Finite Group Databasebeta

← #508441

Subgroups — #508441

Order 512.

580 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×#2243C211yesmaximal, characteristic, complemented
2C2×#2164C211yesmaximal, characteristic, complemented
3#33215C211yesmaximal, characteristic, complemented
4#33182C211yesmaximal, characteristic
5#13477C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13475C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13383C211yesmaximal, characteristic, complemented
10#13382C211yesmaximal, characteristic, complemented
11#13381C211yesmaximal, characteristic
12#13380C211yesmaximal, characteristic
13#13282C211yesmaximal, characteristic, complemented
14#13204C211yesmaximal, characteristic, complemented
15#12908C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#468—21no—
19C2×#467C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2628—21no—
23#2620—21no—
24#2243—24no—
25#2164—24no—
26C2×#367C2211yescharacteristic
27C2×#367C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42C2×#346C2211yescharacteristic
43C2×C4×C16C2211yescharacteristic
44#1993—21no—
45#1973—21no—
46#1970C2211yescharacteristic
47#1969C2211yescharacteristic
48#1959—21no—
49#1944—21no—
50#1942C2211yescharacteristic
51#1942C2211yescharacteristic
52#1846C2211yescharacteristic
53#1845C2211yescharacteristic
54#1809—21no—
55#1808C2211yescharacteristic
56#1808C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C22×M16D411yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C4⋊Q8—21no—
67C2×#124—21no—
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×#117D411yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#500—22no—
76#500—22no—
77#495—22no—
78#495—22no—
79#468—44no—
80#467—41no—
81#467—41no—
82#466—41no—
83#466—41no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×#103—21no—
94C2×#103—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#396—41no—
106#396—41no—
107#395—41no—
108#394—41no—
109#394—41no—
110#393—41no—
111#391—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#384—21no—
116#382—21no—
117#380—21no—
118#380—21no—
119#367—41no—
120#367—41no—
121#366—41no—
122#366—41no—
123C4⋊C16—22no—
124C4⋊C16—22no—
125#358—22no—
126#358—22no—
127#358—41no—
128#358—41no—
129#357—22no—
130#357—22no—
131#357—41no—
132#357—41no—
133C22⋊C16—22no—
134C22⋊C16—22no—
135#346D414yescomplemented
136C4×C16—22no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—21no—
142C22×Q8—41no—
143C22×D4—21no—
144C22×D4—41no—
145C23×C4C2×D411yescharacteristic
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×SD16—42no—
153C2×SD16—42no—
154C2×SD16—42no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2411yescharacteristic
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176#124—42no—
177#124—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180#121—42no—
181#121—42no—
182#117—22no—
183#117—22no—
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×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4D811yescharacteristic
197C2×C4⋊C4D811yescharacteristic
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4D811yescharacteristic
207C2×C22⋊C4D811yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16C2×D412yes—
221C2×C16C2×D412yes—
222#107—21no—
223#107—21no—
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#107—21no—
233#107—21no—
234#107—42no—
235#107—42no—
236#106—42no—
237#103—42no—
238#103—42no—
239#102—42no—
240#102—42no—
241#97—22no—
242#97—22no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8C2×D412yes—
246C4×C8#4112yes—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255C24—41no—
256C24C2×D811yescharacteristic
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—82no—
264C2×Q8—161no—
265C2×Q8—161no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—82no—
273C2×D4—161no—
274C2×D4—161no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C22×D411yescharacteristic
294Q16—41no—
295Q16—41no—
296Q16—41no—
297Q16—41no—
298SD16—84no—
299D8—41no—
300D8—41no—
301D8—41no—
302D8—41no—
303M16—44no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8#12112yes—
315C2×C8AGL(1,ℤ/8)12yes—
316C2×C8#13712yes—
317C2×C8C22×D411yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C8H(5,𝔽2)11yescharacteristic
321C2×C82−1+411yescharacteristic
322C2×C82−1+411yescharacteristic
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—161no—
340C4⋊C4—161no—
341C4⋊C4—161no—
342C4⋊C4—161no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—161no—
350C22⋊C4—161no—
351C22⋊C4—161no—
352C22⋊C4—161no—
353C42—21no—
354C42—21no—
355C42#12112yes—
356C42#13512yes—
357C16—22no—
358C16—22no—
359C16—41no—
360C16—41no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23C22×D811yescharacteristic
376Q8—81no—
377Q8—81no—
378Q8—81no—
379Q8—81no—
380Q8—164no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—164no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
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—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
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—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#46612yes—
425C2×C4#46612yes—
426C2×C4C2×#12111yescharacteristic
427C2×C4#53511yescharacteristic
428C2×C4#53611yescharacteristic
429C2×C4C2×#13511yescharacteristic
430C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
431C2×C4C2×#13711yescharacteristic
432C8—41no—
433C8—41no—
434C8#46814yes—
435C8#53711yescharacteristic
436C8#53711yescharacteristic
437C8#54211yescharacteristic
438C8#54211yescharacteristic
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22C2×#46611yescharacteristic
454C22#320511yescharacteristic
455C22#322611yescharacteristic
456C22#341511yescharacteristic
457C22#341611yescharacteristic
458C22#341611yescharacteristic
459C22#341711yescharacteristic
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4#262011yescharacteristic
473C4C2×#46811yescharacteristic
474C4#321011yescharacteristic
475C4#322811yescharacteristic
476C2—81no—
477C2—321no—
478C2#2984211yescharacteristic
479C2#3361211yescharacteristic
480C2#3361311yescharacteristic
481C2#3379011yescharacteristic
482C2#3379111yescharacteristic
483C2#3380911yescharacteristic
484C2#3381011yescharacteristic
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
1. C2×#2243C2semidirect
2. C2×#2164C2semidirect
3. #33215C2semidirect
5. #13477C2semidirect
6. #13476C2semidirect
7. #13475C2semidirect
8. #13474C2semidirect
9. #13383C2semidirect
10. #13382C2semidirect
13. #13282C2semidirect
14. #13204C2semidirect
15. #12908C2semidirect
135. #346D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22432
C2×#21642
#332152
#331822
#134772
#134762
#134752
#134742
#133832
#133822
#133812
#133802
#132822
#132042
#129082