Finite Group Databasebeta

← C2×#33911 #7623795

Subgroups — C2×#33911 #7623795

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3338C211yesmaximal, characteristic
2C2×#3336C211yesmaximal, characteristic
3#33911C2116yesmaximal, direct product factor
4C2×#2255C211yesmaximal, characteristic, complemented
5C2×#2236C211yesmaximal, characteristic, complemented
6C2×#2237C211yesmaximal, characteristic, complemented
7C2×#2235C211yesmaximal, characteristic, complemented
8C2×#2235C211yesmaximal, characteristic, complemented
9C2×#2229C211yesmaximal, characteristic, complemented
10C2×#2225C211yesmaximal, characteristic, complemented
11C2×#2219C211yesmaximal, characteristic, complemented
12C2×#2216C211yesmaximal, characteristic, complemented
13C2×#2176C211yesmaximal, characteristic, complemented
14C2×#2168C211yesmaximal, characteristic
15C2×#2159C211yesmaximal, characteristic, complemented
16C22×#367C211yesmaximal, characteristic, complemented
17C2×#554—21no—
18C2×#552—21no—
19C22×Q32C2211yescharacteristic
20C22×M32C2211yescharacteristic
21C23×C16C2211yescharacteristic
22#3338—24no—
23#3338C2218yes—
24#3336—24no—
25#3336C2218yes—
26C2×#500C2211yescharacteristic
27C2×#499C2211yescharacteristic
28C2×#484—21no—
29C2×#484—21no—
30C2×#485—21no—
31C2×#483—21no—
32C2×#479—21no—
33C2×#477—21no—
34C2×#467C2211yescharacteristic
35C2×#467C2211yescharacteristic
36C2×#467C2211yescharacteristic
37C2×#467C2211yescharacteristic
38C2×C4×Q16C2211yescharacteristic
39C2×C4×Q16C2211yescharacteristic
40C2×C8×D4C2211yescharacteristic
41C2×#427C2211yescharacteristic
42C22×#107C2211yescharacteristic
43C22×#107C2211yescharacteristic
44C22×#107C2211yescharacteristic
45#2255C2218yescomplemented
46#2237C2218yescomplemented
47#2236C2218yescomplemented
48#2235C2218yescomplemented
49#2235C2218yescomplemented
50#2229C2218yescomplemented
51#2225C2218yescomplemented
52#2219C2218yescomplemented
53#2216C2218yescomplemented
54#2176C2218yescomplemented
55#2168C2218yes—
56#2159C2218yescomplemented
57C2×#367C2211yescharacteristic
58C2×#367C2211yescharacteristic
59C2×#367C2211yescharacteristic
60C2×#367C2211yescharacteristic
61C2×#367C2211yescharacteristic
62C2×#367C2211yescharacteristic
63C2×#367C2218yescomplemented
64C2×#366C2211yescharacteristic
65C2×#366C2211yescharacteristic
66C2×#366C2211yescharacteristic
67C2×C4⋊C16C2211yescharacteristic
68C2×#358C2211yescharacteristic
69C2×#358C2211yescharacteristic
70C2×#358C2211yescharacteristic
71C2×#358C2211yescharacteristic
72C2×#358C2211yescharacteristic
73C2×#358C2211yescharacteristic
74C2×C22⋊C16C2211yescharacteristic
75C2×C22⋊C16C2211yescharacteristic
76C2×#346C2211yescharacteristic
77C22×Q16C2311yescharacteristic
78C22×Q16C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81#554—28no—
82#554—44no—
83#552—28no—
84#552—44no—
85C2×C4⋊Q8—21no—
86C2×#126—41no—
87C2×#125—21no—
88C2×#125—41no—
89C2×#125—41no—
90C2×#123—41no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—41no—
96C2×C22⋊Q8—41no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×D4—41no—
100C2×C4×D4D411yescharacteristic
101C2×#117—21no—
102C2×#117—41no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C2×Q32—24no—
107C2×Q32C2314yes—
108C2×M32—24no—
109C2×M32C2314yes—
110C22×C16—24no—
111C22×C16C2311yescharacteristic
