Finite Group Databasebeta

← #28366172

Subgroups — #28366172

Order 13952.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812669C211yesmaximal, characteristic
2#20294928C211yesmaximal, characteristic, complemented
3#20294927C211yesmaximal, characteristic, complemented
4#20294926C211yesmaximal, characteristic, complemented
5#20294919C211yesmaximal, characteristic, complemented
6#20294918C211yesmaximal, characteristic, complemented
7#20294917C211yesmaximal, characteristic, complemented
8#20294916C211yesmaximal, characteristic, complemented
9#20294915C211yesmaximal, characteristic, complemented
10#20294904C211yesmaximal, characteristic, complemented
11#20294904C211yesmaximal, characteristic, complemented
12#20294902C211yesmaximal, characteristic, complemented
13#20294881C211yesmaximal, characteristic, complemented
14#20294881C211yesmaximal, characteristic, complemented
15#388×C109C211yesmaximal, characteristic, complemented
16C24×D109C2211yescharacteristic
17C22×D436C2211yescharacteristic
18C22×D436C2211yescharacteristic
19C22×C22⋊D109C2211yescharacteristic
20C22×C4×D109C2211yescharacteristic, complemented
21#16812669C418yescomplemented
22C2×C4⋊Dic109C2211yescharacteristic, complemented
23C2×#13182008C2211yescharacteristic, complemented
24C2×C22⋊Dic109C2211yescharacteristic
25C2×C22⋊Dic109C2211yescharacteristic
26C2×#13182006C2211yescharacteristic
27C2×#13182006C2211yescharacteristic
28C2×#13182006C2211yescharacteristic
29C2×#13182006C2211yescharacteristic
30C2×#13182006C2211yescharacteristic
31C2×#13182006C2211yescharacteristic
32C2×#13182006C2211yescharacteristic
33C2×#13182006C2211yescharacteristic
34C2×#13182006C2211yescharacteristic
35C2×#13182006C2211yescharacteristic
36C2×#13182006C2211yescharacteristic, complemented
37C2×#13182006C2211yescharacteristic, complemented
38C2×#13182006C2211yescharacteristic, complemented
39C2×C4×Dic109C2211yescharacteristic, complemented
40C2×C4×Dic109C2211yescharacteristic, complemented
41#16812606C2211yescharacteristic, complemented
42#16812606C2211yescharacteristic, complemented
43#16812606C2211yescharacteristic, complemented
44#16812606C2211yescharacteristic, complemented
45C2×C4⋊C4×C109C2211yescharacteristic, complemented
46C2×C22⋊C4×C109C2211yescharacteristic
47C2×C22⋊C4×C109C2211yescharacteristic
48C2×C22⋊C4×C109C2211yescharacteristic
49C2×C42×C109C2211yescharacteristic, complemented
50#95×C109C2211yescharacteristic, complemented
51#95×C109C2211yescharacteristic, complemented
52C23×D109—41no—
53C23×D109—41no—
54C23×D109—41no—
55C23×D109—41no—
56C23×D109—41no—
57C23×D109—41no—
58C23×D109C2×C414yescomplemented
59C23×D109C2311yescharacteristic
60C23×D109C2311yescharacteristic
61C23×D109C2311yescharacteristic
62C2×D436—41no—
63C2×D436—41no—
64C2×D436—41no—
65C2×D436—41no—
66C2×D436C2×C414yes—
67C2×D436C2×C414yes—
68C2×C22⋊D109—41no—
69C2×C22⋊D109—41no—
70C2×C22⋊D109C2×C414yes—
71C22×Dic109C2311yescharacteristic
72C22×Dic109C2311yescharacteristic
73C22×Dic109C2311yescharacteristic
74C22×Dic109C2311yescharacteristic
75C22×Dic109C2311yescharacteristic
76C2×C4×D109—41no—
77C2×C4×D109—41no—
78C2×C4×D109—41no—
79C4⋊Dic109—41no—
80#13182008—41no—
81C22⋊Dic109—41no—
82C22⋊Dic109—41no—
83#13182006—41no—
84#13182006—41no—
85#13182006—41no—
86#13182006—41no—
87#13182006—41no—
88#13182006—41no—
89#13182006—41no—
90#13182006—41no—
91#13182006—41no—
92#13182006—41no—
93#13182006—41no—
94#13182006C2×C414yescomplemented
95#13182006C2×C414yescomplemented
96C4×Dic109—41no—
97C4×Dic109—41no—
98C24×C109C2311yescharacteristic
