Finite Group Databasebeta

← #34053965

Subgroups — #34053965

Order 19072.

752 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×#18011957C211yesmaximal, characteristic
2C2×#18011943C211yesmaximal, characteristic, complemented
3C2×SD16×D149C211yesmaximal, characteristic, complemented
4#23750198C211yesmaximal, characteristic, complemented
5#23749790C211yesmaximal, characteristic, complemented
6#23749787C211yesmaximal, characteristic, complemented
7#23749786C211yesmaximal, characteristic, complemented
8#23749778C211yesmaximal, characteristic, complemented
9#23749777C211yesmaximal, characteristic, complemented
10#23749775C211yesmaximal, characteristic, complemented
11#23749774C211yesmaximal, characteristic, complemented
12#23749628C211yesmaximal, characteristic, complemented
13#23749620C211yesmaximal, characteristic, complemented
14#23749590C211yesmaximal, characteristic, complemented
15#452×C149C211yesmaximal, characteristic, complemented
16#18011957—24no—
17#18011957—24no—
18C2×#14558737C2211yescharacteristic
19C2×#14558737C2211yescharacteristic
20C2×#14558736—21no—
21C2×#14558736C2211yescharacteristic
22C2×Q8×D149C2211yescharacteristic, complemented
23C22×D596—21no—
24C22×D596C2211yescharacteristic
25C2×D4×D149—21no—
26C2×D4×D149—21no—
27C2×D4×D149C2211yescharacteristic
28C2×D4×D149C2211yescharacteristic
29#18011943—24no—
30C2×#14558726C2211yescharacteristic, complemented
31C2×#14558725C2211yescharacteristic, complemented
32C2×#14558725C2211yescharacteristic, complemented
33C2×#14558724C2211yescharacteristic, complemented
34C2×#14558723C2211yescharacteristic, complemented
35C2×#14558723C2211yescharacteristic, complemented
36SD16×D149—24no—
37C2×#14558719C2211yescharacteristic, complemented
38C2×C8×D149C2211yescharacteristic, complemented
39#18011910—21no—
40#18011897—21no—
41#18011896—21no—
42#18011893—21no—
43#18011891C2211yescharacteristic, complemented
44#18011890C2211yescharacteristic, complemented
45#18011887C2211yescharacteristic, complemented
46#18011869C2211yescharacteristic, complemented
47#18011835C2211yescharacteristic, complemented
48#18011833C2211yescharacteristic, complemented
49#18011833C2211yescharacteristic, complemented
50#18011832C2211yescharacteristic, complemented
51#18011831C2211yescharacteristic, complemented
52#18011831C2211yescharacteristic, complemented
53C22⋊#11924746C2211yescharacteristic, complemented
54#18011823C2211yescharacteristic, complemented
55#18011823C2211yescharacteristic, complemented
56C2×#41×C149C2211yescharacteristic
57C2×Q16×C149C2211yescharacteristic, complemented
58C2×SD16×C149C2211yescharacteristic, complemented
59C22⋊Q8×C149C2211yescharacteristic, complemented
60#103×C149C2211yescharacteristic, complemented
61#102×C149C2211yescharacteristic, complemented
62C22⋊C8×C149C2211yescharacteristic, complemented
63C23×D149—21no—
64C23×D149—21no—
65C23×D149—21no—
66#14558737—22no—
67#14558737—22no—
68#14558737—22no—
69#14558737D414yes—
70#14558736—28no—
71#14558736—22no—
72#14558736—22no—
73C2×Dic298—21no—
74C2×Dic298D411yescharacteristic
75C2×Dic298C2311yescharacteristic
76Q8×D149—22no—
77Q8×D149—22no—
78C2×D596—28no—
79C2×D596—21no—
80C2×D596—21no—
