Finite Group Databasebeta

← #165015

Subgroups — #165015

Order 512.

723 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36285C211yesmaximal, characteristic
2#22736C211yesmaximal, characteristic
3#22734C211yesmaximal, characteristic
4#7733C212yesmaximal
5#7689C212yesmaximal
6#7663C211yesmaximal, characteristic
7#7660C211yesmaximal, characteristic
8#7653C211yesmaximal, characteristic
9#7615C212yesmaximal
10#7577C212yesmaximal
11#7571C211yesmaximal, characteristic
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2863C2212yes—
15#2859C2211yescharacteristic
16#2606—22no—
17#2590—22no—
18#2582C411yescharacteristic
19#2582C411yescharacteristic
20#2570—21no—
21#2565—21no—
22#2507C412yes—
23#2446C412yes—
24#2370—21no—
25#2356—21no—
26#2355C411yescharacteristic
27Q8×C4⋊C4C411yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29#2295C2212yes—
30C42×Q8C2211yescharacteristic
31#1940—22no—
32#1939—22no—
33#1939—21no—
34#1939—21no—
35#1831C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38#1831C2211yescharacteristic
39#1767C2212yes—
40C4×C4⋊C8C2212yes—
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1272C2211yescharacteristic
47#1272C2211yescharacteristic
48#1272C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61#446—44no—
62C8×Q8—44no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×#97—22no—
72C2×C4×C8—22no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75#400—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#391—21no—
82#391—21no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2311yescharacteristic
87#391C2311yescharacteristic
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—22no—
95#389—22no—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2312yes—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#384—21no—
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#384C2×C411yescharacteristic
114#383—22no—
115#382—21no—
116#382—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129#376—22no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×Q8C22×C411yescharacteristic
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—41no—
153#125—42no—
154#125—42no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—24no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8SD1614yes—
163C4×Q8Q1614yes—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
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×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207C4⋊C8—44no—
208C4⋊C8—44no—
209C4⋊C8—42no—
210C4⋊C8—44no—
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8—44no—
214C4⋊C8—42no—
215#97—44no—
216C4×C8—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—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8#10314yes—
248C2×Q8#13112yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—41no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—81no—
310C4⋊C4#10318yes—
311C4⋊C4#10314yes—
312C42—22no—
313C42—21no—
314C42—22no—
315C42—21no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—41no—
321C42—41no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—44no—
326C42—42no—
327C42—42no—
328C42—82no—
329C42C2×SD1612yes—
330C42C2×SD1614yes—
331C42C2×Q1614yes—
332C42C2×Q1612yes—
333C42AGL(1,ℤ/8)12yes—
334C42#13712yes—
335C23C22×C22⋊C411yescharacteristic
336Q8—28no—
337Q8—42no—
338Q8—42no—
339Q8—81no—
340Q8—81no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4#41312yes—
383C2×C4C2×#10312yes—
384C2×C4C2×#10314yes—
385C2×C4C2×#10314yes—
386C2×C4C2×#10312yes—
387C2×C4#41712yes—
388C2×C4#41712yes—
389C2×C4C2×#13111yescharacteristic
390C2×C4#56811yescharacteristic
391C2×C4C22×SD1611yescharacteristic
392C2×C4C22×Q1611yescharacteristic
393C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
394C2×C4C2×#13711yescharacteristic
395C8—42no—
396C8—42no—
397C8—81no—
398C8—82no—
399C8—82no—
400C8—81no—
401C22#288011yescharacteristic
402C22#288411yescharacteristic
403C22C2×#41311yescharacteristic
404C22C22×#10311yescharacteristic
405C22C2×#41711yescharacteristic
406C22#290111yescharacteristic
407C22#290111yescharacteristic
408C4—21no—
409C4—21no—
410C4—24no—
411C4—24no—
412C4—22no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—42no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—82no—
426C4—82no—
427C4#155014yes—
428C2#1742311yescharacteristic
429C2#1750911yescharacteristic
430C2#1751511yescharacteristic
431C2#1761911yescharacteristic
432C2#1762811yescharacteristic
433C2#1768711yescharacteristic
434C2C2×#155011yescharacteristic
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
#362852
#227362
#227342
#77332
#76892
#76632
#76602
#76532
#76152
#75772
#75712