Finite Group Databasebeta

← #165140

Subgroups — #165140

Order 512.

683 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36285C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic
3#22736C211yesmaximal, characteristic
4#7713C212yesmaximal
5#7709C212yesmaximal
6#7663C211yesmaximal, characteristic
7#7660C211yesmaximal, characteristic
8#7653C211yesmaximal, characteristic
9#7630C212yesmaximal
10#7596C212yesmaximal
11#7571C211yesmaximal, characteristic
12C2×#446—21no—
13#2863C2212yes—
14#2859C2211yescharacteristic
15#2606—22no—
16#2590—22no—
17#2582C411yescharacteristic
18#2582C411yescharacteristic
19#2570—21no—
20#2565—21no—
21#2507C412yes—
22#2446C412yes—
23#2370—21no—
24#2356—21no—
25#2355C411yescharacteristic
26Q8×C4⋊C4C411yescharacteristic
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#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1272C2211yescharacteristic
47#1272C2211yescharacteristic
48#1272C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
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#398—22no—
74#398—22no—
75#398—22no—
76#398—22no—
77#398—22no—
78#391—21no—
79#391—21no—
80#391C2×C412yes—
81#391C2×C412yes—
82#391C2×C412yes—
83#391C2311yescharacteristic
84#391C2311yescharacteristic
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—22no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2312yes—
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#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#383—22no—
112#382—21no—
113#382—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
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×Q8C22⋊C411yescharacteristic
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—22no—
151C4⋊Q8—41no—
152#125—42no—
153#125—42no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—24no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2411yescharacteristic
206C4⋊C8—44no—
207C4⋊C8—42no—
208C4⋊C8—44no—
209C4⋊C8—42no—
210C4⋊C8—42no—
211C4⋊C8—42no—
212#97—44no—
213#97—44no—
214C4×C8—42no—
215C4×C8—42no—
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—22no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8#13112yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—42no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—42no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—81no—
309C42—22no—
310C42—21no—
311C42—22no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—42no—
319C42—41no—
320C42—41no—
321C42—42no—
322C42—42no—
323C42—41no—
324C42—44no—
325C42—42no—
326C42—42no—
327C42—82no—
328C42AGL(1,ℤ/8)12yes—
329C42#13712yes—
330C42#13712yes—
331C42#13712yes—
332C23C22×C22⋊C411yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—44no—
336Q8—81no—
337Q8—81no—
338C2×C4—21no—
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—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4#41312yes—
382C2×C4#41712yes—
383C2×C4#41712yes—
384C2×C4#41912yes—
385C2×C4#41912yes—
386C2×C4C2×#13111yescharacteristic
387C2×C4#56811yescharacteristic
388C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
389C2×C4C2×#13711yescharacteristic
390C2×C4C2×#13711yescharacteristic
391C2×C4C2×#13711yescharacteristic
392C8—42no—
393C8—42no—
394C8—81no—
395C8—82no—
396C8—81no—
397C8—82no—
398C22#288011yescharacteristic
399C22#288411yescharacteristic
400C22C2×#41311yescharacteristic
401C22C2×#41911yescharacteristic
402C22C2×#41711yescharacteristic
403C22#290111yescharacteristic
404C22#290111yescharacteristic
405C4—21no—
406C4—21no—
407C4—22no—
408C4—22no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—41no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—82no—
423C4—82no—
424C4#155914yes—
425C2#1742311yescharacteristic
426C2#1751411yescharacteristic
427C2#1751511yescharacteristic
428C2#1762811yescharacteristic
429C2#1763211yescharacteristic
430C2#1768711yescharacteristic
431C2C2×#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
#362852
#228322
#227362
#77132
#77092
#76632
#76602
#76532
#76302
#75962
#75712