81C2×D596—22no—
82C2×D596—22no—
83C2×D596—21no—
84C2×D596D411yescharacteristic
85C2×D596C2311yescharacteristic
86C2×D596C2311yescharacteristic
87C2×C22⋊D149—21no—
88C2×C22⋊D149—21no—
89C2×C22⋊D149D411yescharacteristic
90C2×C22⋊D149D411yescharacteristic
91D4×D149—28no—
92D4×D149—28no—
93D4×D149—22no—
94D4×D149—22no—
95D4×D149—22no—
96D4×D149D414yes—
97C2×C4×D149—21no—
98C2×C4×D149—21no—
99C2×C4×D149—21no—
100C2×C4×D149C2311yescharacteristic
101C2×C4×D149C2311yescharacteristic
102#14558726—22no—
103#14558725—22no—
104#14558725—22no—
105#14558724—22no—
106#14558723—22no—
107#14558723—22no—
108#14558719—22no—
109C2×#11924746C2311yescharacteristic
110C2×#11924746C2311yescharacteristic
111C8×D149—22no—
112C4⋊Dic149—21no—
113C4⋊Dic149C2311yescharacteristic
114#14558715—21no—
115#14558715—21no—
116C22⋊Dic149—21no—
117#14558713—21no—
118#14558713—21no—
119#14558713—21no—
120C4×Dic149—21no—
121#41×C149—22no—
122#41×C149—22no—
123C2×Q8×C149C2311yescharacteristic
124C2×Q8×C149C2311yescharacteristic
125C2×D4×C149—21no—
126C2×D4×C149C2311yescharacteristic
127C22×C4×C149—21no—
128C22×C4×C149C2311yescharacteristic
129Q16×C149—22no—
130SD16×C149—22no—
131C2×C8×C149C2311yescharacteristic
132C2×C8×C149C2311yescharacteristic
133C4⋊C4×C149—21no—
134C4⋊C4×C149C2311yescharacteristic
135C22⋊C4×C149—21no—
136C22×D149—21no—
137C22×D149—21no—
138C22×D149—24no—
139C22×D149—24no—
140C22×D149—24no—
141C22×D149—81no—
142C22×D149—81no—
143C22×D149—81no—
144C22×D149—81no—
145C22×D149—81no—
146C22×D149—81no—
147C22×D149C2×D411yescharacteristic
148C22×D149C2×D411yescharacteristic
149Dic298—22no—
150Dic298—21no—
151Dic298—21no—
152Dic298—42no—
153D596—24no—
154D596—24no—
155D596—24no—
156D596—22no—
157D596—21no—
158D596—21no—
159D596C2×D412yes—
160D596C2×D412yes—
161C22⋊D149—24no—
162C22⋊D149—22no—
163C22⋊D149—22no—
164C22⋊D149—24no—
165C2×Dic149—21no—
166C2×Dic149—21no—
167C2×Dic149C2×D411yescharacteristic
168C2×Dic149C2×D411yescharacteristic
169C4×D149—24no—
170C4×D149—21no—
171C4×D149—21no—
172C4×D149—24no—
173C4×D149—42no—
174C4×D149C2×D412yes—
175C4×D149C2×D412yes—
176#11924746—21no—
177#11924746—21no—
178C23×C149—21no—
179C23×C149C2×D411yescharacteristic
180Q8×C149—21no—
181Q8×C149—21no—
182Q8×C149—21no—
183Q8×C149C2×D412yes—
184D4×C149—24no—
185D4×C149—21no—
186D4×C149C2×D412yes—
187C2×C4×C149—24no—
188C2×C4×C149—21no—
189C2×C4×C149—21no—
190C2×C4×C149—21no—
191C2×C4×C149—21no—
192C2×C4×C149—21no—
193C2×C4×C149C2×D411yescharacteristic
194C2×C4×C149C2411yescharacteristic
195C8×C149—21no—
196C8×C149—21no—
197C22×C149—21no—
198C22×C149—22no—
199C22×C149—21no—
200C22×C149—21no—
201C22×C149—41no—
202C22×C149—41no—
203C22×C149C22×D411yescharacteristic
204C2×D149—21no—
205C2×D149—21no—
206C2×D149—21no—
207C2×D149—22no—
208C2×D149—22no—
209C2×D149—21no—
210C2×D149—21no—
211C2×D149—41no—
212C2×D149—41no—
213C2×D149—41no—
214C2×D149—41no—
215C2×D149—84no—
216C2×D149—84no—
217C2×D149—84no—
218C2×D149#12012yes—
219C4×C149—21no—
220C4×C149—22no—
221C4×C149—41no—
222C4×C149—41no—
223C4×C149C22×D411yescharacteristic
224C4×C149C22×D411yescharacteristic
225Dic149—21no—
226Dic149—41no—
227Dic149—41no—
228Dic149#12012yes—
229C2×C149—41no—
