Finite Group Databasebeta

← #498354

Subgroups — #498354

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2217C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic, complemented
3C2×#2158C211yesmaximal, characteristic, complemented
4#23027C211yesmaximal, characteristic
5#23026C211yesmaximal, characteristic, complemented
6#13238C211yesmaximal, characteristic, complemented
7#13167C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#12994C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic
12#12938C211yesmaximal, characteristic, complemented
13#12937C211yesmaximal, characteristic, complemented
14#12674C211yesmaximal, characteristic, complemented
15#12611C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#427C2211yescharacteristic
23C2×#427C2211yescharacteristic
24#2363—21no—
25#2362—21no—
26#2217C418yescomplemented
27#2175—24no—
28#2158—24no—
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#367C2211yescharacteristic
32C2×#366C2211yescharacteristic
33C2×#366C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42C2×#346C2211yescharacteristic
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1927—21no—
48#1926—21no—
49#1875—21no—
50#1873—21no—
51#1865—21no—
52#1864—21no—
53C4×#107C2211yescharacteristic
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×#117—21no—
70C2×#117—21no—
71C2×SD32—22no—
72C2×SD32—22no—
73C2×SD32C2×C414yes—
74C22×C16—22no—
75C22×C16—22no—
76C22×C16C2×C414yescomplemented
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81#467—22no—
82#467C2×C414yescomplemented
83#466—22no—
84#466C2×C414yes—
85C4×Q16—22no—
86C4×Q16—41no—
87C4×D8—22no—
88C4×D8—41no—
89#427—22no—
90#427—22no—
91#427—41no—
92#427—41no—
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#106—21no—
100C2×#106—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×C22⋊C8D411yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#390—21no—
112#390—41no—
113#389—21no—
114#389—21no—
115#388—41no—
116#388—41no—
117#388—41no—
118#387—41no—
119#386—41no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—41no—
123C4×C22⋊C4—41no—
124C4×C22⋊C4D411yescharacteristic
125#367—22no—
126#367—22no—
127#367C2×C414yescomplemented
128#366—22no—
129#366D414yescomplemented
130#358—22no—
131#358—41no—
132#358C2×C414yescomplemented
133#357—22no—
134#357—41no—
135#357C2×C414yes—
136C22⋊C16—22no—
137#346—22no—
138#346—22no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336D411yescharacteristic
144#336D411yescharacteristic
145C22×Q8—21no—
146C22×D4—21no—
147C22×D4—41no—
148C23×C4—41no—
149C23×C4C2×D411yescharacteristic
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16C22×C412yes—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8C22×C412yes—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C22×C412yes—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8#4112yes—
171C22×C8C2411yescharacteristic
172C22⋊Q8—24no—
173C22⋊Q8—42no—
174#121—24no—
175#121—42no—
176C4×Q8—42no—
177C4×Q8—81no—
178C4×D4—42no—
179C4×D4—81no—
180#117—42no—
181#117—42no—
182#117—81no—
183#117—81no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4D811yescharacteristic
202C2×C22⋊C4D811yescharacteristic
203C2×C42—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210SD32—28no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—28no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16C22×C412yes—
221C2×C16C22×C412yes—
222C2×C16C2×D412yes—
223C2×C16#4112yes—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107C22×C412yes—
233#107C22×C412yes—
234#107C2×D412yes—
235#107C2×D412yes—
236#106—42no—
237#106—42no—
238#103—24no—
239#103—42no—
240#103—81no—
241#102—24no—
242#102—42no—
243#102—81no—
244C22⋊C8—22no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95D811yescharacteristic
256#95D811yescharacteristic
257C24—41no—
258C24C2×D811yescharacteristic
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×D4—22no—
266C2×D4—44no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—161no—
273C2×D4—161no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—161no—
288C22×C4C4×D412yes—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4#13512yes—
295C22×C4C22×D411yescharacteristic
296Q16—22no—
297Q16—22no—
298Q16—41no—
299Q16—41no—
300D8—22no—
301D8—22no—
302D8—41no—
303D8—41no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8C4×D412yes—
317C2×C8C23×C411yescharacteristic
318C2×C8C22×D411yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8H(5,𝔽2)11yescharacteristic
322C2×C82−1+411yescharacteristic
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—44no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—161no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—161no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—82no—
353C42—82no—
354C42—82no—
355C42—82no—
356C42—161no—
357C16—22no—
358C16—41no—
359C16—41no—
360C16C4×D414yes—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—42no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23C4×D812yes—
375C23C22×D811yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—81no—
379Q8—81no—
380D4—42no—
381D4—42no—
382D4—81no—
383D4—81no—
384D4—164no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
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×C4—81no—
424C2×C4—164no—
425C2×C4C4×D812yes—
426C2×C4C2×C4×D411yescharacteristic
427C2×C4D4211yescharacteristic
428C2×C4#54711yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4#57611yescharacteristic
431C2×C4#57811yescharacteristic
432C8—22no—
433C8—41no—
434C8—41no—
435C8C2×C4×D411yescharacteristic
436C8#52011yescharacteristic
437C8#54811yescharacteristic
438C8#55311yescharacteristic
439C22—22no—
440C22—22no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22C2×C4×D811yescharacteristic
454C22D4×D811yescharacteristic
455C22#333411yescharacteristic
456C22#341411yescharacteristic
457C22#341411yescharacteristic
458C22#341611yescharacteristic
459C22#341611yescharacteristic
460C4—22no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4—82no—
470C4—161no—
471C4—161no—
472C4#236111yescharacteristic
473C4#295111yescharacteristic
474C4#328511yescharacteristic
475C4#333911yescharacteristic
476C2—42no—
477C2—162no—
478C2#2302511yescharacteristic
479C2#3345111yescharacteristic
480C2#3345711yescharacteristic
481C2#3383311yescharacteristic
482C2#3383711yescharacteristic
483C2#3391511yescharacteristic
484C2#3391711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2175C2semidirect
3. C2×#2158C2semidirect
5. #23026C2semidirect
6. #13238C2semidirect
7. #13167C2semidirect
8. #13132C2semidirect
9. #13131C2semidirect
10. #12994C2semidirect
12. #12938C2semidirect
13. #12937C2semidirect
14. #12674C2semidirect
15. #12611C2semidirect
26. #2217C4semidirect
76. C22×C16C2×C4semidirect
82. #467C2×C4semidirect
127. #367C2×C4semidirect
129. #366D4semidirect
132. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22172
C2×#21752
C2×#21582
#230272
#230262
#132382
#131672
#131322
#131312
#129942
#129922
#129382
#129372
#126742
#126112