Finite Group Databasebeta

← #162427

Subgroups — #162427

Order 512.

656 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35187C211yesmaximal, characteristic
2#22828C211yesmaximal, characteristic
3#22018C211yesmaximal, characteristic
4#7729C212yesmaximal
5#7717C212yesmaximal
6#7669C212yesmaximal
7#7657C212yesmaximal
8#7588C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12#2864C2212yes—
13#2863C2212yes—
14#2655—22no—
15#2582C411yescharacteristic
16#2571C411yescharacteristic
17#2568C411yescharacteristic
18#2546C411yescharacteristic
19#2524—22no—
20#2514—21no—
21#2514C412yes—
22#2508C412yes—
23#2446—21no—
24#2327—21no—
25#2320—21no—
26C4×C4⋊Q8C2211yescharacteristic
27#2295C2211yescharacteristic
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1916—22no—
33#1916—22no—
34#1839C2212yes—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2211yescharacteristic
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—22no—
64C2×C4×C8—22no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68#401—22no—
69#400—22no—
70#400—22no—
71#400—22no—
72#400—22no—
73#398—22no—
74#398—22no—
75#398—22no—
76#391—21no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391C2×C412yes—
82#391C2×C412yes—
83#391C2×C411yescharacteristic
84#391C2311yescharacteristic
85#389—21no—
86#389—21no—
87#389—21no—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C411yescharacteristic
93#389C2312yes—
94#389C2311yescharacteristic
95#387—22no—
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—21no—
105#387—21no—
106#384C2×C411yescharacteristic
107#383—22no—
108#383—22no—
109#382—21no—
110#382—22no—
111#382C2×C412yes—
112#382C2×C412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123#376—21no—
124C43C2311yescharacteristic
125C43C2312yes—
126#336—21no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—22no—
134#336C2311yescharacteristic
135#336C2312yes—
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×Q8C22⋊C411yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C4⋊Q8—24no—
150C4⋊Q8—44no—
151C4×Q8—24no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—24no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42C22×C412yes—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C412yes—
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D412yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2411yescharacteristic
196C4⋊C8—42no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—82no—
200C4×C8—42no—
201C4×C8—44no—
202C4×C8—42no—
203C4×C8—81no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—22no—
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#95—22no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C2×C8—22no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C42—24no—
297C42—22no—
298C42—22no—
299C42—21no—
300C42—22no—
301C42—21no—
302C42—22no—
303C42—22no—
304C42—42no—
305C42—41no—
306C42—41no—
307C42—42no—
308C42—42no—
309C42—41no—
310C42—42no—
311C42—44no—
312C42—42no—
313C42#13114yes—
314C42#13512yes—
315C42#13512yes—
316C23C22×C22⋊C411yescharacteristic
317Q8—44no—
318Q8—82no—
319Q8—84no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4#40812yes—
364C2×C4#40812yes—
365C2×C4#41612yes—
366C2×C4#41612yes—
367C2×C4#42112yes—
368C2×C4C2×#13112yes—
369C2×C4C2×#13511yescharacteristic
370C2×C4C2×#13511yescharacteristic
371C2×C4#57711yescharacteristic
372C2×C4#57811yescharacteristic
373C8—42no—
374C8—81no—
375C8—82no—
376C8—81no—
377C8—81no—
378C8—82no—
379C22C2×#40811yescharacteristic
380C22#288411yescharacteristic
381C22#288911yescharacteristic
382C22C2×#41611yescharacteristic
383C22#289811yescharacteristic
384C22C2×#42111yescharacteristic
385C22#290111yescharacteristic
386C4—22no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—81no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—82no—
403C4—81no—
404C4#148214yes—
405C2#1739111yescharacteristic
406C2C2×#148211yescharacteristic
407C2#1748411yescharacteristic
408C2#1760111yescharacteristic
409C2#1761611yescharacteristic
410C2#1768311yescharacteristic
411C2#1772511yescharacteristic
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
#351872
#228282
#220182
#77292
#77172
#76692
#76572
#75882
#75772
#75702
#75622