230C2×C149—42no—
231C2×C149C2×#12011yescharacteristic
232C2×C149#57711yescharacteristic
233C2×C149#57711yescharacteristic
234D149—22no—
235D149—41no—
236D149—42no—
237D149—42no—
238C149#300911yescharacteristic, complemented, Sylow
239#3009—1491nomaximal, Sylow
240C2×H(5,𝔽2)—1491no—
241C2×#137—1491no—
242C22×SD16—1491no—
243#536—1491no—
244#452—1491no—
245#452—1491no—
246#452—1491no—
247#452D14911yescharacteristic, complemented
248#450—1491no—
249#450—1491no—
250#450—1491no—
251#450—1491no—
252#418—1491no—
253#416—1491no—
254#408—1491no—
255H(5,𝔽2)—2984no—
256H(5,𝔽2)—2984no—
257C2×#41—2981no—
258C2×#41—1491no—
259C2×#41—1491no—
260C2×#41—1491no—
261C2×#41C2×D14911yescharacteristic
262C22×Q8—1491no—
263C22×D4—2981no—
264C22×D4—2981no—
265C22×D4—2981no—
266C22×D4—1491no—
267C22×D4—1491no—
268C22×D4—1491no—
269#137—2984no—
270C2×Q16—1491no—
271C2×Q16C2×D14911yescharacteristic, complemented
272C2×SD16—2984no—
273C2×SD16—1491no—
274C2×SD16—1491no—
275C2×SD16—1491no—
276C2×SD16—1491no—
277C2×SD16—1491no—
278C2×SD16C2×D14911yescharacteristic, complemented
279C2×M16—1491no—
280C22×C8—1491no—
281C4⋊Q8—2981no—
282#124—2981no—
283#123—2981no—
284#123—2981no—
285C22⋊Q8—1491no—
286C22⋊Q8—1491no—
287C22⋊Q8—1491no—
288C22⋊Q8C2×D14911yescharacteristic, complemented
289#117—1491no—
290#103—1491no—
291#103C2×D14911yescharacteristic, complemented
292#102—1491no—
293#102—1491no—
294#102—1491no—
295#102—1491no—
296#102—1491no—
297#102C2×D14911yescharacteristic, complemented
298C22⋊C8—1491no—
299C22⋊C8—1491no—
300C22⋊C8—1491no—
301C22⋊C8C2×D14911yescharacteristic, complemented
302C24—2981no—
303C24—2981no—
304C24—2981no—
305#41—22no—
306#41—22no—
307#41—2988no—
308#41—2982no—
309#41—2982no—
310#41—2982no—
311#41—2982no—
312#41—2982no—
313#41—1494no—
314C2×Q8—2981no—
315C2×Q8—2982no—
316C2×Q8—2982no—
317C2×Q8—1491no—
318C2×Q8—1491no—
319C2×Q8C22×D14911yescharacteristic
320C2×Q8C22×D14911yescharacteristic
321C2×D4—21no—
322C2×D4—2988no—
323C2×D4—2981no—
324C2×D4—2981no—
325C2×D4—2982no—
326C2×D4—2982no—
327C2×D4—2988no—
328C2×D4—2988no—
329C2×D4—2982no—
330C2×D4—2982no—
331C2×D4—2982no—
332C2×D4—2981no—
333C2×D4—2981no—
334C2×D4—2981no—
335C2×D4—1491no—
336C2×D4—1491no—
337C2×D4—1494no—
338C2×D4—1491no—
339C2×D4—1491no—
340C2×D4—1491no—
341C2×D4C22×D14911yescharacteristic
342C22×C4—21no—
343C22×C4—2981no—
344C22×C4—2981no—
345C22×C4—2981no—
346C22×C4—1491no—
347C22×C4—1491no—
348C22×C4C22×D14911yescharacteristic
349Q16—22no—
350Q16—2982no—
351SD16—22no—
352SD16—2982no—
353SD16—2982no—
354SD16—2982no—
355SD16—2982no—
356SD16—2982no—
357M16—2982no—
358C2×C8—2982no—
359C2×C8—1491no—
360C2×C8—1491no—
361C2×C8C22×D14911yescharacteristic
362C2×C8C22×D14911yescharacteristic
363C4⋊C4—21no—
364C4⋊C4—2981no—
365C4⋊C4—2981no—
366C4⋊C4—2981no—
367C4⋊C4—1491no—
368C4⋊C4C22×D14911yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—2981no—
371C22⋊C4—2981no—
372C22⋊C4—2981no—
373C22⋊C4—2981no—
374C42—2981no—
375C23—21no—
376C23—2984no—
377C23—2984no—
378C23—2984no—
379C23—2981no—
380C23—2981no—
381C23—1491no—
382C23—1491no—
383C23—11921no—
384C23—11921no—
385C23—11921no—
386C23—11921no—
387C23—11921no—
388C23—11921no—
