Finite Group Databasebeta

← #411138

Subgroups — #411138

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1941C211yesmaximal, characteristic
2C2×#1770C211yesmaximal, characteristic
3#15194C211yesmaximal, characteristic
4#15149C211yesmaximal, characteristic
5#11378C212yesmaximal
6#11327C212yesmaximal
7#11228C211yesmaximal, characteristic
8#11227C211yesmaximal, characteristic
9#10881C212yesmaximal
10#10881C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×#106C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#398—21no—
18C2×#389C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#1941—24no—
24#1770—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1445—21no—
35#1444—21no—
36#1442—22no—
37#1441—22no—
38#1441—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#106—22no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—24no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—22no—
67C2×#97—24no—
68C2×#97—42no—
69C2×#97—42no—
70C2×C4×C8—24no—
71C2×C4×C8D414yes—
72#398—44no—
73#389—24no—
74#389—24no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—42no—
80#389—42no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95D412yes—
96#340—22no—
97#340—22no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#340Q818yes—
104#336—24no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C2×D412yes—
129C22×C8C2×D412yes—
130C22×C8C2×Q814yes—
131C22×C8#4114yes—
132C22×C8#4114yes—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
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—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4C2×Q814yes—
178C2×C4⋊C4C2×Q814yes—
179C2×C4⋊C4#4112yes—
180C2×C4⋊C4#4112yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C2×D412yes—
190C2×C42#4112yes—
191#106—28no—
192#106—44no—
193C4⋊C8—28no—
194#97—48no—
195C4×C8—28no—
196#95—24no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—42no—
202#95—42no—
203#95—42no—
204#95—42no—
205#95—81no—
206#95—81no—
207#95—82no—
208#95—82no—
209C24C22×D411yescharacteristic
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
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—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4#12112yes—
267C22×C4#12112yes—
268C22×C4AGL(1,ℤ/8)12yes—
269C22×C4#13712yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×Q812yes—
272C22×C4C2×#4112yes—
273C22×C4C2×#4112yes—
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C42−1+412yes—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8#12314yes—
296C2×C8C4⋊Q814yes—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—44no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—44no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—84no—
322C4⋊C4—88no—
323C4⋊C4—88no—
324C4⋊C4—84no—
325C42—22no—
326C42—22no—
327C42—24no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—42no—
332C23C2×#12111yescharacteristic
333C23#53212yes—
334C23#54611yescharacteristic
335C23#54611yescharacteristic
336C23D4×Q812yes—
337C23C2×AGL(1,ℤ/8)11yescharacteristic
338C23C2×#13711yescharacteristic
339C23#57512yes—
340C23#57711yescharacteristic
341C23#57711yescharacteristic
342C23#57712yes—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4#46814yes—
403C2×C4C2×#12311yescharacteristic
404C2×C4#52511yescharacteristic
405C2×C4#52511yescharacteristic
406C2×C4#52912yes—
407C2×C4C2×C4⋊Q811yescharacteristic
408C2×C4#53312yes—
409C2×C4#53312yes—
410C2×C4#53511yescharacteristic
411C2×C4#53611yescharacteristic
412C2×C4#53912yes—
413C2×C4#53912yes—
414C2×C4#55212yes—
415C2×C4#55212yes—
416C2×C4#55311yescharacteristic
417C2×C4#55311yescharacteristic
418C2×C4#55412yes—
419C2×C4#55412yes—
420C2×C4#55712yes—
421C8—24no—
422C8—24no—
423C8—44no—
424C22#253511yescharacteristic
425C22#254611yescharacteristic
426C22#255612yes—
427C22#255612yes—
428C22#285411yescharacteristic
429C22C2×#46811yescharacteristic
430C22#305411yescharacteristic
431C22#305711yescharacteristic
432C22#306011yescharacteristic
433C22#317312yes—
434C22#317412yes—
435C22#318012yes—
436C22#318112yes—
437C22#332312yes—
438C22#332411yescharacteristic
439C22#332612yes—
440C22#332911yescharacteristic
441C22#333111yescharacteristic
442C22#333211yescharacteristic
443C22#338412yes—
444C22#338712yes—
445C22#338812yes—
446C22#338812yes—
447C4—24no—
448C4—42no—
449C4—84no—
450C4—84no—
451C4—82no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4—161no—
456C4—161no—
457C4#239911yescharacteristic
458C4#248411yescharacteristic
459C4#250211yescharacteristic
460C4#259012yes—
461C4#259012yes—
462C4#263311yescharacteristic
463C2#1510611yescharacteristic
464C2#2814711yescharacteristic
465C2#2815011yescharacteristic
466C2#2868111yescharacteristic
467C2#2868411yescharacteristic
468C2#2895112yes—
469C2#2895312yes—
470C2#2895612yes—
471C2#2895812yes—
472C2#3313411yescharacteristic
473C2#3313711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#19412
C2×#17702
#151942
#151492
#113782
#113272
#112282
#112272
#108812
#108202