Finite Group Databasebeta

← #224073

Subgroups — #224073

Order 512.

816 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2507C211yesmaximal, characteristic
2#22627C212yesmaximal
3#9883C212yesmaximal
4#9883C212yesmaximal
5#9879C212yesmaximal
6#8562C212yesmaximal
7#8560C211yesmaximal, characteristic
8#8154C212yesmaximal
9#8153C211yesmaximal, characteristic
10#2507—22no—
11#2507—22no—
12#2507—22no—
13#2507—22no—
14C2×#398C412yes—
15C2×#398C412yes—
16C2×#389C411yescharacteristic
17C2×#389C411yescharacteristic
18C2×#389C411yescharacteristic
19C2×#389C2212yes—
20C2×#389C2211yescharacteristic
21C2×#387C2212yes—
22C2×#384C411yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1940—22no—
26#1917—22no—
27#1916—22no—
28#1797C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1771—22no—
32#1749C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2212yes—
36C2×#324C2211yescharacteristic
37C2×#324C2211yescharacteristic
38#1280C2211yescharacteristic
39#1280C2211yescharacteristic
40#1280C2212yes—
41#1280C2212yes—
42#1280C2212yes—
43#1280C2212yes—
44#1280C2212yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2312yes—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C22⋊C8C2312yes—
65C2×C22⋊C8C2312yes—
66C2×C22⋊C8C2312yes—
67C2×C22⋊C8C2312yes—
68C2×#97—22no—
69C2×#97—22no—
70C2×C4×C8—22no—
71#398—22no—
72#398—22no—
73#398—22no—
74#398—22no—
75#398—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#389—21no—
80#389—22no—
81#389—21no—
82#389—21no—
83#389—22no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#384—22no—
99#384—21no—
100#384—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2311yescharacteristic
113C2×#95C2312yes—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#324—24no—
121#324—22no—
122#324Q814yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C412yes—
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C2×Q812yes—
179C2×C4⋊C4#4112yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—42no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—42no—
188C2×C42—82no—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C4⋊C8—44no—
196C4⋊C8—82no—
197C4⋊C8—82no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8C2×Q814yes—
206#97—44no—
207#97—82no—
208C4×C8—82no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223C24C23×C411yescharacteristic
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C4×Q812yes—
279C22×C4C4×Q812yes—
280C22×C4#13112yes—
281C22×C4C22×D412yes—
282C22×C4C22×Q811yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C42−1+412yes—
285C2×C8—24no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4#10114yes—
322C42—24no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—42no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23C22×C22⋊C411yescharacteristic
337C23C2×C4×Q811yescharacteristic
338C23#51911yescharacteristic
339C23C2×#13111yescharacteristic
340C23#56811yescharacteristic
341C23#56811yescharacteristic
342C23#56811yescharacteristic
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#41012yes—
407C2×C4C2×#10114yes—
408C2×C4C2×#10112yes—
409C2×C4#54812yes—
410C2×C4D4×Q812yes—
411C8—44no—
412C8—82no—
413C8—82no—
414C8—82no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22Q8×C22⋊C411yescharacteristic
428C22#287911yescharacteristic
429C22#288011yescharacteristic
430C22C2×#41011yescharacteristic
431C22#288311yescharacteristic
432C22C22×#10111yescharacteristic
433C22#288811yescharacteristic
434C22#296111yescharacteristic
435C22#296211yescharacteristic
436C22#298511yescharacteristic
437C22#298711yescharacteristic
438C4—24no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—81no—
451C4—82no—
452C4—82no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2#1739411yescharacteristic
458C2#1740811yescharacteristic
459C2#1741511yescharacteristic
460C2#1742611yescharacteristic
461C2#2251111yescharacteristic
462C2#2251311yescharacteristic
463C2Q8×#10111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25072
#226272
#98832
#98792
#85622
#85602
#81542
#81532