389C23D4×D14911yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—21no—
393Q8—2982no—
394Q8—2981no—
395Q8—2981no—
396Q8—5962no—
397Q8D4×D14912yes—
398D4—24no—
399D4—21no—
400D4—2984no—
401D4—2984no—
402D4—2984no—
403D4—2982no—
404D4—2984no—
405D4—2982no—
406D4—2982no—
407D4—2984no—
408D4—2981no—
409D4—2981no—
410D4—1492no—
411D4—1492no—
412D4D4×D14912yes—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—2984no—
420C2×C4—2984no—
421C2×C4—2981no—
422C2×C4—2981no—
423C2×C4—2981no—
424C2×C4—2981no—
425C2×C4—5962no—
426C2×C4—1491no—
427C2×C4—1491no—
428C2×C4—1492no—
429C2×C4—1492no—
430C2×C4D4×D14911yescharacteristic
431C2×C4C23×D14911yescharacteristic
432C8—21no—
433C8—21no—
434C8—2981no—
435C8—2981no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—2981no—
443C22—2981no—
444C22—2981no—
445C22—2981no—
446C22—2981no—
447C22—2982no—
448C22—2982no—
449C22—5961no—
450C22—5961no—
451C22—5961no—
452C22—5961no—
453C22—1492no—
454C22—11924no—
455C22—11924no—
456C22—11924no—
457C22C2×D4×D14911yescharacteristic
458C4—21no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—2981no—
463C4—5961no—
464C4—5961no—
465C4—1492no—
466C4C2×D4×D14911yescharacteristic
467C4C2×D4×D14911yescharacteristic
468C2—41no—
469C2—42no—
470C2—2982no—
471C2—5961no—
472C2—5962no—
473C2—5962no—
474C2#120×D14911yescharacteristic
475C2#2375043211yescharacteristic
476C2#2375043411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#452×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011957C2
hypercenter#452D149
hypocenterC149#3009
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. C2×#18011943C2semidirect
3. C2×SD16×D149C2semidirect
4. #23750198C2semidirect
5. #23749790C2semidirect
6. #23749787C2semidirect
7. #23749786C2semidirect
8. #23749778C2semidirect
9. #23749777C2semidirect
10. #23749775C2semidirect
11. #23749774C2semidirect
12. #23749628C2semidirect
13. #23749620C2semidirect
14. #23749590C2semidirect
15. #452×C149C2semidirect
22. C2×Q8×D149C22semidirect
30. C2×#14558726C22semidirect
31. C2×#14558725C22semidirect
32. C2×#14558725C22semidirect
33. C2×#14558724C22semidirect
34. C2×#14558723C22semidirect
35. C2×#14558723C22semidirect
37. C2×#14558719C22semidirect
38. C2×C8×D149C22semidirect
43. #18011891C22semidirect
44. #18011890C22semidirect
45. #18011887C22semidirect
46. #18011869C22semidirect
47. #18011835C22semidirect
48. #18011833C22semidirect
49. #18011833C22semidirect
50. #18011832C22semidirect
51. #18011831C22semidirect
52. #18011831C22semidirect
53. C22⋊#11924746C22semidirect
54. #18011823C22semidirect
55. #18011823C22semidirect
57. C2×Q16×C149C22semidirect
58. C2×SD16×C149C22semidirect
59. C22⋊Q8×C149C22semidirect
60. #103×C149C22semidirect
61. #102×C149C22semidirect
62. C22⋊C8×C149C22semidirect
238. C149#3009semidirect
247. #452D149semidirect
271. C2×Q16C2×D149semidirect
278. C2×SD16C2×D149semidirect
288. C22⋊Q8C2×D149semidirect
291. #103C2×D149semidirect
297. #102C2×D149semidirect
301. C22⋊C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119572
C2×#180119432
C2×SD16×D1492
#237501982
#237497902
#237497872
#237497862
#237497782
#237497772
#237497752
#237497742
#237496282
#237496202
#237495902
#452×C1492
#3009149