Finite Group Databasebeta

← #508359

Subgroups — #508359

Order 512.

561 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2220C211yesmaximal, characteristic
2C2×#2151C211yesmaximal, characteristic, complemented
3#33139C211yesmaximal, characteristic, complemented
4#33138C211yesmaximal, characteristic
5#13515C211yesmaximal, characteristic, complemented
6#13514C211yesmaximal, characteristic, complemented
7#13477C211yesmaximal, characteristic, complemented
8#13476C211yesmaximal, characteristic, complemented
9#13475C211yesmaximal, characteristic, complemented
10#13474C211yesmaximal, characteristic, complemented
11#13449C211yesmaximal, characteristic, complemented
12#13449C211yesmaximal, characteristic, complemented
13#13060C211yesmaximal, characteristic, complemented
14#13059C211yesmaximal, characteristic
15#12836C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2625—21no—
23#2624—21no—
24#2220—24no—
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28#2151—24no—
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1969C2211yescharacteristic
42#1969C2211yescharacteristic
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1942C2211yescharacteristic
46#1942C2211yescharacteristic
47#1936—21no—
48#1936—21no—
49#1935—21no—
50#1935—21no—
51#1895—21no—
52#1894—21no—
53#1808C2211yescharacteristic
54#1808C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C22×M16D411yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×#117D411yescharacteristic
68C2×SD32—22no—
69C2×SD32—41no—
70C2×SD32—41no—
71C2×M32—41no—
72C2×M32—41no—
73C2×M32D414yes—
74C22×C16—22no—
75C22×C16—41no—
76C22×C16—41no—
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81#467—22no—
82#467—22no—
83#466—22no—
84#466—22no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97#405—22no—
98#405—22no—
99C2×#97D411yescharacteristic
100C2×#97D411yescharacteristic
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#400—41no—
104#399—41no—
105#398—41no—
106#397—41no—
107#397—41no—
108#397—41no—
109#394—41no—
110#394—41no—
111#391—21no—
112#391—21no—
113#390—21no—
114#390—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#380—21no—
120#380—21no—
121#367—41no—
122#366—22no—
123#366—41no—
124C4⋊C16—22no—
125C4⋊C16—22no—
126C4⋊C16—22no—
127C4⋊C16—22no—
128#358—22no—
129#358—41no—
130#358—41no—
131#357—22no—
132#357—41no—
133#357—41no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4—41no—
143C23×C4C2×D411yescharacteristic
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×M16—22no—
157C2×M16—41no—
158C2×M16—41no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D412yes—
167C22×C8#4112yes—
168C22×C8C2411yescharacteristic
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171#121—42no—
172#121—42no—
173#117—22no—
174#117—22no—
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×C4⋊C4—41no—
186C2×C4⋊C4D811yescharacteristic
187C2×C4⋊C4D811yescharacteristic
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4D811yescharacteristic
197C2×C22⋊C4D811yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200SD32—44no—
201M32—44no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—44no—
213C2×C16C2×D412yes—
214C2×C16C2×D412yes—
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#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#103—42no—
228#103—42no—
229#103—42no—
230#102—42no—
231#102—42no—
232#102—42no—
233#97—22no—
234#97—22no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248#95—41no—
249C24—41no—
250C24C2×D811yescharacteristic
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—82no—
264C2×D4—161no—
265C2×D4—161no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4#12112yes—
278C22×C4C2×D811yescharacteristic
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4#13512yes—
284C22×C4C22×D411yescharacteristic
285Q16—41no—
286Q16—41no—
287Q16—41no—
288Q16—41no—
289D8—41no—
290D8—41no—
291D8—41no—
292D8—41no—
293M16—44no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8#12112yes—
307C2×C8C22×D411yescharacteristic
308C2×C8C2×#4111yescharacteristic
309C2×C8H(5,𝔽2)11yescharacteristic
310C2×C8H(5,𝔽2)11yescharacteristic
311C2×C82−1+411yescharacteristic
312C2×C82−1+411yescharacteristic
313C4⋊C4—22no—
314C4⋊C4—22no—
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—41no—
327C4⋊C4—82no—
328C4⋊C4—161no—
329C4⋊C4—161no—
330C4⋊C4—161no—
331C4⋊C4—161no—
332C22⋊C4—22no—
333C22⋊C4—22no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—161no—
337C22⋊C4—161no—
338C22⋊C4—161no—
339C22⋊C4—161no—
340C22⋊C4—161no—
341C22⋊C4—161no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C16—22no—
347C16—22no—
348C16—41no—
349C16—41no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23#46612yes—
364C23C22×D811yescharacteristic
365Q8—81no—
366Q8—81no—
367Q8—81no—
368Q8—81no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—164no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
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—81no—
412C2×C4—81no—
413C2×C4#46612yes—
414C2×C4C2×#12111yescharacteristic
415C2×C4#53711yescharacteristic
416C2×C4#53711yescharacteristic
417C2×C4C2×#13511yescharacteristic
418C2×C4#57511yescharacteristic
419C2×C4#57511yescharacteristic
420C8—22no—
421C8—41no—
422C8—41no—
423C8#53411yescharacteristic
424C8#53611yescharacteristic
425C8#53911yescharacteristic
426C8#53911yescharacteristic
427C22—22no—
428C22—22no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22C2×#46611yescharacteristic
442C22#324511yescharacteristic
443C22#324511yescharacteristic
444C22#341511yescharacteristic
445C22#341611yescharacteristic
446C22#341611yescharacteristic
447C22#341711yescharacteristic
448C4—22no—
449C4—41no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4#261811yescharacteristic
461C4#305411yescharacteristic
462C4#325011yescharacteristic
463C4#325011yescharacteristic
464C2—42no—
465C2—321no—
466C2#2977011yescharacteristic
467C2#3361211yescharacteristic
468C2#3361311yescharacteristic
469C2#3382311yescharacteristic
470C2#3382411yescharacteristic
471C2#3382611yescharacteristic
472C2#3382811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2151C2semidirect
3. #33139C2semidirect
5. #13515C2semidirect
6. #13514C2semidirect
7. #13477C2semidirect
8. #13476C2semidirect
9. #13475C2semidirect
10. #13474C2semidirect
11. #13449C2semidirect
12. #13449C2semidirect
13. #13060C2semidirect
15. #12836C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22202
C2×#21512
#331392
#331382
#135152
#135142
#134772
#134762
#134752
#134742
#134492
#130602
#130592
#128362