Finite Group Databasebeta

← #164855

Subgroups — #164855

Order 512.

675 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36497C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic
3#22022C211yesmaximal, characteristic
4#7717C212yesmaximal
5#7717C212yesmaximal
6#7683C211yesmaximal, characteristic
7#7680C211yesmaximal, characteristic
8#7652C212yesmaximal
9#7636C211yesmaximal, characteristic
10#7604C212yesmaximal
11#7568C211yesmaximal, characteristic
12#2864C2212yes—
13#2859C2211yescharacteristic
14#2570C412yes—
15#2552C411yescharacteristic
16#2551C411yescharacteristic
17#2550C412yes—
18#2520—22no—
19#2462—22no—
20#2453—21no—
21#2451—21no—
22#2370—21no—
23#2365—21no—
24#2355C411yescharacteristic
25#2355C411yescharacteristic
26C4×C4⋊Q8C2212yes—
27C4×#125C2211yescharacteristic
28C42×Q8C2211yescharacteristic
29#1984—21no—
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1917—22no—
34#1917—22no—
35#1839C2212yes—
36#1839C2212yes—
37#1831C2212yes—
38#1831C2212yes—
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125C2×C411yescharacteristic
58C2×C4×Q8—22no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×#97—21no—
67C2×#97—21no—
68C2×C4×C8—21no—
69C2×C4×C8—21no—
70#400—22no—
71#398—22no—
72#398—22no—
73#398—22no—
74#395—22no—
75#395—22no—
76#391—21no—
77#391—21no—
78#391—22no—
79#391—22no—
80#389—21no—
81#389—21no—
82#389—22no—
83#389—21no—
84#389—21no—
85#389—22no—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2311yescharacteristic
90#389C2311yescharacteristic
91#389C2312yes—
92#387—22no—
93#387—22no—
94#387—21no—
95#387—21no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#384C2×C412yes—
103#384C2×C411yescharacteristic
104#384C2×C411yescharacteristic
105#382—21no—
106#382C2×C411yescharacteristic
107#382C2×C412yes—
108#382C2×C412yes—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C411yescharacteristic
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D412yes—
124C43C2311yescharacteristic
125C43C2312yes—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8—22no—
141C22×Q8C22⋊C411yescharacteristic
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—24no—
149C4⋊Q8—42no—
150C4⋊Q8—44no—
151#125—22no—
152#125—41no—
153#125—44no—
154#125—81no—
155#125—81no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—82no—
160C4×Q8—82no—
161C2×C4⋊C4—22no—
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—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C411yescharacteristic
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×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C22×C412yes—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2411yescharacteristic
205C4⋊C8—42no—
206C4⋊C8—42no—
207C4⋊C8—82no—
208C4⋊C8—82no—
209#97—81no—
210#97—81no—
211C4×C8—42no—
212C4×C8—42no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—21no—
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—21no—
227#95—22no—
228#95—21no—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—82no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4C23×C411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C4⋊C4—21no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—82no—
301C4⋊C4—81no—
302C4⋊C4—82no—
303C4⋊C4—81no—
304C4⋊C4#10114yes—
305C4⋊C4#13112yes—
306C42—21no—
307C42—22no—
308C42—21no—
309C42—22no—
310C42—21no—
311C42—22no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—21no—
316C42—22no—
317C42—21no—
318C42—22no—
319C42—24no—
320C42—22no—
321C42—22no—
322C42—42no—
323C42—42no—
324C42—41no—
325C42—42no—
326C42—42no—
327C42—82no—
328C42—82no—
329C23C22×C22⋊C411yescharacteristic
330Q8—44no—
331Q8—88no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4C2×#10114yes—
377C2×C4C2×#10112yes—
378C2×C4#42112yes—
379C2×C4#42112yes—
380C2×C4C2×#13111yescharacteristic
381C2×C4#56811yescharacteristic
382C2×C4#57611yescharacteristic
383C2×C4#57711yescharacteristic
384C2×C4#57711yescharacteristic
385C2×C4#57811yescharacteristic
386C8—81no—
387C8—82no—
388C8—81no—
389C8—81no—
390C8—82no—
391C8—81no—
392C22#288011yescharacteristic
393C22#288411yescharacteristic
394C22C22×#10111yescharacteristic
395C22#289711yescharacteristic
396C22#289811yescharacteristic
397C22C2×#42111yescharacteristic
398C22C2×#42111yescharacteristic
399C4—21no—
400C4—21no—
401C4—24no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—42no—
409C4—42no—
410C4—41no—
411C4—42no—
412C4—42no—
413C4—82no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—82no—
418C4#153714yes—
419C2#1743011yescharacteristic
420C2#1752211yescharacteristic
421C2#1752511yescharacteristic
422C2#1760111yescharacteristic
423C2#1760211yescharacteristic
424C2C2×#153711yescharacteristic
425C2#1777411yescharacteristic
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
#364972
#220222
#77172
#76832
#76802
#76522
#76362
#76042
#75682