Finite Group Databasebeta

← #164845

Subgroups — #164845

Order 512.

657 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36313C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic
3#22022C211yesmaximal, characteristic
4#7733C212yesmaximal
5#7729C212yesmaximal
6#7682C211yesmaximal, characteristic
7#7682C211yesmaximal, characteristic
8#7644C212yesmaximal
9#7636C211yesmaximal, characteristic
10#7596C212yesmaximal
11#7568C211yesmaximal, characteristic
12#2863C2212yes—
13#2862C2211yescharacteristic
14#2568C411yescharacteristic
15#2546C411yescharacteristic
16#2528—22no—
17#2524—22no—
18#2514—21no—
19#2514C412yes—
20#2453C412yes—
21#2446—21no—
22#2365—21no—
23#2365C411yescharacteristic
24#2365C411yescharacteristic
25#2356—21no—
26C4×#125C2211yescharacteristic
27#2295C2212yes—
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#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C411yescharacteristic
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
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#401—22no—
71#400—22no—
72#400—22no—
73#400—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×C412yes—
84#391C2×C412yes—
85#391C2312yes—
86#389—22no—
87#389—21no—
88#389C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2311yescharacteristic
92#389C2311yescharacteristic
93#387—22no—
94#387—21no—
95#387—21no—
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#383—22no—
105#383—22no—
106#382—21no—
107#382—21no—
108#382—21no—
109#382C2×C411yescharacteristic
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D412yes—
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123#376—22no—
124C43C2311yescharacteristic
125C43C2312yes—
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×Q8—21no—
141C22×Q8C22⋊C411yescharacteristic
142C22×Q8C22×C412yes—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—44no—
152#125—22no—
153#125—41no—
154C4×Q8—24no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—44no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C412yes—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2411yescharacteristic
199C4⋊C8—42no—
200C4⋊C8—42no—
201C4⋊C8—82no—
202C4⋊C8—82no—
203#97—81no—
204#97—81no—
205C4×C8—42no—
206C4×C8—42no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4#13112yes—
301C42—22no—
302C42—21no—
303C42—22no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—21no—
311C42—21no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—41no—
320C42—44no—
321C42—44no—
322C42—42no—
323C42—42no—
324C23C22×C22⋊C411yescharacteristic
325Q8—44no—
326Q8—82no—
327Q8—82no—
328Q8—84no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4#41312yes—
375C2×C4#42112yes—
376C2×C4#42112yes—
377C2×C4C2×#13111yescharacteristic
378C2×C4#56811yescharacteristic
379C2×C4#57711yescharacteristic
380C2×C4#57711yescharacteristic
381C2×C4#57811yescharacteristic
382C2×C4#57811yescharacteristic
383C8—81no—
384C8—81no—
385C8—82no—
386C8—82no—
387C8—81no—
388C8—81no—
389C22#288011yescharacteristic
390C22#288411yescharacteristic
391C22C2×#41311yescharacteristic
392C22#289811yescharacteristic
393C22#289811yescharacteristic
394C22C2×#42111yescharacteristic
395C22C2×#42111yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—42no—
404C4—41no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—41no—
409C4—42no—
410C4—82no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—82no—
415C4#153514yes—
416C2#1742811yescharacteristic
417C2#1752311yescharacteristic
418C2#1752511yescharacteristic
419C2#1760111yescharacteristic
420C2#1760111yescharacteristic
421C2C2×#153511yescharacteristic
422C2#1777811yescharacteristic
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
#363132
#220222
#77332
#77292
#76822
#76442
#76362
#75962
#75682