Finite Group Databasebeta

← #164852

Subgroups — #164852

Order 512.

671 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36447C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic
3#22022C211yesmaximal, characteristic
4#7723C212yesmaximal
5#7720C212yesmaximal
6#7683C211yesmaximal, characteristic
7#7680C211yesmaximal, characteristic
8#7650C212yesmaximal
9#7636C211yesmaximal, characteristic
10#7602C212yesmaximal
11#7568C211yesmaximal, characteristic
12#2859C2211yescharacteristic
13#2859C2212yes—
14#2571C411yescharacteristic
15#2571C412yes—
16#2552C412yes—
17#2552C411yescharacteristic
18#2508—22no—
19#2508—21no—
20#2506—22no—
21#2506—21no—
22#2351—21no—
23Q8×C4⋊C4—21no—
24Q8×C4⋊C4C411yescharacteristic
25Q8×C4⋊C4C411yescharacteristic
26C4×#125C2211yescharacteristic
27C4×#125C2212yes—
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1917—22no—
34#1917—22no—
35#1839C2212yes—
36#1839C2212yes—
37#1831C2212yes—
38#1831C2212yes—
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×#125—21no—
54C2×#125—22no—
55C2×#125C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×#97—21no—
67C2×#97—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70#400—22no—
71#400—22no—
72#400—22no—
73#398—22no—
74#398—22no—
75#398—22no—
76#389—21no—
77#389—22no—
78#389—22no—
79#389—21no—
80#389—22no—
81#389—21no—
82#389—21no—
83#389—22no—
84#389—21no—
85#389—22no—
86#389—21no—
87#389—22no—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2311yescharacteristic
92#389C2311yescharacteristic
93#389C2312yes—
94#387—21no—
95#387—22no—
96#387—21no—
97#387—22no—
98#384—21no—
99#384—22no—
100#384C2×C411yescharacteristic
101#384C2×C412yes—
102#384C2×C411yescharacteristic
103#384C2×C412yes—
104#382—21no—
105#382—22no—
106#382—21no—
107#382—22no—
108#382C2×C411yescharacteristic
109#382C2×C412yes—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C43C2312yes—
125C43C2311yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8C22⋊C411yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147#125—22no—
148#125—24no—
149#125—41no—
150#125—42no—
151#125—81no—
152#125—82no—
153#125—81no—
154#125—82no—
155C4×Q8—24no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—44no—
159C4×Q8—44no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2411yescharacteristic
207C4⋊C8—42no—
208C4⋊C8—42no—
209C4⋊C8—82no—
210C4⋊C8—82no—
211#97—81no—
212#97—81no—
213C4×C8—42no—
214C4×C8—42no—
215#95—21no—
216#95—22no—
217#95—21no—
218#95—22no—
219#95—21no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8#10114yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C4⋊C4—21no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C4⋊C4—82no—
300C4⋊C4—81no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4#13112yes—
306C42—22no—
307C42—21no—
308C42—22no—
309C42—21no—
310C42—21no—
311C42—22no—
312C42—21no—
313C42—22no—
314C42—21no—
315C42—22no—
316C42—21no—
317C42—22no—
318C42—24no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—44no—
326C42—44no—
327C42—42no—
328C42—42no—
329C23C22×C22⋊C411yescharacteristic
330Q8—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4C2×#10114yes—
376C2×C4C2×#10112yes—
377C2×C4#42112yes—
378C2×C4#42112yes—
379C2×C4C2×#13111yescharacteristic
380C2×C4#56811yescharacteristic
381C2×C4#57611yescharacteristic
382C2×C4#57711yescharacteristic
383C2×C4#57711yescharacteristic
384C2×C4#57811yescharacteristic
385C8—81no—
386C8—82no—
387C8—81no—
388C8—81no—
389C8—82no—
390C8—81no—
391C22#288011yescharacteristic
392C22#288411yescharacteristic
393C22C22×#10111yescharacteristic
394C22#289711yescharacteristic
395C22#289811yescharacteristic
396C22C2×#42111yescharacteristic
397C22C2×#42111yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—24no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—42no—
409C4—42no—
410C4—41no—
411C4—42no—
412C4—82no—
413C4—81no—
414C4—81no—
415C4—82no—
416C4—81no—
417C4#153414yes—
418C2#1743011yescharacteristic
419C2#1752311yescharacteristic
420C2#1752611yescharacteristic
421C2#1760111yescharacteristic
422C2#1760211yescharacteristic
423C2C2×#153411yescharacteristic
424C2#1777711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#364472
#220222
#77232
#77202
#76832
#76802
#76502
#76362
#76022
#75682