Finite Group Databasebeta

← #165121

Subgroups — #165121

Order 512.

705 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36535C211yesmaximal, characteristic
2#22736C211yesmaximal, characteristic
3#22736C211yesmaximal, characteristic
4#7697C212yesmaximal
5#7697C212yesmaximal
6#7663C211yesmaximal, characteristic
7#7660C211yesmaximal, characteristic
8#7652C211yesmaximal, characteristic
9#7636C212yesmaximal
10#7602C212yesmaximal
11#7570C211yesmaximal, characteristic
12C2×#446—21no—
13C2×#446—21no—
14#2864C2212yes—
15#2859C2211yescharacteristic
16#2590—22no—
17#2588—22no—
18#2581—21no—
19#2571C411yescharacteristic
20#2570C412yes—
21#2565—21no—
22#2552C411yescharacteristic
23#2550C412yes—
24#2370—21no—
25#2355—21no—
26#2355C411yescharacteristic
27#2355C411yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29C4×#125C2212yes—
30C42×Q8C2211yescharacteristic
31#1940—22no—
32#1940—22no—
33#1939—21no—
34#1939—21no—
35#1831C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1831C2211yescharacteristic
39#1767C2212yes—
40#1767C2212yes—
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2211yescharacteristic
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×#125C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60#446—44no—
61#446—44no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×#97—22no—
71C2×#97—22no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74#400—22no—
75#400—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#395—22no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—22no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2311yescharacteristic
94#389C2312yes—
95#389C2311yescharacteristic
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#384—21no—
107#384C2×C411yescharacteristic
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#382—21no—
111#382—21no—
112#382C2×C412yes—
113#382C2×C412yes—
114#382C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C43C2312yes—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8C22⋊C411yescharacteristic
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—41no—
152#125—24no—
153#125—42no—
154#125—42no—
155#125—42no—
156C4×Q8—24no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—42no—
162C4×Q8—42no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
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—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207C4⋊C8—44no—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C4⋊C8—44no—
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214C4⋊C8—42no—
215#97—44no—
216#97—44no—
217C4×C8—42no—
218C4×C8—42no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—21no—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—81no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—42no—
279C4⋊C4—21no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4#10114yes—
311C4⋊C4#13112yes—
312C42—22no—
313C42—21no—
314C42—22no—
315C42—21no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—24no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C42—82no—
330C42—82no—
331C42AGL(1,ℤ/8)12yes—
332C42AGL(1,ℤ/8)12yes—
333C42#13712yes—
334C42#13712yes—
335C23C22×C22⋊C411yescharacteristic
336Q8—44no—
337Q8—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4C2×#10114yes—
381C2×C4C2×#10112yes—
382C2×C4#41712yes—
383C2×C4#41712yes—
384C2×C4#41712yes—
385C2×C4#41712yes—
386C2×C4C2×#13111yescharacteristic
387C2×C4#56811yescharacteristic
388C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
389C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
390C2×C4C2×#13711yescharacteristic
391C2×C4C2×#13711yescharacteristic
392C8—42no—
393C8—42no—
394C8—81no—
395C8—81no—
396C8—82no—
397C8—82no—
398C22#288011yescharacteristic
399C22#288411yescharacteristic
400C22C22×#10111yescharacteristic
401C22C2×#41711yescharacteristic
402C22C2×#41711yescharacteristic
403C22#290111yescharacteristic
404C22#290111yescharacteristic
405C4—21no—
406C4—21no—
407C4—24no—
408C4—22no—
409C4—22no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—81no—
420C4—82no—
421C4—81no—
422C4—81no—
423C4—82no—
424C4#156114yes—
425C2#1743011yescharacteristic
426C2#1751211yescharacteristic
427C2#1751511yescharacteristic
428C2#1762811yescharacteristic
429C2#1762811yescharacteristic
430C2#1768911yescharacteristic
431C2C2×#156111yescharacteristic
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
#365352
#227362
#76972
#76632
#76602
#76522
#76362
#76022
#75702