99C22×C4×C109C2311yescharacteristic
100C22×C4×C109C2311yescharacteristic
101C22×C4×C109C2311yescharacteristic
102C22×C4×C109C2311yescharacteristic
103C22×C4×C109C2311yescharacteristic
104C22×C4×C109C2311yescharacteristic
105C4⋊C4×C109—41no—
106C22⋊C4×C109—41no—
107C22⋊C4×C109—41no—
108C22⋊C4×C109C2×C414yescomplemented
109C42×C109—41no—
110C23×C109—21no—
111C23×C109—21no—
112C23×C109—21no—
113C23×C109—21no—
114C23×C109—21no—
115C23×C109—21no—
116C23×C109C22×C412yes—
117C23×C109C2411yescharacteristic
118C22×D109—21no—
119C22×D109—21no—
120C22×D109—21no—
121C22×D109—21no—
122C22×D109—21no—
123C22×D109—21no—
124C22×D109—21no—
125C22×D109—21no—
126C22×D109—21no—
127C22×D109—21no—
128C22×D109—21no—
129C22×D109—21no—
130C22×D109—21no—
131C22×D109—21no—
132C22×D109—21no—
133C22×D109—21no—
134C22×D109—21no—
135C22×D109—21no—
136C22×D109—44no—
137C22×D109—44no—
138C22×D109—44no—
139C22×D109—44no—
140C22×D109—44no—
141C22×D109—44no—
142C22×D109—44no—
143C22×D109C22×C412yes—
144C22×D109C22×C412yes—
145C22×D109C22×C412yes—
146C2×C4×C109—21no—
147C2×C4×C109—21no—
148C2×C4×C109—21no—
149C2×C4×C109—21no—
150C2×C4×C109—21no—
151C2×C4×C109—21no—
152C2×C4×C109—21no—
153C2×C4×C109—21no—
154C2×C4×C109—21no—
155C2×C4×C109—21no—
156C2×C4×C109—21no—
157C2×C4×C109—21no—
158C2×C4×C109—21no—
159C2×C4×C109—21no—
160C2×C4×C109—21no—
161C2×C4×C109—21no—
162C2×C4×C109C22×C412yes—
163C2×C4×C109C22×C412yes—
164C22⋊D109—44no—
165C2×Dic109—21no—
166C2×Dic109—21no—
167C2×Dic109—21no—
168C2×Dic109—21no—
169C2×Dic109—21no—
170C2×Dic109—21no—
171C2×Dic109—21no—
172C2×Dic109—21no—
173C2×Dic109—21no—
174C2×Dic109—21no—
175C2×Dic109—21no—
176C2×Dic109—21no—
177C2×Dic109—21no—
178C2×Dic109—21no—
179C2×Dic109C22×C412yes—
180D436—44no—
181D436—44no—
182C4×D109—44no—
183C22×C109—22no—
184C22×C109—22no—
185C22×C109—22no—
186C22×C109—41no—
187C22×C109—41no—
188C22×C109—41no—
189C22×C109—41no—
190C22×C109C23×C411yescharacteristic
191C22×C109H(5,𝔽2)11yescharacteristic
192C22×C109H(5,𝔽2)11yescharacteristic
193C22×C109H(5,𝔽2)11yescharacteristic
194C22×C109H(5,𝔽2)11yescharacteristic
195C22×C109H(5,𝔽2)11yescharacteristic
196C22×C109H(5,𝔽2)11yescharacteristic
197C2×D109—22no—
198C2×D109—22no—
199C2×D109—22no—
200C2×D109—22no—
201C2×D109—22no—
202C2×D109—22no—
203C2×D109—22no—
204C2×D109—22no—
205C2×D109—22no—
206C2×D109—416no—
207C2×D109—41no—
208C2×D109—41no—
209C2×D109—41no—
210C2×D109—41no—
211C2×D109—41no—
212C2×D109—41no—
213C2×D109—41no—
214C2×D109—41no—
215C2×D109—41no—
216C2×D109—41no—
217C2×D109—41no—
218C2×D109—41no—
219C4×C109—22no—
220C4×C109—22no—
221C4×C109—41no—
222C4×C109—41no—
223C4×C109—41no—
224C4×C109—41no—
225Dic109—22no—
226Dic109—41no—
227Dic109—41no—
228Dic109—41no—
229Dic109—41no—
230C2×C109—42no—
231C2×C109#51811yescharacteristic
232C2×C109#51811yescharacteristic
233C2×C109#51811yescharacteristic
234C2×C109#56011yescharacteristic
235C2×C109#56011yescharacteristic
236C2×C109#56011yescharacteristic
237C2×C109H(3,𝔽4)11yescharacteristic
238D109—42no—
239D109—42no—
240D109—42no—
241#2375—1091nomaximal, Sylow
242C109#237511yescharacteristic, complemented, Sylow
243C2×#120—1091no—
244#390—1091no—
245#390—1091no—
246#390—1091no—
247#388—1091no—
248#388—1091no—
249#388—1091no—
250#388—1091no—
251#388—1091no—
252#388D10911yescharacteristic, complemented
253#386—1091no—
254#386—1091no—
255#386—1091no—
256#379—1091no—
257#379—1091no—
258C25—1091no—
259C22×D4—1091no—
260C22×D4—1091no—
261C22×D4—1091no—
262C23×C4—1091no—
263#120—1098no—
264C2×C4⋊C4—1091no—
265C2×C4⋊C4—1091no—
266C2×C4⋊C4C2×D10911yescharacteristic, complemented
