Finite Group Databasebeta

← #165114

Subgroups — #165114

Order 512.

687 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36286C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic
3#22736C211yesmaximal, characteristic
4#7713C212yesmaximal
5#7709C212yesmaximal
6#7662C211yesmaximal, characteristic
7#7662C211yesmaximal, characteristic
8#7652C211yesmaximal, characteristic
9#7630C212yesmaximal
10#7596C212yesmaximal
11#7570C211yesmaximal, characteristic
12C2×#446—21no—
13#2863C2212yes—
14#2862C2211yescharacteristic
15#2605—22no—
16#2588—22no—
17#2581—21no—
18#2570—21no—
19#2568C411yescharacteristic
20#2567C411yescharacteristic
21#2511C412yes—
22#2451C412yes—
23#2369C411yescharacteristic
24#2365C411yescharacteristic
25#2356—21no—
26#2355—21no—
27C4×C4⋊Q8C2211yescharacteristic
28#2295C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1917—22no—
34#1839C2212yes—
35#1831C2212yes—
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38#1770—21no—
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42#1745C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61#446—44no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×#97—22no—
69C2×#97—22no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72#400—22no—
73#400—22no—
74#400—22no—
75#398—22no—
76#395—22no—
77#395—22no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—22no—
84#391—22no—
85#391C2×C412yes—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2312yes—
90#389—21no—
91#389—21no—
92#389—21no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2311yescharacteristic
97#389C2311yescharacteristic
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#384—21no—
108#384C2×C411yescharacteristic
109#383—22no—
110#382—21no—
111#382—21no—
112#382—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126#376—22no—
127C43C2311yescharacteristic
128C43C2312yes—
129#336—21no—
130#336—22no—
131#336—21no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8C22⋊C411yescharacteristic
144C22×Q8C22×C412yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—41no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—24no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—42no—
164C2×C4⋊C4—21no—
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⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
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×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2411yescharacteristic
204C4⋊C8—44no—
205C4⋊C8—42no—
206C4⋊C8—44no—
207C4⋊C8—42no—
208C4⋊C8—42no—
209C4⋊C8—42no—
210#97—44no—
211#97—44no—
212C4×C8—42no—
213C4×C8—42no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—21no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235C2×Q8—22no—
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—81no—
246C2×Q8—81no—
247C2×Q8—81no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C23×C411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—42no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4#13112yes—
310C42—22no—
311C42—21no—
312C42—22no—
313C42—21no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—41no—
321C42—41no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—44no—
326C42—42no—
327C42—42no—
328C42—82no—
329C42AGL(1,ℤ/8)12yes—
330C42#13712yes—
331C42#13712yes—
332C42#13712yes—
333C23C22×C22⋊C411yescharacteristic
334Q8—44no—
335Q8—84no—
336Q8—82no—
337Q8—82no—
338Q8—162no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
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—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4#41312yes—
383C2×C4#41712yes—
384C2×C4#41712yes—
385C2×C4#41912yes—
386C2×C4#41912yes—
387C2×C4C2×#13111yescharacteristic
388C2×C4#56811yescharacteristic
389C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
390C2×C4C2×#13711yescharacteristic
391C2×C4C2×#13711yescharacteristic
392C2×C4C2×#13711yescharacteristic
393C8—42no—
394C8—42no—
395C8—81no—
396C8—82no—
397C8—81no—
398C8—82no—
399C22#288011yescharacteristic
400C22#288411yescharacteristic
401C22C2×#41311yescharacteristic
402C22C2×#41911yescharacteristic
403C22C2×#41711yescharacteristic
404C22#290111yescharacteristic
405C22#290111yescharacteristic
406C4—21no—
407C4—21no—
408C4—22no—
409C4—22no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—41no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—82no—
421C4—82no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4#155914yes—
426C2#1742811yescharacteristic
427C2#1751211yescharacteristic
428C2#1751411yescharacteristic
429C2#1762811yescharacteristic
430C2#1763211yescharacteristic
431C2#1769111yescharacteristic
432C2C2×#155911yescharacteristic
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
#362862
#228322
#227362
#77132
#77092
#76622
#76522
#76302
#75962
#75702