Finite Group Databasebeta

← #101693

Subgroups — #101693

Order 512.

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

normal subgroups only

This group has 510 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#28571C211yesmaximal, characteristic
2#23120C211yesmaximal, characteristic
3#9884C211yesmaximal, characteristic, complemented
4#8577C211yesmaximal, characteristic
5#7665C211yesmaximal, characteristic, complemented
6#7029C211yesmaximal, characteristic, complemented
7#7025C211yesmaximal, characteristic, complemented
8C2×#465—21no—
9C2×#435—21no—
10#2857C2211yescharacteristic
11C4×#121C2211yescharacteristic
12#2084C411yescharacteristic, complemented
13#2071C411yescharacteristic, complemented
14#2043C411yescharacteristic, complemented
15#2013C411yescharacteristic, complemented
16#1927—21no—
17#1923—21no—
18#1917—21no—
19#1892—21no—
20#1877—21no—
21#1870—21no—
22#1869—21no—
23#1797C2211yescharacteristic
24#1753—21no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#327—21no—
29C4⋊#95C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1271—21no—
34#1271—21no—
35C22×SD16—21no—
36C22×M16—21no—
37C23×C8—21no—
38C23×C8—21no—
39C2×C4⋊Q8C2×C411yescharacteristic
40C2×C22⋊Q8—21no—
41C2×#121D411yescharacteristic
42C2×C4×D4—21no—
43C2×C4×D4D411yescharacteristic
44C22×C4⋊C4—21no—
45C22×C42C2311yescharacteristic
46#465—42no—
47#465—42no—
48#435—42no—
49#435—42no—
50C2×#106—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×#103—21no—
54C2×#103—21no—
55C2×#103—21no—
56C2×#102—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8D411yescharacteristic
62C2×C22⋊C8D411yescharacteristic
63C2×#97—21no—
64C2×C4×C8—21no—
65#391—21no—
66#390C2×C411yescharacteristic
67#389—21no—
68#389—21no—
69#389C2×C411yescharacteristic
70#388—21no—
71#386—21no—
72#384C2×C411yescharacteristic
73#380—21no—
74#380—21no—
75#380D411yescharacteristic
76C4×C4⋊C4D411yescharacteristic
77C4×C22⋊C4—21no—
78C2×#95—21no—
79C2×#95—21no—
80#340—21no—
81#340—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336C2×C411yescharacteristic
89#336C2×C411yescharacteristic
90#336D411yescharacteristic
91#336D411yescharacteristic
92#327—42no—
93C22×Q8—21no—
94C22×D4—21no—
95C22×D4C22⋊C411yescharacteristic
96C23×C4—21no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×SD16—42no—
102C2×SD16—42no—
103C2×SD16—81no—
104C2×M16—42no—
105C2×M16—42no—
106C2×M16—81no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—81no—
121C22×C8—81no—
122C4⋊Q8—81no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127#121—21no—
128#121—21no—
129#121—21no—
130#121—21no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—81no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—81no—
146C2×C4⋊C4—81no—
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—41no—
156C2×C42—21no—
157C2×C42—81no—
158C2×C42C22×C411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42#4111yescharacteristic
162#106—42no—
163C4⋊C8—42no—
164C4⋊C8—81no—
165#103—42no—
166#103—42no—
167#103—81no—
168#102—42no—
169C22⋊C8—21no—
170C22⋊C8—21no—
171C22⋊C8—21no—
172C22⋊C8—42no—
173C22⋊C8—42no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—81no—
177C22⋊C8SD1612yescomplemented
178#97—42no—
179C4×C8—81no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189C24—21no—
190C24#12111yescharacteristic
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
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—
249M16—84no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
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—41no—
268C2×C8—82no—
269C2×C8—82no—
270C2×C8—82no—
271C2×C8—82no—
272C2×C8—81no—
273C2×C8—81no—
274C2×C8—81no—
275C4⋊C4—21no—
276C4⋊C4—21no—
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—81no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42—41no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23#38611yescharacteristic
321C23#46111yescharacteristic
322C23#46311yescharacteristic
323C23#46811yescharacteristic
324C23#46911yescharacteristic
325C23#47111yescharacteristic
326C23#47311yescharacteristic
327Q8—81no—
328Q8—81no—
329Q8—81no—
330Q8—81no—
331D4—42no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—81no—
337C2×C4—21no—
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—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—41no—
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—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—161no—
385C2×C4—161no—
386C2×C4—161no—
387C2×C4—161no—
388C2×C4#41711yescharacteristic
389C2×C4#41811yescharacteristic
390C2×C4C4×SD1611yescharacteristic
391C2×C4#44011yescharacteristic
392C2×C4#44311yescharacteristic
393C2×C4#44411yescharacteristic
394C2×C4#45011yescharacteristic
395C2×C4#45211yescharacteristic
396C2×C4#46811yescharacteristic
397C2×C4#46911yescharacteristic
398C8—42no—
399C8—42no—
400C8—42no—
401C8—81no—
402C8—81no—
403C8—81no—
404C8—161no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22#169111yescharacteristic
425C22#169311yescharacteristic
426C22#169911yescharacteristic
427C22#170111yescharacteristic
428C22#187611yescharacteristic
429C22#187711yescharacteristic
430C22#188411yescharacteristic
431C22#189511yescharacteristic
432C22#190611yescharacteristic
433C22#190811yescharacteristic
434C22#191311yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4—162no—
450C4—161no—
451C4—161no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—161no—
457C2#905511yescharacteristic
458C2#921011yescharacteristic
459C2#922511yescharacteristic
460C2#926311yescharacteristic
461C2#927511yescharacteristic
462C2#938611yescharacteristic
463C2#943111yescharacteristic
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. #9884C2semidirect
5. #7665C2semidirect
6. #7029C2semidirect
7. #7025C2semidirect
12. #2084C4semidirect
13. #2071C4semidirect
14. #2043C4semidirect
15. #2013C4semidirect
177. C22⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#285712
#231202
#98842
#85772
#76652
#70292
#70252