Finite Group Databasebeta

← #495973

Subgroups — #495973

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33475C211yesmaximal, characteristic, complemented
2#33326C211yesmaximal, characteristic
3#21627C211yesmaximal, characteristic
4#13265C211yesmaximal, characteristic, complemented
5#13163C211yesmaximal, characteristic, complemented
6#13106C211yesmaximal, characteristic, complemented
7#13106C211yesmaximal, characteristic, complemented
8#12917C211yesmaximal, characteristic
9#12917C211yesmaximal, characteristic
10#12787C211yesmaximal, characteristic
11#12665C211yesmaximal, characteristic, complemented
12#12664C211yesmaximal, characteristic, complemented
13#12663C211yesmaximal, characteristic, complemented
14#12615C211yesmaximal, characteristic
15#12614C211yesmaximal, characteristic
16C2×#504C2211yescharacteristic
17C2×#504C2211yescharacteristic
18#3076C411yescharacteristic
19#2960C2211yescharacteristic
20C4×#131C2211yescharacteristic
21#2183—21no—
22#2183—21no—
23#2183—21no—
24#2183—21no—
25C2×#364C2211yescharacteristic
26C2×#364C2211yescharacteristic
27#2151C2211yescharacteristic
28#2151C2211yescharacteristic
29#2150C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×#350C2211yescharacteristic
32C2×#350C2211yescharacteristic
33#2108—21no—
34#2108—21no—
35#2108C2211yescharacteristic
36#2108C2211yescharacteristic
37#2108C2211yescharacteristic
38#2108C2211yescharacteristic
39C4×M32—21no—
40C4×M32C2211yescharacteristic
41C2×#346—21no—
42C2×#346C2211yescharacteristic
43C2×#346C2211yescharacteristic
44C2×C4×C16C2211yescharacteristic
45#1986C411yescharacteristic
46#1950C411yescharacteristic
47#1922C411yescharacteristic
48#1922C411yescharacteristic
49#1866C411yescharacteristic
50#1866C411yescharacteristic
51#1806C411yescharacteristic
52#1772C2211yescharacteristic
53#1772C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1764C2211yescharacteristic
56#1764C2211yescharacteristic
57#1379C2211yescharacteristic
58#1379C2211yescharacteristic
59#1379C2211yescharacteristic
60#1379C2211yescharacteristic
61#1379C2211yescharacteristic
62#1379C2211yescharacteristic
63#1376C2211yescharacteristic
64#1376C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#131C2×C411yescharacteristic
68C2×#131C2311yescharacteristic
69#533C2×C411yescharacteristic
70C4×#41C2×C411yescharacteristic
71#504—22no—
72#504—22no—
73#504—22no—
74#504Q814yescomplemented
75C2×M32—21no—
76C2×M32—21no—
77C2×M32C2311yescharacteristic
78C2×M32C2311yescharacteristic
79C2×M32C2311yescharacteristic
80C2×M32C2311yescharacteristic
81C22×C16—21no—
82C22×C16—21no—
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#479C811yescharacteristic
88#479C811yescharacteristic
89#478C811yescharacteristic
90#478C811yescharacteristic
91#477C811yescharacteristic
92#476C811yescharacteristic
93#475C811yescharacteristic
94#474C811yescharacteristic
95#446—21no—
96#446—21no—
97C8×Q8—21no—
98C8×Q8—21no—
99C2×#108C2×C411yescharacteristic
100C2×#108C2×C411yescharacteristic
101#427C2×C411yescharacteristic
102#427C2×C411yescharacteristic
103#424C2×C411yescharacteristic
104#424C2×C411yescharacteristic
105#423C2×C411yescharacteristic
106C2×C4⋊C8C2×C411yescharacteristic
107C2×#104C2×C411yescharacteristic
108C2×#104C2×C411yescharacteristic
109#416C2×C411yescharacteristic
110#416C2×C411yescharacteristic
111#405—21no—
112#405—21no—
113#405C2×C411yescharacteristic
114#405C2311yescharacteristic