112C22×C16C2311yescharacteristic
113C22×C16C2311yescharacteristic
114C22×C16C2314yes—
115C22×C16C2311yescharacteristic
116#500—22no—
117#500C2314yes—
118#499—22no—
119#499C2314yes—
120#485—28no—
121#484—28no—
122#484—28no—
123#483—28no—
124#479—28no—
125#477—28no—
126#467—22no—
127#467—22no—
128#467—22no—
129#467—22no—
130#467C2314yes—
131#467C2314yes—
132#467C2314yes—
133#467C2314yes—
134C4×Q16—22no—
135C4×Q16—22no—
136C4×Q16C2314yes—
137C4×Q16C2314yes—
138C8×D4D414yes—
139C8×D4C2314yes—
140#427—22no—
141#427C2314yes—
142C2×#107—21no—
143C2×#107—21no—
144C2×#107—21no—
145C2×#107—22no—
146C2×#107—22no—
147C2×#107—22no—
148C2×#107—21no—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#107C2311yescharacteristic
153C2×#107C2311yescharacteristic
154C2×#107C2311yescharacteristic
155C2×#107C2311yescharacteristic
156C2×#107C2311yescharacteristic
157C2×#107C2311yescharacteristic
158C2×#106—21no—
159C2×#106—21no—
160C2×C4⋊C8D411yescharacteristic
161C2×#103—21no—
162C2×#103—21no—
163C2×#103—21no—
164C2×#103—21no—
165C2×#103—21no—
166C2×#103—21no—
167C2×C22⋊C8D411yescharacteristic
168C2×C22⋊C8D411yescharacteristic
169C2×C4×C8C2311yescharacteristic
170#367C2314yes—
171#367C2314yes—
172#367C2314yes—
173#367C2314yes—
174#367C2314yes—
175#367C2314yes—
176#366C2314yes—
177#366C2314yes—
178#366C2314yes—
179C4⋊C16C2314yes—
180#358C2314yes—
181#358C2314yes—
182#358C2314yes—
183#358C2314yes—
184#358C2314yes—
185#358C2314yes—
186C22⋊C16C2314yes—
187C22⋊C16C2314yes—
188#346C2314yes—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—41no—
192C22×D4D811yescharacteristic
193C23×C4—41no—
194C23×C4—41no—
195C23×C4—41no—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C2×Q16—21no—
199C2×Q16—21no—
200C2×Q16—22no—
201C2×Q16—22no—
202C2×Q16—22no—
203C2×Q16—22no—
204C2×Q16C2412yes—
205C2×Q16C2412yes—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D411yescharacteristic
215C22×C8C2412yes—
216C22×C8C2412yes—
217C22×C8C2411yescharacteristic
218C4⋊Q8—24no—
219C4⋊Q8—42no—
220#126—48no—
221#125—24no—
222#125—48no—
223#125—48no—
224#125—42no—
225#123—48no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—42no—
231C22⋊Q8—42no—
232C22⋊Q8—42no—
233C22⋊Q8—42no—
234C22⋊Q8—48no—
235C22⋊Q8—48no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—42no—
239C4×Q8—42no—
240C4×D4—48no—
241C4×D4D814yes—
242C4×D4C2×D414yes—
243#117—24no—
244#117—42no—
245#117—48no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4D811yescharacteristic
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4D811yescharacteristic
277C2×C22⋊C4D811yescharacteristic
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42C2×D411yescharacteristic
282Q32—28no—
283M32—28no—
284C2×C16—22no—
285C2×C16—22no—
286C2×C16—22no—
287C2×C16—28no—
288C2×C16#4114yes—
289C2×C16C2412yes—
290C2×C16C2412yes—
291C2×C16C2412yes—
292C2×C16C2412yes—
293#107—21no—
294#107—21no—
295#107—21no—
296#107—21no—
297#107—21no—
298#107—24no—
299#107—24no—
300#107—24no—
301#107—24no—
302#107—21no—
303#107C2412yes—
304#107C2412yes—
305#107C2412yes—
306#107C2412yes—
307#107C2412yes—
308#107C2412yes—
309#106—24no—
310#106—24no—
311C4⋊C8C2×D414yes—
312#103—24no—
313#103—24no—
314#103—24no—
315#103—24no—
316#103—24no—
317#103—24no—
318C22⋊C8C2×D414yes—
319C22⋊C8C2×D414yes—
320C4×C8C2×D412yes—
