Finite Group Databasebeta

← #101694

Subgroups — #101694

Order 512.

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

normal subgroups only

This group has 459 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#28575C211yesmaximal, characteristic
2#23241C211yesmaximal, characteristic
3#9884C211yesmaximal, characteristic
4#8578C211yesmaximal, characteristic
5#7665C211yesmaximal, characteristic
6#7029C211yesmaximal, characteristic
7#7025C211yesmaximal, characteristic
8C2×#467—21no—
9#2857C2211yescharacteristic
10C4×C22⋊Q8C2211yescharacteristic
11#2083C411yescharacteristic, complemented
12#2072C411yescharacteristic, complemented
13#2043C411yescharacteristic, complemented
14#2016C411yescharacteristic, complemented
15#1985—21no—
16#1928—21no—
17#1917—21no—
18#1916—21no—
19#1892—21no—
20#1878—21no—
21#1850—21no—
22#1797C2211yescharacteristic
23#1771—21no—
24#1752—21no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#328—21no—
29C4⋊#95C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1271—21no—
34#1271—21no—
35C22×Q16—21no—
36C23×C8—21no—
37C23×C8—21no—
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8D411yescharacteristic
41C2×C4×Q8D411yescharacteristic
42C22×C4⋊C4—21no—
43C22×C42C2311yescharacteristic
44#467—42no—
45#467—42no—
46C2×#107—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×#103—21no—
51C2×#103—21no—
52C2×#103—21no—
53C2×#103—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8D411yescharacteristic
56C2×C22⋊C8D411yescharacteristic
57C2×#97—21no—
58C2×#97—21no—
59C2×C4×C8—21no—
60#391—21no—
61#391C2×C411yescharacteristic
62#389—21no—
63#389—21no—
64#389C2×C411yescharacteristic
65#387—21no—
66#385—21no—
67#384C2×C411yescharacteristic
68#380—21no—
69#380—21no—
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—
91C22×Q8—21no—
92C22×Q8C22⋊C411yescharacteristic
93C23×C4—41no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4#4111yescharacteristic
97C2×Q16—42no—
98C2×Q16—42no—
99C2×Q16—81no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—42no—
107C22×C8—42no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—81no—
114C22×C8—81no—
115C4⋊Q8—81no—
116C22⋊Q8—21no—
117C22⋊Q8—21no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8—41no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C4×Q8—81no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—81no—
137C2×C4⋊C4—81no—
138C2×C4⋊C4C22⋊C411yescharacteristic
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—41no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—81no—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42#4111yescharacteristic
151#107—42no—
152C4⋊C8—42no—
153C4⋊C8—81no—
154C4⋊C8—81no—
155#103—42no—
156#103—42no—
157#103—42no—
158#103—81no—
159C22⋊C8—21no—
160C22⋊C8—21no—
161C22⋊C8—21no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8Q1612yescomplemented
165#97—42no—
166#97—81no—
167C4×C8—81no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—41no—
174#95—41no—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179C24#12111yescharacteristic
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4C4×D411yescharacteristic
215C22×C4#12011yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12311yescharacteristic
218C22×C4C2×Q1611yescharacteristic
219C22×C4#13511yescharacteristic
220C22×C4AGL(1,ℤ/8)11yescharacteristic
221C22×C4AGL(1,ℤ/8)11yescharacteristic
222C22×C4AGL(1,ℤ/8)11yescharacteristic
223C22×C4#13711yescharacteristic
224C22×C4#13711yescharacteristic
225C22×C4#13711yescharacteristic
226Q16—84no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C2×C8—81no—
250C2×C8—81no—
251C2×C8—81no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42—41no—
284C42—41no—
285C42—41no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23#38611yescharacteristic
291C23#46211yescharacteristic
292C23#46411yescharacteristic
293C23#46811yescharacteristic
294C23#46911yescharacteristic
295C23#47111yescharacteristic
296C23#47311yescharacteristic
297Q8—42no—
298Q8—81no—
299Q8—81no—
300Q8—81no—
301Q8—81no—
302Q8—81no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—161no—
350C2×C4—161no—
351C2×C4—161no—
352C2×C4—161no—
353C2×C4#41711yescharacteristic
354C2×C4#41811yescharacteristic
355C2×C4C4×Q1611yescharacteristic
356C2×C4#44111yescharacteristic
357C2×C4#44311yescharacteristic
358C2×C4#44411yescharacteristic
359C2×C4#45111yescharacteristic
360C2×C4#45211yescharacteristic
361C2×C4#46811yescharacteristic
362C2×C4#46911yescharacteristic
363C8—42no—
364C8—42no—
365C8—42no—
366C8—81no—
367C8—81no—
368C8—81no—
369C8—161no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22#169211yescharacteristic
383C22#169411yescharacteristic
384C22#170011yescharacteristic
385C22#170211yescharacteristic
386C22#187611yescharacteristic
387C22#187811yescharacteristic
388C22#188411yescharacteristic
389C22#189511yescharacteristic
390C22#190611yescharacteristic
391C22#190811yescharacteristic
392C22#191311yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—162no—
409C4—161no—
410C4—161no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2#905611yescharacteristic
416C2#921011yescharacteristic
417C2#922611yescharacteristic
418C2#926311yescharacteristic
419C2#927611yescharacteristic
420C2#938611yescharacteristic
421C2#943111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
11. #2083C4semidirect
12. #2072C4semidirect
13. #2043C4semidirect
14. #2016C4semidirect
164. C22⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#285752
#232412
#98842
#85782
#76652
#70292
#70252