Finite Group Databasebeta

← #502003

Subgroups — #502003

Order 512.

580 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2217C211yesmaximal, characteristic
2C2×C4×SD32C211yesmaximal, characteristic, complemented
3#25033C211yesmaximal, characteristic, complemented
4#25032C211yesmaximal, characteristic
5#13483C211yesmaximal, characteristic, complemented
6#13482C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic
9#13237C211yesmaximal, characteristic, complemented
10#13166C211yesmaximal, characteristic, complemented
11#12936C211yesmaximal, characteristic, complemented
12#12935C211yesmaximal, characteristic, complemented
13#12907C211yesmaximal, characteristic, complemented
14#12835C211yesmaximal, characteristic, complemented
15C16×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22#2421—21no—
23#2420—21no—
24#2217—24no—
25C4×SD32—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C4×C16C2211yescharacteristic
39C2×C4×C16C2211yescharacteristic
40#2074—21no—
41#2073—21no—
42#2035—21no—
43#2034—21no—
44#1970C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1865—21no—
49#1864—21no—
50#1846C2211yescharacteristic
51#1846C2211yescharacteristic
52#1808C2211yescharacteristic
53#1808C2211yescharacteristic
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×C22⋊Q8—21no—
65C2×#121—21no—
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C2×SD32—22no—
69C2×SD32—22no—
70C2×SD32D414yes—
71C22×C16—22no—
72C22×C16—22no—
73C22×C16—22no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#467—22no—
79#467—41no—
80#466—22no—
81#466—41no—
82C4×Q16—22no—
83C4×Q16—41no—
84C4×D8—22no—
85C4×D8—41no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×#106—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#397—41no—
105#397—41no—
106#394—41no—
107#391—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#388—41no—
112#387—41no—
113#386—41no—
114#382—21no—
115#382—21no—
116#381—41no—
117#380—21no—
118#380—21no—
119C4×C22⋊C4D411yescharacteristic
120#367—22no—
121#367—22no—
122#367—22no—
123#366—41no—
124#366—41no—
125#358—22no—
126#358—22no—
127#358—22no—
128#357—22no—
129#357—22no—
130#357—22no—
131C22⋊C16—22no—
132C4×C16—22no—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336D411yescharacteristic
139#336D411yescharacteristic
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4—41no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16C2×D412yes—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8C2×D412yes—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2411yescharacteristic
167C22⋊Q8—42no—
168C22⋊Q8—81no—
169#121—42no—
170#121—81no—
171C4×Q8—42no—
172C4×Q8—81no—
173C4×D4—42no—
174C4×D4—81no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4D811yescharacteristic
193C2×C22⋊C4D811yescharacteristic
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197SD32—28no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—44no—
207C2×C16C2×D412yes—
208C2×C16#4112yes—
209C2×C16#4112yes—
210C2×C16#4112yes—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#106—81no—
224#106—81no—
225#103—42no—
226#103—42no—
227#103—42no—
228#102—42no—
229#102—42no—
230#102—42no—
231C22⋊C8—22no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95D811yescharacteristic
247#95D811yescharacteristic
248C24—41no—
249C24C2×D811yescharacteristic
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—161no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—161no—
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C22×D411yescharacteristic
287Q16—22no—
288Q16—22no—
289Q16—41no—
290Q16—41no—
291D8—22no—
292D8—22no—
293D8—41no—
294D8—41no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8#13512yes—
308C2×C8C22×D411yescharacteristic
309C2×C8C2×#4111yescharacteristic
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C82−1+411yescharacteristic
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—161no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—161no—
337C22⋊C4—161no—
338C22⋊C4—161no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—82no—
344C16—22no—
345C16—41no—
346C16—41no—
347C16#12314yes—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—42no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23#50812yes—
362C23C22×D811yescharacteristic
363Q8—42no—
364Q8—42no—
365Q8—81no—
366Q8—81no—
367D4—42no—
368D4—42no—
369D4—81no—
370D4—81no—
371D4—164no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—164no—
412C2×C4#50812yes—
413C2×C4#53211yescharacteristic
414C2×C4D4211yescharacteristic
415C2×C4#54711yescharacteristic
416C2×C4C2×#13511yescharacteristic
417C2×C4#57611yescharacteristic
418C2×C4#57811yescharacteristic
419C8—22no—
420C8—41no—
421C8—41no—
422C8C2×#12311yescharacteristic
423C8#54211yescharacteristic
424C8#54711yescharacteristic
425C8#55311yescharacteristic
426C22—22no—
427C22—22no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22#316911yescharacteristic
441C22D4×D811yescharacteristic
442C22#333411yescharacteristic
443C22C2×#50811yescharacteristic
444C22C2×#50811yescharacteristic
445C22#341611yescharacteristic
446C22#341611yescharacteristic
447C4—22no—
448C4—41no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—82no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4#242211yescharacteristic
460C4#317511yescharacteristic
461C4#328311yescharacteristic
462C4#333911yescharacteristic
463C2—42no—
464C2—162no—
465C2#2503411yescharacteristic
466C2#3373511yescharacteristic
467C2#3374311yescharacteristic
468C2#3384811yescharacteristic
469C2#3384811yescharacteristic
470C2#3390411yescharacteristic
471C2#3390711yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×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×C4×SD32C2semidirect
3. #25033C2semidirect
5. #13483C2semidirect
6. #13482C2semidirect
7. #13383C2semidirect
9. #13237C2semidirect
10. #13166C2semidirect
11. #12936C2semidirect
12. #12935C2semidirect
13. #12907C2semidirect
14. #12835C2semidirect
15. C16×C22⋊C4C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22172
C2×C4×SD322
#250332
#250322
#134832
#134822
#133832
#133802
#132372
#131662
#129362
#129352
#129072
#128352
C16×C22⋊C42