Finite Group Databasebeta

← #162444

Subgroups — #162444

Order 512.

706 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35199C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic
3#22018C211yesmaximal, characteristic
4#7733C212yesmaximal
5#7722C212yesmaximal
6#7673C212yesmaximal
7#7662C212yesmaximal
8#7590C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic
10#7571C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2863C2212yes—
14#2862C2212yes—
15#2666—22no—
16#2582C411yescharacteristic
17#2568C411yescharacteristic
18#2551C411yescharacteristic
19#2546C411yescharacteristic
20#2520—22no—
21#2508—21no—
22#2451—21no—
23#2451C412yes—
24#2446C412yes—
25#2331—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#1272C2212yes—
46#1272C2212yes—
47#1272C2212yes—
48#1272C2211yescharacteristic
49#1272C2212yes—
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2211yescharacteristic
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2×C412yes—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C8×Q8—44no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4×C8—21no—
70C2×C4×C8—22no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73#400—22no—
74#400—22no—
75#398—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#395—22no—
80#395—22no—
81#391—21no—
82#391—21no—
83#391—22no—
84#391—22no—
85#391—22no—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2×C412yes—
90#391C2×C411yescharacteristic
91#391C2312yes—
92#391C2311yescharacteristic
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C411yescharacteristic
102#389C2311yescharacteristic
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—21no—
109#387—21no—
110#387—22no—
111#384C2×C411yescharacteristic
112#383—22no—
113#382—21no—
114#382—22no—
115#382—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126#376—21no—
127C43C2311yescharacteristic
128C43C2312yes—
129#336—22no—
130#336—21no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8—22no—
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×Q8C22×C412yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C4⋊Q8—24no—
152C4⋊Q8—44no—
153C4⋊Q8—82no—
154C4⋊Q8—82no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—82no—
159C4×Q8SD1614yes—
160C4×Q8Q1614yes—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C412yes—
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2411yescharacteristic
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8—44no—
203C4⋊C8—44no—
204C4⋊C8—42no—
205C4⋊C8—82no—
206C4×C8—42no—
207C4×C8—44no—
208C4×C8—42no—
209C4×C8—81no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
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—21no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—42no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—82no—
242C2×Q8—82no—
243C2×Q8#10314yes—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—22no—
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×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C2×C8—22no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—41no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—42no—
286C4⋊C4—41no—
287C4⋊C4—44no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—81no—
295C4⋊C4—82no—
296C4⋊C4—81no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4#10318yes—
302C4⋊C4#10314yes—
303C42—24no—
304C42—21no—
305C42—22no—
306C42—21no—
307C42—22no—
308C42—22no—
309C42—41no—
310C42—42no—
311C42—41no—
312C42—42no—
313C42—42no—
314C42—41no—
315C42—42no—
316C42—42no—
317C42—82no—
318C42#13114yes—
319C42C2×SD1614yes—
320C42C2×SD1612yes—
321C42C2×Q1614yes—
322C42C2×Q1612yes—
323C23C22×C22⋊C411yescharacteristic
324Q8—28no—
325Q8—82no—
326Q8—84no—
327Q8—164no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4#40812yes—
370C2×C4#40812yes—
371C2×C4C2×#10312yes—
372C2×C4C2×#10314yes—
373C2×C4C2×#10314yes—
374C2×C4C2×#10312yes—
375C2×C4#42112yes—
376C2×C4C2×#13112yes—
377C2×C4C22×SD1611yescharacteristic
378C2×C4C22×Q1611yescharacteristic
379C2×C4#57711yescharacteristic
380C2×C4#57811yescharacteristic
381C8—42no—
382C8—81no—
383C8—82no—
384C8—81no—
385C8—81no—
386C8—82no—
387C22C2×#40811yescharacteristic
388C22#288411yescharacteristic
389C22#288811yescharacteristic
390C22C22×#10311yescharacteristic
391C22#289811yescharacteristic
392C22C2×#42111yescharacteristic
393C22#290111yescharacteristic
394C4—24no—
395C4—22no—
396C4—24no—
397C4—22no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—41no—
404C4—42no—
405C4—42no—
406C4—82no—
407C4—81no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—82no—
412C4#148114yes—
413C2#1739011yescharacteristic
414C2C2×#148111yescharacteristic
415C2#1748411yescharacteristic
416C2#1760111yescharacteristic
417C2#1761911yescharacteristic
418C2#1768111yescharacteristic
419C2#1772211yescharacteristic
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
#351992
#227342
#220182
#77332
#77222
#76732
#76622
#75902
#75772
#75712
#75622