Finite Group Databasebeta

← #147077

Subgroups — #147077

Order 512.

742 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1812C211yesmaximal, characteristic
2#20662C211yesmaximal, characteristic
3C2×#1757C211yesmaximal, characteristic
4#12583C212yesmaximal
5#12575C212yesmaximal
6C22×#131C2211yescharacteristic
7C2×#427—21no—
8C2×#427C411yescharacteristic, complemented
9C22×#106C411yescharacteristic, complemented
10C2×#413C2211yescharacteristic
11C2×#410C2212yes—
12C2×#410C2211yescharacteristic
13C2×#405—21no—
14#2311C2212yes—
15#1812—22no—
16#1812C414yescomplemented
17#1809—21no—
18#1809—21no—
19#1807—21no—
20#1807—21no—
21#1757—22no—
22#1757C414yescomplemented
23C2×#344—21no—
24#1736C412yescomplemented
25C2×H(3,ℤ/4)—21no—
26#1393—24no—
27#1390—24no—
28C22×#41C2311yescharacteristic
29C2×#131—41no—
30C2×#131—41no—
31C2×#131—42no—
32C2×#131C2×C412yes—
33C2×#131C2×C412yes—
34C2×#131D412yes—
35C2×#131Q812yes—
36C22×M16—21no—
37C22×M16C2×C411yescharacteristic
38C22×M16D411yescharacteristic
39C22×M16D411yescharacteristic
40C23×C8C2×C411yescharacteristic
41C23×C8D411yescharacteristic
42C23×C8D411yescharacteristic
43C2×#117—21no—
44C2×#117—22no—
45C2×#117C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47#427—24no—
48#427—42no—
49#427—42no—
50C2×#107—21no—
51C2×#107—41no—
52C2×#106—21no—
53C2×#106—21no—
54C2×#106—21no—
55C2×#106—41no—
56C2×#106—41no—
57C2×#106—41no—
58#413—21no—
59#413—21no—
60#413C2×C412yes—
61#413Q812yes—
62C2×#101—21no—
63C2×#100D412yes—
64#410—21no—
65#410—21no—
66#410—22no—
67#410—22no—
68#410—22no—
69#410C2×C414yes—
70#410C2×C412yes—
71#410Q812yes—
72C2×#99—21no—
73C2×#99—22no—
74C2×#99—22no—
75C2×#99D412yes—
76#405—24no—
77#405—42no—
78C2×#97—41no—
79C2×C4×C8—41no—
80#391—22no—
81#390—22no—
82#386—22no—
83#386—22no—
84#380—22no—
85#380—22no—
86#344—24no—
87#340—22no—
88#340—22no—
89H(3,ℤ/4)—24no—
90C2×#41—41no—
91C2×#41—41no—
92C2×#41—41no—
93C2×#41—41no—
94C2×#41C22×C411yescharacteristic
95C2×#41C22×C411yescharacteristic
96C2×#41C2×D411yescharacteristic
97C2×#41C2×Q811yescharacteristic
98C2×#41C2×Q811yescharacteristic
99C2×#41C2×Q811yescharacteristic
100C2×#41#4111yescharacteristic
101C2×#41#4111yescharacteristic
102C22×Q8#4111yescharacteristic
103C22×D4C2×D411yescharacteristic
104C22×D4C2×D411yescharacteristic
105C22×D4#4111yescharacteristic
106C23×C4—22no—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110#131—42no—
111#131—42no—
112#131—42no—
113#131—42no—
114#131—42no—
115#131—42no—
116#131—42no—
117#131—42no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—41no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16C4212yes—
131C2×M16C22⋊C412yes—
132C2×M16C22⋊C412yes—
133C2×M16C4⋊C412yes—
134C2×M16C4⋊C412yes—
135C2×M16C4⋊C412yes—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—41no—
146C22×C8C4212yes—
147C22×C8C22⋊C412yes—
148C22×C8C22⋊C412yes—
149C22×C8C22⋊C411yescharacteristic
150C22×C8C22⋊C411yescharacteristic
151C22×C8C22⋊C411yescharacteristic
152C22×C8C22⋊C411yescharacteristic
153C22×C8C4⋊C412yes—
154C22×C8C4⋊C412yes—
155C22×C8C4⋊C412yes—
156#117—22no—
157#117—22no—
158#117—24no—
159#117—41no—
160#117—41no—
161#117—42no—
162#117—42no—
163#117—42no—
164#117—42no—
165#117—42no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C42—22no—
179C2×C42—41no—
180C2×C42—42no—
181#107—42no—
182#107—44no—
183#106—42no—
184#106—44no—
185#106—41no—
186#106—41no—
187#106—42no—
188#101—22no—
189#101—22no—
190#100—22no—
191#100C4⋊C414yes—
192#99—22no—
193#99—22no—
194#99—22no—
195#99—24no—
196#99—24no—
197#99—42no—
198#99—42no—
199#99C4⋊C414yes—
200#97—44no—
201C4×C8—44no—
202#95—42no—
203#95—42no—
204C24C4×D411yescharacteristic
205C24#12111yescharacteristic
206C24#12711yescharacteristic
207#41—41no—
208#41—41no—
209#41—41no—
210#41—41no—
