Finite Group Databasebeta

← #101690

Subgroups — #101690

Order 512.

502 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 98 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#28574C211yesmaximal, characteristic
2#23241C211yesmaximal, characteristic, complemented
3#9680C211yesmaximal, characteristic
4#8578C211yesmaximal, characteristic, complemented
5#7664C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic, complemented
7#7025C211yesmaximal, characteristic, complemented
8C2×#465—21no—
9#2856C2211yescharacteristic
10C4×C22⋊Q8C2211yescharacteristic
11#2084C411yescharacteristic, complemented
12#2072C411yescharacteristic, complemented
13#2041C411yescharacteristic, complemented
14#2014C411yescharacteristic, complemented
15#1985—21no—
16#1926—21no—
17#1917—21no—
18#1916—21no—
19#1893—21no—
20#1880—21no—
21#1850—21no—
22#1797C2211yescharacteristic
23#1771—21no—
24#1752—21no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#328—21no—
29C2×#327—21no—
30C2×#327—21no—
31C4⋊#95C2211yescharacteristic
32#1294—21no—
33#1270—21no—
34#1270—21no—
35C22×SD16—21no—
36C23×C8—21no—
37C23×C8—21no—
38C2×#127C2×C411yescharacteristic
39C2×C22⋊Q8D411yescharacteristic
40C2×#121—21no—
41C2×#121—21no—
42C2×#121—21no—
43C2×C4×Q8D411yescharacteristic
44C22×C4⋊C4—21no—
45C22×C42C2311yescharacteristic
46#465—42no—
47#465—42no—
48C2×#106—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×#103—21no—
53C2×#102—21no—
54C2×#102—21no—
55C2×#102—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8D411yescharacteristic
58C2×C22⋊C8D411yescharacteristic
59C2×#97—21no—
60C2×#97—21no—
61C2×C4×C8—21no—
62#391C2×C411yescharacteristic
63#390—21no—
64#390—21no—
65#390—21no—
66#389C2×C411yescharacteristic
67#387—21no—
68#385—21no—
69#384C2×C411yescharacteristic
70#380D411yescharacteristic
71C4×C4⋊C4—21no—
72C4×C4⋊C4D411yescharacteristic
73C4×C22⋊C4—21no—
74C2×#95—21no—
75C2×#95—21no—
76#340—21no—
77#340—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336C2×C411yescharacteristic
87#336C2×C411yescharacteristic
88#336D411yescharacteristic
89#336D411yescharacteristic
90#328—42no—
91#327—42no—
92#327—42no—
93C22×Q8C22⋊C411yescharacteristic
94C22×D4—21no—
95C22×D4—21no—
96C22×D4—21no—
97C22×D4—41no—
98C23×C4—41no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4#4111yescharacteristic
102C2×SD16—42no—
103C2×SD16—42no—
104C2×SD16—81no—
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—81no—
119C22×C8—81no—
120#127—81no—
121C22⋊Q8—21no—
122C22⋊Q8—21no—
123C22⋊Q8—21no—
124C22⋊Q8—21no—
125#121—41no—
126#121—41no—
127#121—41no—
128#121—41no—
129#121—41no—
130#121—41no—
131#121—41no—
132#121—41no—
133#121—81no—
134#121—81no—
135C4×Q8—81no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—81no—
145C2×C4⋊C4—81no—
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—81no—
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4111yescharacteristic
161#106—42no—
162C4⋊C8—42no—
163C4⋊C8—81no—
164C4⋊C8—81no—
165#103—42no—
166#102—42no—
167#102—42no—
168#102—81no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—21no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8SD1612yescomplemented
175#97—42no—
176#97—81no—
177C4×C8—81no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—41no—
187C24—41no—
188C24#12111yescharacteristic
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4—81no—
206C2×D4—81no—
207C2×D4—81no—
208C2×D4—81no—
209C2×D4—81no—
210C2×D4—81no—
211C2×D4—81no—
212C2×D4—81no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4C2×SD1611yescharacteristic
241C22×C4#13511yescharacteristic
242C22×C4AGL(1,ℤ/8)11yescharacteristic
243C22×C4AGL(1,ℤ/8)11yescharacteristic
244C22×C4AGL(1,ℤ/8)11yescharacteristic
245C22×C4AGL(1,ℤ/8)11yescharacteristic
246C22×C4#13711yescharacteristic
247C22×C4#13711yescharacteristic
248SD16—84no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—82no—
268C2×C8—82no—
269C2×C8—82no—
270C2×C8—82no—
271C2×C8—81no—
272C2×C8—81no—
273C2×C8—81no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—82no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23#38611yescharacteristic
319C23#46011yescharacteristic
320C23#46411yescharacteristic
321C23#46811yescharacteristic
322C23#46911yescharacteristic
323C23#47111yescharacteristic
324C23#47211yescharacteristic
325Q8—42no—
326Q8—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4—161no—
332D4—161no—
333D4—161no—
334D4—161no—
335D4—161no—
336D4—161no—
337D4—161no—
338D4—161no—
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—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—161no—
383C2×C4—161no—
384C2×C4—161no—
385C2×C4—161no—
386C2×C4#41711yescharacteristic
387C2×C4#41811yescharacteristic
388C2×C4C4×SD1611yescharacteristic
389C2×C4#44011yescharacteristic
390C2×C4#44311yescharacteristic
391C2×C4#44411yescharacteristic
392C2×C4#44811yescharacteristic
393C2×C4#44911yescharacteristic
394C2×C4#46811yescharacteristic
395C2×C4#46911yescharacteristic
396C8—42no—
397C8—42no—
398C8—42no—
399C8—81no—
400C8—81no—
401C8—81no—
402C8—161no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—161no—
416C22—161no—
417C22—161no—
418C22—161no—
419C22—161no—
420C22—161no—
421C22—161no—
422C22#168611yescharacteristic
423C22#168811yescharacteristic
424C22#169811yescharacteristic
425C22#170211yescharacteristic
426C22#187611yescharacteristic
427C22#188011yescharacteristic
428C22#188411yescharacteristic
429C22#189511yescharacteristic
430C22#190611yescharacteristic
431C22#190711yescharacteristic
432C22#191311yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—41no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—162no—
449C4—161no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—321no—
455C2#905411yescharacteristic
456C2#921011yescharacteristic
457C2#923011yescharacteristic
458C2#926411yescharacteristic
459C2#927811yescharacteristic
460C2#938611yescharacteristic
461C2#943211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radical#2856C22
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
2. #23241C2semidirect
4. #8578C2semidirect
6. #7029C2semidirect
7. #7025C2semidirect
11. #2084C4semidirect
12. #2072C4semidirect
13. #2041C4semidirect
14. #2014C4semidirect
174. C22⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#285742
#232412
#96802
#85782
#76642
#70292
#70252