115#405C2311yescharacteristic
116#405C2311yescharacteristic
117C4×M16—21no—
118C4×M16C2×C411yescharacteristic
119C2×C4×C8—21no—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124#364—22no—
125#364—22no—
126C4⋊C16—21no—
127C4⋊C16—21no—
128C4⋊C16—21no—
129C4⋊C16—21no—
130C4⋊C16—21no—
131C4⋊C16—21no—
132C4⋊C16—21no—
133C4⋊C16—21no—
134#350—22no—
135#350—22no—
136#346—21no—
137#346—21no—
138#346—21no—
139#346—21no—
140#346—22no—
141#346—21no—
142#346D411yescharacteristic
143#346D411yescharacteristic
144#346D411yescharacteristic
145#346D411yescharacteristic
146C4×C16—21no—
147C4×C16—21no—
148C4×C16—21no—
149C4×C16—21no—
150C4×C16—22no—
151C4×C16—21no—
152#341C2×C411yescharacteristic
153#341C2×C411yescharacteristic
154#339C2×C411yescharacteristic
155#339C2×C411yescharacteristic
156#339C2×C411yescharacteristic
157#339C2×C411yescharacteristic
158#339C2×C411yescharacteristic
159#339C2×C411yescharacteristic
160#336C2×C411yescharacteristic
161#336C2×C411yescharacteristic
162C2×#41C22×C411yescharacteristic
163#131—22no—
164#131—22no—
165#131—21no—
166#131—21no—
167#131C2×Q812yes—
168#131#4112yes—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16C22×C411yescharacteristic
172C2×M16C22×C411yescharacteristic
173C2×M16C22×C411yescharacteristic
174C2×M16C22×C411yescharacteristic
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C2411yescharacteristic
181C4⋊Q8C2×C811yescharacteristic
182C4⋊Q8C2×C811yescharacteristic
183#125C2×C811yescharacteristic
184#125C2×C811yescharacteristic
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8C2×C811yescharacteristic
190C4×Q8C2×C811yescharacteristic
191C4×D4—21no—
192C4×D4—21no—
193C4×D4C2×C811yescharacteristic
194C4×D4C2×C811yescharacteristic
195#117—21no—
196#117C22×C411yescharacteristic
197#117C22×C411yescharacteristic
198#117C22×C411yescharacteristic
199C2×C4⋊C4—21no—
200C2×C42—21no—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204M32—24no—
205M32—21no—
206M32—21no—
207M32—24no—
208M32—21no—
209M32—21no—
210M32—21no—
211M32—21no—
212M32C2×Q812yes—
213M32#4112yes—
214C2×C16—24no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—24no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16C2×D411yescharacteristic
229C2×C16C2×D411yescharacteristic
230C2×C16C2×D411yescharacteristic
231C2×C16C2×D411yescharacteristic
232C2×C16C2×Q812yes—
233C2×C16#4112yes—
234#108—22no—
235#108—22no—
236#107C2×C811yescharacteristic
237#107C2×C811yescharacteristic
238#107C2×C811yescharacteristic
239#107C2×C811yescharacteristic
240#106C2×C811yescharacteristic
241#106C2×C811yescharacteristic
242#106C2×C811yescharacteristic
243#106C2×C811yescharacteristic
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C2×C811yescharacteristic
251C4⋊C8C2×C811yescharacteristic
252C4⋊C8C2×C811yescharacteristic
253C4⋊C8C2×C811yescharacteristic
254#104—22no—
255#104—22no—
256#103C2×C811yescharacteristic
257#103C2×C811yescharacteristic
258#103C2×C811yescharacteristic
259#103C2×C811yescharacteristic
260#102C2×C811yescharacteristic
261#102C2×C811yescharacteristic
262#102C2×C811yescharacteristic
263#102C2×C811yescharacteristic
264#97—21no—
265#97—21no—
266#97—21no—
267#97—21no—
268#97—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8C2×D411yescharacteristic
279C4×C8C2×D411yescharacteristic
280C4×C8#4111yescharacteristic
281C4×C8#4111yescharacteristic
282#95—21no—
283#41—21no—
284#41—21no—
285#41C4×Q812yes—
