Finite Group Databasebeta

← #101403

Subgroups — #101403

Order 512.

511 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27678C211yesmaximal, characteristic
2#22176C211yesmaximal, characteristic
3#11357C211yesmaximal, characteristic, complemented
4#8725C211yesmaximal, characteristic, complemented
5#8571C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic, complemented
7#7020C211yesmaximal, characteristic, complemented
8C2×#435—21no—
9C4×#121C2211yescharacteristic
10C2×#384C2211yescharacteristic
11#2100C412yescomplemented
12#2094C412yescomplemented
13#1948—21no—
14#1924—21no—
15#1917—21no—
16#1874—21no—
17#1870—21no—
18#1869—21no—
19#1853—21no—
20#1851—21no—
21#1802—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25C2×#340—21no—
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#327—21no—
29C4⋊#95C2211yescharacteristic
30#1296—21no—
31#1296—21no—
32#1294—21no—
33C22×M16—21no—
34C23×C8D411yescharacteristic
35C23×C8D411yescharacteristic
36C2×#121D411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4D411yescharacteristic
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C42C2311yescharacteristic
44#435—42no—
45#435—42no—
46C2×#106—22no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D411yescharacteristic
51C2×C4⋊C8D411yescharacteristic
52C2×#102—22no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×#97—21no—
60#390C2×C411yescharacteristic
61#389C2×C411yescharacteristic
62#388—21no—
63#386—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384C2×C412yes—
68#380D411yescharacteristic
69C4×C4⋊C4D411yescharacteristic
70C4×C22⋊C4—21no—
71C2×#95—21no—
72C2×#95—21no—
73#340—22no—
74#340—22no—
75#340—42no—
76#340—42no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336C2×C412yes—
84#327—42no—
85C22×D4—21no—
86C22×D4C22⋊C411yescharacteristic
87C23×C4—21no—
88C23×C4—41no—
89C23×C4—41no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4#4111yescharacteristic
93C2×M16—42no—
94C2×M16—81no—
95C2×M16—81no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—22no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8SD1612yes—
110#121—21no—
111#121—21no—
112#121—21no—
113#121—21no—
114C4×D4—41no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—82no—
138C2×C4⋊C4—81no—
139C2×C4⋊C4—81no—
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C42—21no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155#106—82no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161C4⋊C8—41no—
162C4⋊C8—42no—
163#102—82no—
164C22⋊C8—42no—
165C22⋊C8—41no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171C22⋊C8—81no—
172C22⋊C8—81no—
173#97—42no—
174#95—21no—
175#95—21no—
176#95—21no—
177#95—41no—
178#95—41no—
179#95—81no—
180C24—21no—
181C24#12111yescharacteristic
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4#12011yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4C2×SD1611yescharacteristic
232C22×C4#13511yescharacteristic
233C22×C4AGL(1,ℤ/8)11yescharacteristic
234C22×C4AGL(1,ℤ/8)11yescharacteristic
235C22×C4#13711yescharacteristic
236C22×C4#13711yescharacteristic
237C22×C4#13711yescharacteristic
238C22×C4#13711yescharacteristic
239M16—84no—
240C2×C8—22no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—81no—
258C2×C8—81no—
259C2×C8—81no—
260C2×C8—81no—
261C2×C8—81no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—82no—
275C4⋊C4—81no—
276C4⋊C4—81no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—81no—
288C22⋊C4—81no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—41no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23#38611yescharacteristic
309C23#46111yescharacteristic
310C23#46311yescharacteristic
311C23#46811yescharacteristic
312C23#47011yescharacteristic
313C23#47111yescharacteristic
314C23#47211yescharacteristic
315D4—42no—
316D4—81no—
317D4—81no—
318D4—81no—
319D4—81no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—161no—
367C2×C4—161no—
368C2×C4—161no—
369C2×C4—161no—
370C2×C4—161no—
371C2×C4—161no—
372C2×C4—161no—
373C2×C4—161no—
374C2×C4#41711yescharacteristic
375C2×C4#41911yescharacteristic
376C2×C4C4×SD1611yescharacteristic
377C2×C4#44011yescharacteristic
378C2×C4#45011yescharacteristic
379C2×C4#45211yescharacteristic
380C2×C4#45311yescharacteristic
381C2×C4#45411yescharacteristic
382C2×C4#45811yescharacteristic
383C2×C4#45811yescharacteristic
384C2×C4#46811yescharacteristic
385C2×C4#47011yescharacteristic
386C8—42no—
387C8—82no—
388C8—81no—
389C8—81no—
390C8—81no—
391C8—161no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22#169111yescharacteristic
412C22#169511yescharacteristic
413C22#169711yescharacteristic
414C22#170311yescharacteristic
415C22#187711yescharacteristic
416C22#188311yescharacteristic
417C22#189011yescharacteristic
418C22#189411yescharacteristic
419C22#190211yescharacteristic
420C22#190711yescharacteristic
421C22#191011yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—41no—
429C4—41no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4—162no—
434C4—162no—
435C4#161311yescharacteristic
436C4#161511yescharacteristic
437C4#161611yescharacteristic
438C4#161811yescharacteristic
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—161no—
444C2#904511yescharacteristic
445C2#922311yescharacteristic
446C2#928911yescharacteristic
447C2#932111yescharacteristic
448C2#937811yescharacteristic
449C2#947311yescharacteristic
450C2#952211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11357C2semidirect
4. #8725C2semidirect
6. #7039C2semidirect
7. #7020C2semidirect
11. #2100C4semidirect
12. #2094C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#276782
#221762
#113572
#87252
#85712
#70392
#70202