Finite Group Databasebeta

← #164106

Subgroups — #164106

Order 512.

702 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35313C211yesmaximal, characteristic
2#22636C211yesmaximal, characteristic
3#22632C211yesmaximal, characteristic
4#7733C212yesmaximal, complemented
5#7709C212yesmaximal, complemented
6#7693C212yesmaximal, complemented
7#7669C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic
9#7575C211yesmaximal, characteristic
10#7575C211yesmaximal, characteristic
11#7564C211yesmaximal, characteristic
12#2863C2212yes—
13#2863C2212yes—
14#2645C412yes—
15#2645C412yes—
16#2514—21no—
17#2514—21no—
18#2511—21no—
19#2511—21no—
20#2493—22no—
21#2493—22no—
22#2317C411yescharacteristic
23#2316C411yescharacteristic
24#2310C411yescharacteristic
25#2310C411yescharacteristic
26#2296C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1924—22no—
32#1923—22no—
33#1831C2212yes—
34#1831C2212yes—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1770—21no—
38C8×C4⋊C4—21no—
39C4×C22⋊C8C2211yescharacteristic
40C4×C22⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2211yescharacteristic
53C23×Q8C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×#97—22no—
69C2×C4×C8—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72#401—22no—
73#401—22no—
74#400—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#400—22no—
79#400—22no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2×C412yes—
90#391C2×C412yes—
91#391C2312yes—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C411yescharacteristic
99#389C2×C411yescharacteristic
100#389C2311yescharacteristic
101#389C2311yescharacteristic
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#385—21no—
109#385—21no—
110#385—22no—
111#384—21no—
112#383—22no—
113#383—22no—
114#382—21no—
115#381—22no—
116#381—22no—
117#380—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4D412yes—
125C4×C22⋊C4D412yes—
126#376C2×C412yes—
127C2×#95—21no—
128C2×#95—21no—
129C43C2312yes—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336C2312yes—
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×Q8—24no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8C22×C412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156#117—24no—
157#117—24no—
158#117—42no—
159#117—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—42no—
186C2×C42C22×C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2411yescharacteristic
196C4⋊C8—42no—
197C4⋊C8—42no—
198C4⋊C8—82no—
199C4⋊C8—82no—
200C22⋊C8—42no—
201C22⋊C8—42no—
202C22⋊C8—81no—
203C22⋊C8—81no—
204#97—44no—
205C4×C8—42no—
206C4×C8—42no—
207C4×C8—44no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—21no—
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—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232C24C2×C22⋊C411yescharacteristic
233C2×Q8—24no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—88no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4#13114yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C22⋊C4—24no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C42—24no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—41no—
326C42—42no—
327C42—42no—
328C42—42no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23#40812yes—
336C23C22×C22⋊C411yescharacteristic
337Q8—84no—
338Q8—84no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—84no—
387C2×C4—162no—
388C2×C4#40812yes—
389C2×C4#41012yes—
390C2×C4#41312yes—
391C2×C4C2×#13112yes—
392C2×C4#57711yescharacteristic
393C2×C4#57711yescharacteristic
394C2×C4#57811yescharacteristic
395C2×C4#57811yescharacteristic
396C8—42no—
397C8—42no—
398C8—81no—
399C8—81no—
400C8—82no—
401C8—82no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22C2×#40811yescharacteristic
410C22C2×#41011yescharacteristic
411C22C2×#41311yescharacteristic
412C22#289811yescharacteristic
413C22#289811yescharacteristic
414C22#290111yescharacteristic
415C22#290111yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—42no—
425C4—41no—
426C4—41no—
427C4—82no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—82no—
433C4#147314yes—
434C2—42no—
435C2C2×#147311yescharacteristic
436C2#1746911yescharacteristic
437C2#1748411yescharacteristic
438C2#1760511yescharacteristic
439C2#1760511yescharacteristic
440C2#1769111yescharacteristic
441C2#1777811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7733C2semidirect
5. #7709C2semidirect
6. #7693C2semidirect
7. #7669C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#353132
#226362
#226322
#77332
#77092
#76932
#76692
#75982
#75752
#75642