Finite Group Databasebeta

← #164099

Subgroups — #164099

Order 512.

704 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35265C211yesmaximal, characteristic
2#22639C211yesmaximal, characteristic
3#22636C211yesmaximal, characteristic
4#7717C212yesmaximal, complemented
5#7703C212yesmaximal, complemented
6#7677C212yesmaximal, complemented
7#7663C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic
9#7576C211yesmaximal, characteristic
10#7573C211yesmaximal, characteristic
11#7564C211yesmaximal, characteristic
12#2864C2212yes—
13#2859C2212yes—
14#2647C412yes—
15#2647C412yes—
16#2509—21no—
17#2508—21no—
18#2507—21no—
19#2506—21no—
20#2505—22no—
21#2473—22no—
22#2317C411yescharacteristic
23#2317C411yescharacteristic
24#2317C411yescharacteristic
25#2316C411yescharacteristic
26#2296C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1939—21no—
31#1924—22no—
32#1923—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1770—21no—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2211yescharacteristic
42#1745C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×#125—22no—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×#97—22no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—22no—
74#400—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#398—22no—
79#398—22no—
80#398—22no—
81#398—22no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—22no—
89#389—22no—
90#389—21no—
91#389—21no—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C411yescharacteristic
98#389C2×C411yescharacteristic
99#389C2311yescharacteristic
100#389C2312yes—
101#389C2311yescharacteristic
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—21no—
107#385—21no—
108#385—22no—
109#385—22no—
110#385—22no—
111#384—21no—
112#384C2×C412yes—
113#384C2×C412yes—
114#382—21no—
115#382—22no—
116#382C2×C412yes—
117#382C2×C412yes—
118#380—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4D412yes—
126C4×C22⋊C4D412yes—
127#376C2×C412yes—
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
140C23×C4—21no—
141C23×C4—21no—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151#125—82no—
152#125—82no—
153#117—24no—
154#117—24no—
155#117—24no—
156#117—42no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—81no—
176C2×C4⋊C4—81no—
177C2×C4⋊C4C22⋊C411yescharacteristic
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×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22⋊C412yes—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—42no—
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2411yescharacteristic
204C4⋊C8—42no—
205C4⋊C8—42no—
206C4⋊C8—44no—
207C4⋊C8—82no—
208C22⋊C8—42no—
209C22⋊C8—42no—
210C22⋊C8—81no—
211C22⋊C8—81no—
212#97—44no—
213C4×C8—42no—
214C4×C8—42no—
215C4×C8—82no—
216#95—22no—
217#95—21no—
218#95—21no—
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—21no—
229#95—21no—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C24C2×C22⋊C411yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4#13114yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4#10114yes—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C42—24no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—24no—
327C42—42no—
328C42—42no—
329C42—41no—
330C42—41no—
331C42—42no—
332C42—42no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23#40812yes—
340C23C22×C22⋊C411yescharacteristic
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—84no—
388C2×C4—162no—
389C2×C4#40812yes—
390C2×C4#41012yes—
391C2×C4C2×#10114yes—
392C2×C4C2×#10112yes—
393C2×C4C2×#13112yes—
394C2×C4#57611yescharacteristic
395C2×C4#57711yescharacteristic
396C2×C4#57711yescharacteristic
397C2×C4#57811yescharacteristic
398C8—42no—
399C8—42no—
400C8—81no—
401C8—82no—
402C8—82no—
403C8—81no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22C2×#40811yescharacteristic
412C22C2×#41011yescharacteristic
413C22C22×#10111yescharacteristic
414C22#289711yescharacteristic
415C22#289811yescharacteristic
416C22#290111yescharacteristic
417C22#290111yescharacteristic
418C4—24no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—42no—
425C4—42no—
426C4—42no—
427C4—41no—
428C4—41no—
429C4—81no—
430C4—82no—
431C4—81no—
432C4—81no—
433C4—81no—
434C4—82no—
435C4#147014yes—
436C2—42no—
437C2C2×#147011yescharacteristic
438C2#1746811yescharacteristic
439C2#1748411yescharacteristic
440C2#1760511yescharacteristic
441C2#1760711yescharacteristic
442C2#1769211yescharacteristic
443C2#1777411yescharacteristic
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. #7717C2semidirect
5. #7703C2semidirect
6. #7677C2semidirect
7. #7663C2semidirect

Maximal subgroups

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

subgroupindex
#352652
#226392
#226362
#77172
#77032
#76772
#76632
#75982
#75762
#75732
#75642