Finite Group Databasebeta

← #414488

Subgroups — #414488

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1933C211yesmaximal, characteristic
2C2×C8×C4⋊C4C211yesmaximal, characteristic
3#15194C211yesmaximal, characteristic
4#15192C211yesmaximal, characteristic
5#11316C212yesmaximal
6#11228C211yesmaximal, characteristic
7#11077C211yesmaximal, characteristic
8#11067C212yesmaximal
9#10926C212yesmaximal
10#10880C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#103C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2212yes—
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2212yes—
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#1933—24no—
24C8×C4⋊C4—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#336C2212yes—
33#1445—21no—
34#1445—21no—
35#1444—21no—
36#1443—22no—
37#1442—21no—
38#1441—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—24no—
59C2×C4⋊C8—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—42no—
68C2×C4×C8—24no—
69C2×C4×C8—24no—
70C2×C4×C8—42no—
71C2×C4×C8—42no—
72C2×C4×C8D414yes—
73#391—24no—
74#391—22no—
75#391—22no—
76#391—42no—
77#389—24no—
78#389—22no—
79#389—22no—
80#389—42no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95D412yes—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#336—24no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8#4112yes—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2411yescharacteristic
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8#4114yes—
143C22×C8#4114yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4Q1614yes—
178C2×C4⋊C4#4114yes—
179C2×C4⋊C4#4112yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C2×D412yes—
189C2×C42#4112yes—
190C4⋊C8—28no—
191#103—28no—
192#103—48no—
193#103—44no—
194C4×C8—28no—
195C4×C8—48no—
196#95—24no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—42no—
201#95—42no—
202#95—42no—
203#95—42no—
204#95—44no—
205#95—42no—
206#95—81no—
207#95—82no—
208#95—82no—
209#95—81no—
210#95—82no—
211C24C22×D411yescharacteristic
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—42no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4#12112yes—
283C22×C4#12112yes—
284C22×C4C2×Q1614yes—
285C22×C4C2×Q1612yes—
286C22×C4#13514yes—
287C22×C4#13512yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4112yes—
293C22×C4C2×#4112yes—
294C22×C4C2×#4111yescharacteristic
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C4H(5,𝔽2)11yescharacteristic
297C22×C42−1+412yes—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8#12314yes—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—88no—
336C4⋊C4—84no—
337C42—22no—
338C42—22no—
339C42—24no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C23C2×#12111yescharacteristic
345C23#53212yes—
346C23#54611yescharacteristic
347C23#54611yescharacteristic
348C23#54812yes—
349C23C22×Q1611yescharacteristic
350C23C2×#13511yescharacteristic
351C23C2×#13512yes—
352C23#57711yescharacteristic
353C23#57712yes—
354C23#57711yescharacteristic
355Q8—44no—
356Q8—42no—
357Q8—42no—
358Q8—82no—
359Q8—81no—
360Q8—81no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4#46714yes—
419C2×C4C2×#12311yescharacteristic
420C2×C4#52511yescharacteristic
421C2×C4#52511yescharacteristic
422C2×C4#52912yes—
423C2×C4#52912yes—
424C2×C4#52912yes—
425C2×C4#53211yescharacteristic
426C2×C4#53511yescharacteristic
427C2×C4#53611yescharacteristic
428C2×C4#53912yes—
429C2×C4#53912yes—
430C2×C4#55112yes—
431C2×C4#55212yes—
432C2×C4#55311yescharacteristic
433C2×C4#55311yescharacteristic
434C2×C4#55312yes—
435C2×C4#55612yes—
436C2×C4#55612yes—
437C8—24no—
438C8—44no—
439C8—42no—
440C22#253511yescharacteristic
441C22#254511yescharacteristic
442C22#255812yes—
443C22#256012yes—
444C22#285411yescharacteristic
445C22C2×#46711yescharacteristic
446C22#305011yescharacteristic
447C22#305711yescharacteristic
448C22#306011yescharacteristic
449C22#316812yes—
450C22#317012yes—
451C22#317312yes—
452C22#317412yes—
453C22#329612yes—
454C22#329911yescharacteristic
455C22#329911yescharacteristic
456C22#330112yes—
457C22#330411yescharacteristic
458C22#330411yescharacteristic
459C22#336012yes—
460C22#336012yes—
461C22#336312yes—
462C22#336512yes—
463C4—24no—
464C4—42no—
465C4—44no—
466C4—44no—
467C4—82no—
468C4—84no—
469C4—82no—
470C4—162no—
471C4—161no—
472C4—161no—
473C4#239511yescharacteristic
474C4#257712yes—
475C4#259412yes—
476C4#259911yescharacteristic
477C4#260311yescharacteristic
478C4#262311yescharacteristic
479C2#1517411yescharacteristic
480C2#2814211yescharacteristic
481C2#2814411yescharacteristic
482C2#2863811yescharacteristic
483C2#2864211yescharacteristic
484C2#2900912yes—
485C2#2901412yes—
486C2#2907312yes—
487C2#2907612yes—
488C2#3313011yescharacteristic
489C2#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×#19332
C2×C8×C4⋊C42
#151942
#151922
#113162
#112282
#110772
#110672
#109262
#108802
#108202