321C4×C8C2412yes—
322C24C2×D811yescharacteristic
323C24C2×D811yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—44no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—162no—
334C2×D4—24no—
335C2×D4C2×D814yes—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—84no—
352C22×C4—162no—
353C22×C4—162no—
354C22×C4C2×D812yes—
355C22×C4C2×D812yes—
356C22×C4C2×D811yescharacteristic
357C22×C4C2×D811yescharacteristic
358C22×C4C2×D811yescharacteristic
359C22×C4C2×D811yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D411yescharacteristic
363Q16—22no—
364Q16—22no—
365Q16—22no—
366Q16—22no—
367Q16—42no—
368Q16—42no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8C22×D412yes—
379C2×C8C22×D412yes—
380C2×C8C22×D412yes—
381C2×C8C22×D411yescharacteristic
382C2×C8C2×#4112yes—
383C2×C82−1+412yes—
384C2×C8C2511yescharacteristic
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—44no—
399C4⋊C4—41no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4C2×D814yes—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4C2×D814yes—
416C22⋊C4C2×D814yes—
417C42—44no—
418C42—44no—
419C42—44no—
420C42C2×D812yes—
421C42C22×D412yes—
422C16—22no—
423C16—22no—
424C16—22no—
425C16C2×#4114yes—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23#51014yes—
432C23C22×D812yes—
433C23C22×D812yes—
434C23C22×D811yescharacteristic
435Q8—42no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—82no—
440Q8—82no—
441Q8—164no—
442D4—28no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—88no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—164no—
471C2×C4—164no—
472C2×C4#54712yes—
473C2×C4C22×D812yes—
474C2×C4C22×D812yes—
475C2×C4C22×D812yes—
476C2×C4C22×D812yes—
477C2×C4C22×D811yescharacteristic
478C2×C4#57812yes—
479C2×C4C23×D411yescharacteristic
480C8—22no—
481C8—22no—
482C8—22no—
483C8#54712yes—
484C8C22×#4111yescharacteristic
485C8C2×2−1+411yescharacteristic
486C22—22no—
487C22—28no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—44no—
493C22#333412yes—
494C22C2×#51014yes—
495C22C2×#51012yes—
496C22#341612yes—
497C22C23×D811yescharacteristic
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4#333412yes—
511C4C2×#54711yescharacteristic
512C4C2×#57811yescharacteristic
513C2—24no—
514C2—42no—
515C2#3391114yesdirect product factor
516C2C2×#333411yescharacteristic
517C2C22×#51011yescharacteristic
518C2C2×#341611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC22×D4D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33911C2direct (N × H)
4. C2×#2255C2semidirect
5. C2×#2236C2semidirect
6. C2×#2237C2semidirect
7. C2×#2235C2semidirect
8. C2×#2235C2semidirect
9. C2×#2229C2semidirect
10. C2×#2225C2semidirect
11. C2×#2219C2semidirect
12. C2×#2216C2semidirect
13. C2×#2176C2semidirect
15. C2×#2159C2semidirect
16. C22×#367C2semidirect
45. #2255C22semidirect
46. #2237C22semidirect
47. #2236C22semidirect
48. #2235C22semidirect
49. #2235C22semidirect
50. #2229C22semidirect
51. #2225C22semidirect
52. #2219C22semidirect
53. #2216C22semidirect
54. #2176C22semidirect
56. #2159C22semidirect
63. C2×#367C22semidirect
515. C2#33911direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#33382
C2×#33362
#339112
C2×#22552
C2×#22362
C2×#22372
C2×#22352
C2×#22292
C2×#22252
C2×#22192
C2×#22162
C2×#21762
C2×#21682
C2×#21592
C22×#3672