Finite Group Databasebeta

← #100411

Subgroups — #100411

Order 512.

842 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 116 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#32953C211yesmaximal, characteristic
2#13920C211yesmaximal, characteristic
3#9884C211yesmaximal, characteristic, complemented
4#9711C211yesmaximal, characteristic, complemented
5#8310C211yesmaximal, characteristic
6#7039C211yesmaximal, characteristic, complemented
7#7028C211yesmaximal, characteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C2×#474C412yescomplemented
10#2857C2211yescharacteristic
11C2×#390—21no—
12C2×#389—21no—
13C2×#380—21no—
14#1862—22no—
15#1852—21no—
16#1802C412yescomplemented
17#1797C2211yescharacteristic
18#1797C2211yescharacteristic
19#1786—21no—
20C4×C22⋊C8C2211yescharacteristic
21C2×#328—21no—
22C2×#328—21no—
23C2×D4⋊C8—22no—
24#1439—22no—
25C4⋊#95C2211yescharacteristic
26C22⋊#95—21no—
27C4×#95C2211yescharacteristic
28#1294—22no—
29C23×D4D411yescharacteristic
30C24×C4—21no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×C22⋊Q8—21no—
33C2×C4×D4—21no—
34C2×C4×D4—21no—
35C2×C4×D4—44no—
36C2×C4×D4C2×C412yes—
37C2×C4×D4D412yes—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4D411yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—22no—
43C22×C22⋊C4—22no—
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#474—44no—
47C2×#107—22no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8C2×C412yes—
50C2×#102—22no—
51C2×#102—22no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C4×C8—21no—
56#391—21no—
57#390—42no—
58#390—42no—
59#389—21no—
60#389—21no—
61#389—44no—
62#384C2×C411yescharacteristic
63#380—21no—
64#380—21no—
65#380—42no—
66#380—42no—
67C2×#95—22no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95D411yescharacteristic
71C2×#95D411yescharacteristic
72C2×#95D412yes—
73#340—22no—
74#336—21no—
75#336—21no—
76#336—21no—
77#328—42no—
78#328—42no—
79D4⋊C8—44no—
80C25—21no—
81C22×Q8—21no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4—42no—
86C22×D4—42no—
87C22×D4C22⋊C412yes—
88C22×D4SD1612yes—
89C22×D4Q1612yes—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—44no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—82no—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C22×C8—22no—
106C22×C8—41no—
107C22×C8—41no—
108C4⋊Q8—42no—
109C22⋊Q8—41no—
110C22⋊Q8—41no—
111C22⋊Q8—41no—
112C22⋊Q8—41no—
113C4×D4—24no—
114C4×D4—24no—
115C4×D4—42no—
116C4×D4—42no—
117C4×D4—44no—
118C4×D4—44no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4C22⋊C412yes—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4—44no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—82no—
144C2×C22⋊C4—82no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—42no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42C22×C411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154#107—44no—
155C4⋊C8—42no—
156C4⋊C8—42no—
157#102—44no—
158#102—44no—
159C22⋊C8—22no—
160C22⋊C8—41no—
161C22⋊C8—41no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—41no—
165C4×C8—42no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—42no—
170#95—42no—
171#95—44no—
172#95—41no—
173#95—41no—
174#95—41no—
175#95—41no—
176#95D812yes—
177C24—22no—
178C24—22no—
179C24—22no—
180C24—42no—
181C24—42no—
182C24—42no—
183C24—42no—
184C24—41no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24#12011yescharacteristic
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4#10314yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C4×D411yescharacteristic
257C22×C4C4×D411yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4C2×D811yescharacteristic
262C22×C4C2×SD1611yescharacteristic
263C22×C4C2×Q1611yescharacteristic
264C22×C4#13511yescharacteristic
265C22×C4AGL(1,ℤ/8)11yescharacteristic
266C22×C4AGL(1,ℤ/8)11yescharacteristic
267C22×C4AGL(1,ℤ/8)11yescharacteristic
268C22×C4#13711yescharacteristic
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—81no—
274C2×C8—81no—
275C2×C8—81no—
276C2×C8—81no—
277C2×C8—81no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—82no—
287C4⋊C4—82no—
288C4⋊C4—88no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4#9914yes—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—88no—
295C22⋊C4—88no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C42—22no—
299C42—22no—
300C42—41no—
301C42—41no—
302C42—82no—
303C42—82no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—24no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—82no—
324C23—82no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—82no—
331C23#38611yescharacteristic
332C23#44711yescharacteristic
333C23#44911yescharacteristic
334C23#45011yescharacteristic
335C23#45111yescharacteristic
336C23#45511yescharacteristic
337C23#45611yescharacteristic
338Q8—161no—
339Q8—161no—
340D4—24no—
341D4—24no—
342D4—44no—
343D4—44no—
344D4—42no—
345D4—42no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—24no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4C2×#9912yes—
405C2×C4C2×#10311yescharacteristic
406C2×C4#41711yescharacteristic
407C2×C4#42112yes—
408C2×C4C4×D811yescharacteristic
409C2×C4#44211yescharacteristic
410C2×C4#45911yescharacteristic
411C2×C4#46311yescharacteristic
412C2×C4#48311yescharacteristic
413C2×C4#48411yescharacteristic
414C8—82no—
415C8—82no—
416C8—161no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—44no—
430C22—44no—
431C22—44no—
432C22—84no—
433C22—84no—
434C22—84no—
435C22—84no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22#163911yescharacteristic
441C22#164411yescharacteristic
442C22#164911yescharacteristic
443C22#165411yescharacteristic
444C22#187211yescharacteristic
445C22#187911yescharacteristic
446C22#188211yescharacteristic
447C22#189711yescharacteristic
448C22#189811yescharacteristic
449C22#190411yescharacteristic
450C22#190611yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—44no—
458C4—41no—
459C4—41no—
460C4—82no—
461C4—82no—
462C4—162no—
463C4—161no—
464C4—161no—
465C4#151312yes—
466C4#152012yes—
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2—84no—
472C2#898111yescharacteristic
473C2#915111yescharacteristic
474C2#917511yescharacteristic
475C2#923711yescharacteristic
476C2#925911yescharacteristic
477C2#932711yescharacteristic
478C2#936411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4D4
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
4. #9711C2semidirect
6. #7039C2semidirect
7. #7028C2semidirect
9. C2×#474C4semidirect
16. #1802C4semidirect

Maximal subgroups

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

subgroupindex
#329532
#139202
#98842
#97112
#83102
#70392
#70282