M16—42no—
257M16—42no—
258M16—44no—
259M16—44no—
260M16—44no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4#12412yes—
297C4⋊C4#12412yes—
298C22⋊C4—24no—
299C22⋊C4—42no—
300C42—24no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—42no—
306C42#12112yes—
307C42#12112yes—
308C42#12112yes—
309C42#12112yes—
310C42C22⋊Q812yes—
311C42C22⋊Q812yes—
312C42C22⋊Q812yes—
313C42C22⋊Q812yes—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23C2×#12111yescharacteristic
319C23C2×C22⋊Q811yescharacteristic
320C23C2×#12611yescharacteristic
321C23#52912yes—
322C23#54112yes—
323Q8—88no—
324Q8—88no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—44no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—84no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—164no—
354C2×C4#39612yes—
355C2×C4#39612yes—
356C2×C4#39612yes—
357C2×C4#39612yes—
358C2×C4#47112yes—
359C2×C4#47112yes—
360C2×C4#47212yes—
361C2×C4#47312yes—
362C2×C4C2×#12111yescharacteristic
363C2×C4C2×#12111yescharacteristic
364C2×C4C2×C22⋊Q811yescharacteristic
365C2×C4C2×C22⋊Q811yescharacteristic
366C2×C4#52512yes—
367C2×C4C2×#12411yescharacteristic
368C2×C4#53211yescharacteristic
369C2×C4#53611yescharacteristic
370C2×C4#53611yescharacteristic
371C2×C4#53612yes—
372C2×C4#54712yes—
373C2×C4#54812yes—
374C2×C4#55212yes—
375C2×C4#55612yes—
376C2×C4#55612yes—
377C2×C4#55612yes—
378C2×C4#55712yes—
379C2×C4#55712yes—
380C8—44no—
381C8—44no—
382C8—44no—
383C8—82no—
384C22—22no—
385C22—22no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—22no—
392C22C2×#39611yescharacteristic
393C22#245312yes—
394C22#251412yes—
395C22#262111yescharacteristic
396C22#285011yescharacteristic
397C22C2×#47111yescharacteristic
398C22C2×#47111yescharacteristic
399C22C2×#47211yescharacteristic
400C22C2×#47311yescharacteristic
401C4—22no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—84no—
406C4—81no—
407C4—81no—
408C4—162no—
409C4#206314yes—
410C4#206414yes—
411C4#241512yes—
412C4#254511yescharacteristic
413C4#254711yescharacteristic
414C4#256511yescharacteristic
415C4#256811yescharacteristic
416C4#258112yes—
417C2—22no—
418C2—22no—
419C2#1523711yescharacteristic
420C2C2×#206311yescharacteristic
421C2C2×#206411yescharacteristic
422C2#2988611yescharacteristic
423C2#2988711yescharacteristic
424C2#3306011yescharacteristic
425C2#3306111yescharacteristic
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×#19972
C2×#19902
#213132
#211712
C2×#18372
C4×#4242
#113032
#112162
#108572