286#41#13112yes—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8C22×C811yescharacteristic
290C2×D4—21no—
291C2×D4C22×C811yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4#13111yescharacteristic
298C22×C4#13111yescharacteristic
299C22×C4C23×C411yescharacteristic
300M16—24no—
301M16—21no—
302M16—21no—
303M16—22no—
304M16—22no—
305M16—21no—
306M16—21no—
307M16—21no—
308M16—21no—
309M16C4×Q812yes—
310M16#13112yes—
311C2×C8—24no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8C4×D411yescharacteristic
325C2×C8C4×D411yescharacteristic
326C2×C8C22×C811yescharacteristic
327C2×C8C22×C811yescharacteristic
328C2×C8C22×C811yescharacteristic
329C2×C8C22×C811yescharacteristic
330C2×C8C22×D411yescharacteristic
331C2×C8C22×Q811yescharacteristic
332C2×C8C2×#4111yescharacteristic
333C2×C8C2×#4111yescharacteristic
334C2×C8C2×#4111yescharacteristic
335C2×C8C2×#4111yescharacteristic
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—42no—
345C4⋊C4C22×C811yescharacteristic
346C4⋊C4C22×C811yescharacteristic
347C4⋊C4C22×C811yescharacteristic
348C4⋊C4C22×C811yescharacteristic
349C4⋊C4C22×C811yescharacteristic
350C4⋊C4C22×C811yescharacteristic
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42C4×D411yescharacteristic
366C42C4×D411yescharacteristic
367C42C22×C811yescharacteristic
368C42C22×C811yescharacteristic
369C16—22no—
370C16—21no—
371C16—22no—
372C16—21no—
373C16—41no—
374C16—41no—
375C16—41no—
376C16—41no—
377C16C22⋊Q812yes—
378C16C22⋊Q812yes—
379C23—21no—
380C23C2×#13111yescharacteristic
381Q8—21no—
382Q8—81no—
383Q8—81no—
384Q8C8×Q812yes—
385D4—22no—
386D4—22no—
387D4—21no—
388D4C8×Q812yes—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4C8×D411yescharacteristic
409C2×C4C8×D411yescharacteristic
410C2×C4C8×D411yescharacteristic
411C2×C4C8×D411yescharacteristic
412C2×C4C2×C4×D411yescharacteristic
413C2×C4C2×C4×Q811yescharacteristic
414C2×C4C4×#4111yescharacteristic
415C2×C4C23×C811yescharacteristic
416C2×C4C2×#13111yescharacteristic
417C2×C4C2×#13111yescharacteristic
418C8—21no—
419C8—22no—
420C8—21no—
421C8—21no—
422C8—21no—
423C8—21no—
424C8—21no—
425C8—22no—
426C8—41no—
427C8—41no—
428C8C2×C22⋊Q811yescharacteristic
429C8#52511yescharacteristic
430C8#52911yescharacteristic
431C8#53311yescharacteristic
432C22—21no—
433C22—21no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22C2×C8×D411yescharacteristic
438C22#296011yescharacteristic
439C22#296611yescharacteristic
440C4—22no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—42no—
446C4—41no—
447C4—41no—
448C4—41no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—81no—
453C4C4×C22⋊Q811yescharacteristic
454C4#292911yescharacteristic
455C4C2×C8×Q811yescharacteristic
456C4C8×#4111yescharacteristic
457C2—21no—
458C2—21no—
459C2—42no—
460C2C8×C22⋊Q811yescharacteristic
461C2#3344211yescharacteristic
462C2#3344311yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radicalC2×#41C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×#346 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33475C2semidirect
4. #13265C2semidirect
5. #13163C2semidirect
6. #13106C2semidirect
7. #13106C2semidirect
11. #12665C2semidirect
12. #12664C2semidirect
13. #12663C2semidirect
74. #504Q8semidirect

Maximal subgroups

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

subgroupindex
#334752
#333262
#216272
#132652
#131632
#131062
#129172
#127872
#126652
#126642
#126632
#126152
#126142