Finite Group Databasebeta

← #162439

Subgroups — #162439

Order 512.

702 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35208C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic
3#22018C211yesmaximal, characteristic
4#7733C212yesmaximal
5#7720C212yesmaximal
6#7673C212yesmaximal
7#7660C212yesmaximal
8#7591C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic
10#7570C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2863C2212yes—
14#2859C2212yes—
15#2666—22no—
16#2582C411yescharacteristic
17#2567C411yescharacteristic
18#2567C411yescharacteristic
19#2552C411yescharacteristic
20#2511C412yes—
21#2507C412yes—
22#2506—21no—
23#2462—22no—
24#2453—21no—
25#2330—21no—
26#2320—21no—
27C4×C4⋊Q8C2211yescharacteristic
28#2295C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1939—22no—
34#1916—22no—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2212yes—
38#1831C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×#125—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C8×Q8—44no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4×C8—21no—
69C2×C4×C8—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72#400—22no—
73#400—22no—
74#398—22no—
75#398—22no—
76#398—22no—
77#398—22no—
78#395—22no—
79#395—22no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C411yescharacteristic
86#391C2311yescharacteristic
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C411yescharacteristic
100#389C2312yes—
101#389C2311yescharacteristic
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—21no—
108#387—21no—
109#387—22no—
110#384—21no—
111#384C2×C411yescharacteristic
112#384C2×C412yes—
113#384C2×C412yes—
114#383—22no—
115#382—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126#376—21no—
127C43C2311yescharacteristic
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—21no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8—21no—
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C4⋊Q8—24no—
151#125—44no—
152#125—82no—
153#125—82no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—42no—
157C4×Q8—81no—
158C4×Q8—81no—
159C4×Q8SD1614yes—
160C4×Q8Q1614yes—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—42no—
204C4⋊C8—42no—
205C4⋊C8—44no—
206C4⋊C8—44no—
207C4⋊C8—42no—
208C4⋊C8—82no—
209C4×C8—42no—
210C4×C8—44no—
211C4×C8—42no—
212C4×C8—81no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
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#95—22no—
229#95—21no—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—81no—
240C2×Q8—81no—
241C2×Q8#10314yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C2×C8—22no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—42no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—41no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—81no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—81no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4#10318yes—
304C4⋊C4#10314yes—
305C42—24no—
306C42—21no—
307C42—22no—
308C42—21no—
309C42—22no—
310C42—22no—
311C42—41no—
312C42—42no—
313C42—41no—
314C42—42no—
315C42—42no—
316C42—41no—
317C42—42no—
318C42—42no—
319C42—82no—
320C42#13114yes—
321C42C2×SD1614yes—
322C42C2×SD1612yes—
323C42C2×Q1614yes—
324C42C2×Q1612yes—
325C23C22×C22⋊C411yescharacteristic
326Q8—28no—
327Q8—82no—
328Q8—84no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4#40812yes—
371C2×C4#40812yes—
372C2×C4C2×#10312yes—
373C2×C4C2×#10314yes—
374C2×C4C2×#10314yes—
375C2×C4C2×#10312yes—
376C2×C4#42112yes—
377C2×C4C2×#13112yes—
378C2×C4C22×SD1611yescharacteristic
379C2×C4C22×Q1611yescharacteristic
380C2×C4#57611yescharacteristic
381C2×C4#57711yescharacteristic
382C8—42no—
383C8—81no—
384C8—82no—
385C8—81no—
386C8—81no—
387C8—82no—
388C22C2×#40811yescharacteristic
389C22#288411yescharacteristic
390C22#288911yescharacteristic
391C22C22×#10311yescharacteristic
392C22#289711yescharacteristic
393C22C2×#42111yescharacteristic
394C22#290111yescharacteristic
395C4—24no—
396C4—22no—
397C4—24no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—22no—
404C4—41no—
405C4—42no—
406C4—42no—
407C4—81no—
408C4—81no—
409C4—82no—
410C4—82no—
411C4—81no—
412C4—81no—
413C4#148114yes—
414C2#1739111yescharacteristic
415C2C2×#148111yescharacteristic
416C2#1748311yescharacteristic
417C2#1760211yescharacteristic
418C2#1761911yescharacteristic
419C2#1768211yescharacteristic
420C2#1772311yescharacteristic
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
#352082
#227342
#220182
#77332
#77202
#76732
#76602
#75912
#75772
#75702
#75622