Finite Group Databasebeta

← #134534

Subgroups — #134534

Order 512.

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

normal subgroups only

This group has 794 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#14176C211yesmaximal, characteristic
2C2×#1345C212yesmaximal
3#10847C211yesmaximal, characteristic, complemented
4#8560C211yesmaximal, characteristic, complemented
5#7251C212yesmaximal, complemented
6C22×C4×D4C411yescharacteristic
7C2×#423—21no—
8C2×#394C2211yescharacteristic
9C2×#388—21no—
10C2×#385—21no—
11C2×C4×C4⋊C4C2211yescharacteristic
12#1797—21no—
13C2×#343—21no—
14C2×#343C2212yes—
15C2×#324C2212yes—
16#1444C411yescharacteristic
17#1441C2211yescharacteristic
18#1438C411yescharacteristic
19#1437—21no—
20#1434C411yescharacteristic
21C22⋊#95—21no—
22#1345—24no—
23#1280—21no—
24C23×D4C2×C411yescharacteristic
25C24×C4—21no—
26C22×M16—22no—
27C2×C4×D4—41no—
28C2×C4×D4—41no—
29C2×C4×D4—44no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4C2311yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C42C2×C411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42#423—24no—
43#423—42no—
44C2×C4⋊C8—41no—
45C2×C4⋊C8—41no—
46C2×C4⋊C8—41no—
47C2×C22⋊C8—22no—
48#394—21no—
49#394—21no—
50#394D412yes—
51#394Q812yes—
52#388—44no—
53#385—42no—
54#385—42no—
55C4×C4⋊C4—41no—
56C4×C4⋊C4—41no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95C2×C411yescharacteristic
63C2×#95C2×C411yescharacteristic
64C2×#95C2×C411yescharacteristic
65C2×#95D411yescharacteristic
66#343—22no—
67#343—22no—
68#343—42no—
69#336—41no—
70#324—22no—
71#324—22no—
72C25—21no—
73C22×D4—21no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4—42no—
77C22×D4—42no—
78C22×D4C4⋊C412yes—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—81no—
88C23×C4—81no—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C2×M16—24no—
97C2×M16—42no—
98C2×M16—42no—
99C2×M16—42no—
100C2×M16—42no—
101C2×M16—42no—
102C22×C8—42no—
103C22×C8—42no—
104C4×D4—44no—
105C4×D4—44no—
106C4×D4—44no—
107C4×D4—44no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—81no—
117C2×C4⋊C4D812yes—
118C2×C4⋊C4SD1612yes—
119C2×C4⋊C4C2×D411yescharacteristic
120C2×C4⋊C4C2×Q811yescharacteristic
121C2×C4⋊C4#4111yescharacteristic
122C2×C4⋊C4#4111yescharacteristic
123C2×C22⋊C4—24no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—81no—
142C2×C22⋊C4—81no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—42no—
153C2×C42—42no—
154C4⋊C8—44no—
155C4⋊C8—44no—
156C4⋊C8—82no—
157C22⋊C8—42no—
158C22⋊C8—42no—
159#95—21no—
160#95—41no—
161#95—41no—
162#95—41no—
163#95—42no—
164#95—42no—
165#95—42no—
166#95—41no—
167#95—41no—
168#95—81no—
169#95C4⋊C412yes—
170C24—22no—
171C24—22no—
172C24—22no—
173C24—41no—
174C24—41no—
175C24—41no—
176C24—41no—
177C24—41no—
178C24—41no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24—41no—
184C24—41no—
185C24C2×C22⋊C411yescharacteristic
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—48no—
193C2×D4—42no—
194C2×D4—44no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
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—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—48no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4—84no—
249C22×C4—84no—
250C22×C4—84no—
251C22×C4#9911yescharacteristic
252C22×C4#9911yescharacteristic
253C22×C4#10212yes—
254C22×C4#10212yes—
255C22×C4#10412yes—
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4#11711yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C2×D811yescharacteristic
263C22×C4C2×SD1611yescharacteristic
264C22×C4AGL(1,ℤ/8)11yescharacteristic
265C22×C4#13711yescharacteristic
266M16—44no—
267M16—44no—
268M16—82no—
269M16—82no—
270C2×C8—44no—
271C2×C8—82no—
272C2×C8—82no—
273C2×C8—82no—
274C2×C8—82no—
275C2×C8—82no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—84no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—84no—
294C22⋊C4—84no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—81no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—24no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23#38011yescharacteristic
343C23C2×#9911yescharacteristic
344C23#41011yescharacteristic
345C23C2×#10211yescharacteristic
346C23#41711yescharacteristic
347C23C2×#10411yescharacteristic
348C23#42111yescharacteristic
349D4—44no—
350D4—44no—
351D4—44no—
352D4—44no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—816no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4AΣL(1,𝔽16)12yes—
424C2×C4#45911yescharacteristic
425C2×C4#46111yescharacteristic
426C2×C4#47411yescharacteristic
427C2×C4#47611yescharacteristic
428C8—84no—
429C8—162no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22#150211yescharacteristic
458C22#150411yescharacteristic
459C22#151211yescharacteristic
460C22#151911yescharacteristic
461C22#179911yescharacteristic
462C22#180211yescharacteristic
463C22#180611yescharacteristic
464C22C2×AΣL(1,𝔽16)11yescharacteristic
465C22#212011yescharacteristic
466C22#213311yescharacteristic
467C22#213411yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—22no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—82no—
482C4—82no—
483C4—161no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—84no—
489C2#857711yescharacteristic
490C2#1176111yescharacteristic
491C2#1176211yescharacteristic
492C2#1187911yescharacteristic
493C2#1188011yescharacteristic
494C2#1279311yescharacteristic
495C2#1280311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10847C2semidirect
4. #8560C2semidirect
5. #7251C2semidirect

Maximal subgroups

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

subgroupindex
#141762
C2×#13452
#108472
#85602
#72512