211#41—41no—
212#41—41no—
213#41—41no—
214#41—41no—
215#41—41no—
216#41—41no—
217#41—41no—
218#41—41no—
219#41—41no—
220#41—41no—
221#41—41no—
222#41—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8#11711yescharacteristic
226C2×Q8C4×Q811yescharacteristic
227C2×Q8C22⋊Q811yescharacteristic
228C2×Q8#12511yescharacteristic
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4C2×C4⋊C411yescharacteristic
236C2×D4C2×C4⋊C411yescharacteristic
237C2×D4#11711yescharacteristic
238C2×D4C4×D411yescharacteristic
239C2×D4C4×D411yescharacteristic
240C2×D4C4×Q811yescharacteristic
241C2×D4#12111yescharacteristic
242C2×D4#12111yescharacteristic
243C2×D4C22⋊Q811yescharacteristic
244C2×D4#12511yescharacteristic
245C2×D4C4⋊Q811yescharacteristic
246C2×D4C4⋊Q811yescharacteristic
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—162no—
265C22×C4C2×C4211yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C4⋊C411yescharacteristic
270C22×C4C2×C4⋊C411yescharacteristic
271C22×C4C2×C4⋊C411yescharacteristic
272C22×C4C4×D411yescharacteristic
273C22×C4C4×Q811yescharacteristic
274C22×C4C4×Q811yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4C22⋊Q811yescharacteristic
277C22×C4C22⋊Q811yescharacteristic
278C22×C4#12411yescharacteristic
279C22×C4#12411yescharacteristic
280C22×C4C4⋊Q811yescharacteristic
281M16—24no—
282M16—42no—
283M16—42no—
284M16—42no—
285M16—42no—
286M16—42no—
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8#9512yes—
313C2×C8#9512yes—
314C2×C8#9512yes—
315C2×C8#9512yes—
316C2×C8#9912yes—
317C2×C8#9911yescharacteristic
318C2×C8#9911yescharacteristic
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—162no—
334C22⋊C4—162no—
335C42—42no—
336C42—42no—
337C42—44no—
338C42—82no—
339C42—82no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23C4×C4⋊C411yescharacteristic
350C23#38011yescharacteristic
351C23#38411yescharacteristic
352C23#38911yescharacteristic
353C23#38911yescharacteristic
354C23#39011yescharacteristic
355C23#39011yescharacteristic
356Q8—41no—
357Q8—41no—
358Q8—41no—
359Q8—41no—
360D4—41no—
361D4—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370D4—41no—
371D4—41no—
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—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—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—44no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—164no—
412C2×C4—164no—
413C2×C4C2×#9511yescharacteristic
414C2×C4C4×C4⋊C411yescharacteristic
415C2×C4#38011yescharacteristic
416C2×C4#38211yescharacteristic
417C2×C4#38911yescharacteristic
418C2×C4#38911yescharacteristic
419C2×C4#39111yescharacteristic
420C2×C4#39111yescharacteristic
421C2×C4C2×#9911yescharacteristic
422C2×C4C2×#9911yescharacteristic
423C2×C4#41011yescharacteristic
424C2×C4#41011yescharacteristic
425C8—22no—
426C8—22no—
427C8—22no—
428C8—44no—
429C8#34212yes—
430C8#34212yes—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—81no—
449C22—81no—
450C22C4⋊#9511yescharacteristic
451C22#196411yescharacteristic
452C22#196411yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—84no—
460C4—162no—
461C4—164no—
462C4—162no—
463C4C2×#34211yescharacteristic
464C4#174011yescharacteristic
465C4#199611yescharacteristic
466C4#199611yescharacteristic
467C2—21no—
468C2—21no—
469C2—41no—
470C2—41no—
471C2—41no—
472C2—41no—
473C2—41no—
474C2—41no—
475C2#1085411yescharacteristic
476C2#1322912yes—
subgroupG/H
center, socleC22#1964
commutatorC22×C4C2×C42
Frattini, prime radical, cosocleC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G
LECSG ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#427C4semidirect
9. C22×#106C4semidirect
16. #1812C4semidirect
22. #1757C4semidirect
24. #1736C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18122
#206622
C2×#17572
#125832
#125752