Finite Group Databasebeta

← #417072

Subgroups — #417072

Order 512.

944 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3#14348C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#11316C212yesmaximal
6#11316C212yesmaximal
7#11315C212yesmaximal
8#11315C212yesmaximal
9#10843C211yesmaximal, characteristic
10#10842C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1993—24no—
20#1990—24no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1446—22no—
32#1445—22no—
33#1442—22no—
34#1442—22no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×Q8—21no—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8D414yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#97—22no—
63C2×#97—22no—
64C2×#97—22no—
65C2×C4×C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68#384—22no—
69#384—22no—
70#382—22no—
71#382—22no—
72#382—22no—
73#382—22no—
74#376—22no—
75#376—22no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95D412yes—
89C2×#95D412yes—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#336—24no—
99#336—24no—
100C22×Q8—24no—
101C22×Q8—42no—
102C22×Q8—42no—
103C22×Q8—82no—
104C22×Q8—82no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8#4114yes—
129C22×C8#4114yes—
130C4⋊Q8—28no—
131C4⋊Q8—28no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4C2×D414yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42C2×D412yes—
178C2×C42#4112yes—
179C2×C42#4112yes—
180C2×C42#4112yes—
181C2×C42#4112yes—
182C2×C42#4112yes—
183#97—44no—
184C4×C8—44no—
185#95—24no—
186#95—24no—
187#95—42no—
188#95—42no—
189#95—42no—
190#95—42no—
191#95—82no—
192#95—82no—
193#95—82no—
194#95—82no—
195#95—82no—
196#95—82no—
197#95—82no—
198#95—82no—
199C24C22×D411yescharacteristic
200C2×Q8—48no—
201C2×Q8—84no—
202C2×Q8—84no—
203C2×Q8—88no—
204C2×Q8—88no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4#12112yes—
258C22×C4#12112yes—
259C22×C4#12312yes—
260C22×C4#12312yes—
261C22×C4#12412yes—
262C22×C4#12412yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C2×#4111yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C42−1+412yes—
272C22×C42−1+412yes—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—48no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42#12314yes—
311C42#12314yes—
312C42#12614yes—
313C42#12614yes—
314C23C2×#12111yescharacteristic
315C23C2×#12311yescharacteristic
316C23C2×#12411yescharacteristic
317C23#53211yescharacteristic
318C23#53211yescharacteristic
319C23#53611yescharacteristic
320C23#53611yescharacteristic
321C23#57612yes—
322C23#57712yes—
323C23#57712yes—
324C23#57812yes—
325Q8—816no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4#39714yes—
383C2×C4#39714yes—
384C2×C4C2×#12311yescharacteristic
385C2×C4C2×#12311yescharacteristic
386C2×C4C2×#12611yescharacteristic
387C2×C4C2×#12611yescharacteristic
388C2×C4#52912yes—
389C2×C4#52912yes—
390C2×C4#54112yes—
391C2×C4#54112yes—
392C2×C4#54812yes—
393C2×C4#54812yes—
394C2×C4#55212yes—
395C2×C4#55212yes—
396C2×C4#55612yes—
397C2×C4#55612yes—
398C2×C4#55612yes—
399C2×C4#55612yes—
400C8—44no—
401C8—44no—
402C22C2×#39711yescharacteristic
403C22#249911yescharacteristic
404C22#249911yescharacteristic
405C22#254511yescharacteristic
406C22#254511yescharacteristic
407C22#264211yescharacteristic
408C22#286111yescharacteristic
409C22#304111yescharacteristic
410C22#304211yescharacteristic
411C22#304311yescharacteristic
412C22#304411yescharacteristic
413C22#309612yes—
414C22#309712yes—
415C22#313612yes—
416C22#313712yes—
417C22#317312yes—
418C22#317512yes—
419C22#317812yes—
420C22#317912yes—
421C22#324112yes—
422C22#324212yes—
423C22#324312yes—
424C22#324412yes—
425C4—24no—
426C4—24no—
427C4—84no—
428C4—82no—
429C4—82no—
430C4—162no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4#242112yes—
436C4#242112yes—
437C4#251412yes—
438C4#251412yes—
439C2#1524111yescharacteristic
440C2#2363211yescharacteristic
441C2#2363311yescharacteristic
442C2#2788612yes—
443C2#2788712yes—
444C2#2866211yescharacteristic
445C2#2866311yescharacteristic
446C2#2866611yescharacteristic
447C2#2866711yescharacteristic
448C2#3023212yes—
449C2#3328312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ 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×#19932
C2×#19902
#143482
#142822
#113162
#113152
#108432
#108422
#106942