Finite Group Databasebeta

← #164351

Subgroups — #164351

Order 512.

757 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36313C211yesmaximal, characteristic
2#20573C211yesmaximal, characteristic
3#20572C211yesmaximal, characteristic
4#7693C212yesmaximal
5#7689C212yesmaximal
6#7682C211yesmaximal, characteristic
7#7682C211yesmaximal, characteristic
8#7636C211yesmaximal, characteristic
9#7630C212yesmaximal
10#7568C211yesmaximal, characteristic
11#7562C212yesmaximal
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2863C2212yes—
15#2862C2211yescharacteristic
16#2568C411yescharacteristic
17#2546C411yescharacteristic
18#2528—22no—
19#2524—22no—
20#2514—21no—
21#2514C412yes—
22#2453C412yes—
23#2446—21no—
24#2365—21no—
25#2365C411yescharacteristic
26#2365C411yescharacteristic
27#2356—21no—
28C4×#125C2211yescharacteristic
29#2295C2212yes—
30C42×Q8C2211yescharacteristic
31#1771—22no—
32#1770—21no—
33#1770—22no—
34C8×C4⋊C4—21no—
35#1767C2211yescharacteristic
36#1767C2211yescharacteristic
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39#1745C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2211yescharacteristic
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
62#446—28no—
63C8×Q8—28no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×#97—22no—
69C2×#97—21no—
70C2×#97—22no—
71C2×#97—21no—
72C2×C4×C8—22no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—22no—
76#401—22no—
77#400—22no—
78#400—22no—
79#400—22no—
80#398—22no—
81#398—22no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2×C412yes—
90#391C2×C412yes—
91#391C2312yes—
92#389—22no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2311yescharacteristic
98#389C2311yescharacteristic
99#387—22no—
100#387—21no—
101#387—21no—
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#383—22no—
111#383—22no—
112#382—21no—
113#382—21no—
114#382—21no—
115#382C2×C411yescharacteristic
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D412yes—
129#376—22no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—21no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2312yes—
142C22×Q8—21no—
143C22×Q8C22⋊C411yescharacteristic
144C22×Q8C22×C412yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—44no—
154#125—22no—
155#125—41no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2411yescharacteristic
201C4⋊C8—28no—
202C4⋊C8—24no—
203C4⋊C8—24no—
204C4⋊C8—28no—
205#97—28no—
206#97—81no—
207#97—81no—
208#97—82no—
209C4×C8—28no—
210C4×C8—24no—
211C4×C8—24no—
212C4×C8—82no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95—21no—
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—21no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8#10414yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—24no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—24no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C4⋊C4—21no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4#10414yes—
306C4⋊C4#13112yes—
307C42—22no—
308C42—21no—
309C42—22no—
310C42—21no—
311C42—22no—
312C42—21no—
313C42—21no—
314C42—22no—
315C42—22no—
316C42—21no—
317C42—21no—
318C42—22no—
319C42—22no—
320C42—24no—
321C42—22no—
322C42—22no—
323C42—24no—
324C42—42no—
325C42—41no—
326C42—44no—
327C42—42no—
328C42—44no—
329C42—42no—
330C23C22×C22⋊C411yescharacteristic
331Q8—28no—
332Q8—82no—
333Q8—82no—
334Q8—84no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4#41312yes—
379C2×C4C2×#10414yes—
380C2×C4C2×#10414yes—
381C2×C4C2×#10412yes—
382C2×C4C2×#10412yes—
383C2×C4C2×#13111yescharacteristic
384C2×C4#56811yescharacteristic
385C2×C4#57711yescharacteristic
386C2×C4#57711yescharacteristic
387C2×C4#57811yescharacteristic
388C2×C4#57811yescharacteristic
389C8—42no—
390C8—44no—
391C8—42no—
392C8—44no—
393C8—42no—
394C8—42no—
395C22#288011yescharacteristic
396C22#288411yescharacteristic
397C22C2×#41311yescharacteristic
398C22#289811yescharacteristic
399C22#289811yescharacteristic
400C22C22×#10411yescharacteristic
401C22C22×#10411yescharacteristic
402C4—21no—
403C4—21no—
404C4—24no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—24no—
410C4—42no—
411C4—41no—
412C4—42no—
413C4—41no—
414C4—41no—
415C4—42no—
416C4—82no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—82no—
421C4#152414yes—
422C2#1742811yescharacteristic
423C2#1751811yescharacteristic
424C2#1751911yescharacteristic
425C2#1754911yescharacteristic
426C2#1754911yescharacteristic
427C2C2×#152411yescharacteristic
428C2#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
#205732
#205722
#76932
#76892
#76822
#76362
#76302
#75682
#75622