267C2×C22⋊C4—1091no—
268C2×C22⋊C4—1091no—
269C2×C22⋊C4—1091no—
270C2×C22⋊C4—1091no—
271C2×C22⋊C4—1091no—
272C2×C22⋊C4—1091no—
273C2×C22⋊C4—1091no—
274C2×C22⋊C4—1091no—
275C2×C22⋊C4—1091no—
276C2×C22⋊C4—1091no—
277C2×C22⋊C4—1091no—
278C2×C22⋊C4—1091no—
279C2×C22⋊C4—1091no—
280C2×C22⋊C4—1091no—
281C2×C22⋊C4—1091no—
282C2×C22⋊C4C2×D10911yescharacteristic
283C2×C22⋊C4C2×D10911yescharacteristic
284C2×C22⋊C4C2×D10911yescharacteristic
285C2×C42—1091no—
286C2×C42—1091no—
287C2×C42C2×D10911yescharacteristic, complemented
288#95—1091no—
289#95—1091no—
290#95—1091no—
291#95—1091no—
292#95C2×D10911yescharacteristic, complemented
293#95C2×D10911yescharacteristic, complemented
294C24—1094no—
295C24—1091no—
296C24—1091no—
297C24—1091no—
298C24—4361no—
299C24—4361no—
300C24—4361no—
301C24—4361no—
302C24—4361no—
303C24—4361no—
304C24C22×D10911yescharacteristic
305C2×D4—1094no—
306C2×D4—1094no—
307C2×D4—1094no—
308C2×D4—4361no—
309C2×D4—4361no—
310C2×D4—4361no—
311C2×D4—4361no—
312C2×D4—4361no—
313C2×D4—4361no—
314C22×C4—1091no—
315C22×C4—1091no—
316C22×C4—1091no—
317C22×C4—1091no—
318C22×C4—1091no—
319C22×C4—4361no—
320C22×C4—4361no—
321C22×C4—4361no—
322C22×C4C22×D10911yescharacteristic
323C22×C4C22×D10911yescharacteristic
324C22×C4C22×D10911yescharacteristic
325C22×C4C22×D10911yescharacteristic
326C22×C4C22×D10911yescharacteristic
327C22×C4C22×D10911yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—4361no—
330C4⋊C4—4361no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—1094no—
334C22⋊C4—1094no—
335C22⋊C4—4361no—
336C22⋊C4—4361no—
337C22⋊C4—4361no—
338C22⋊C4—4361no—
339C22⋊C4—4361no—
340C22⋊C4—4361no—
341C22⋊C4—4361no—
342C22⋊C4—4361no—
343C22⋊C4—4361no—
344C22⋊C4—4361no—
345C22⋊C4—4361no—
346C22⋊C4—4361no—
347C22⋊C4—4361no—
348C22⋊C4C4×D10914yescomplemented
349C42—41no—
350C42—4361no—
351C42—4361no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—1092no—
359C23—1092no—
360C23—1092no—
361C23—4364no—
362C23—4364no—
363C23—4364no—
364C23—4364no—
365C23—4364no—
366C23—4364no—
367C23—4364no—
368C23—2181no—
369C23—2181no—
370C23—2181no—
371C23—2181no—
372C23—2181no—
373C23—2181no—
374C23—2181no—
375C23—2181no—
376C23—2181no—
377C23—2181no—
378C23—2181no—
379C23—2181no—
380C23—2181no—
381C23—2181no—
382C23—2181no—
383C23—2181no—
384C23—2181no—
385C23—2181no—
386C23C2×C4×D10912yes—
387C23C23×D10911yescharacteristic
388D4—4364no—
389D4—4364no—
390D4—4364no—
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—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—1092no—
408C2×C4—4364no—
409C2×C4—2181no—
410C2×C4—2181no—
411C2×C4—2181no—
412C2×C4—2181no—
413C2×C4—2181no—
414C2×C4—2181no—
415C2×C4—2181no—
416C2×C4—2181no—
417C2×C4—2181no—
418C2×C4—2181no—
419C2×C4—2181no—
420C2×C4—2181no—
421C2×C4—2181no—
422C2×C4—2181no—
423C2×C4C2×C4×D10912yes—
424C2×C4C2×C4×D10912yes—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—43616no—
433C22—4361no—
434C22—4361no—
435C22—4361no—
436C22—4361no—
437C22—4361no—
438C22—4361no—
439C22—4361no—
440C22—4361no—
441C22—4361no—
442C22—4361no—
443C22—4361no—
444C22—4361no—
445C22—2182no—
446C22—2182no—
447C22—2182no—
448C22—2182no—
449C22—2182no—
450C22—2182no—
451C22—2182no—
452C22—2182no—
453C22—2182no—
454C22C22×C4×D10911yescharacteristic
455C22#1681274411yescharacteristic
456C22#1681274411yescharacteristic
457C22#1681274511yescharacteristic
458C22#1681274511yescharacteristic
459C22#1681274511yescharacteristic
460C22#1681274511yescharacteristic
461C4—22no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—4361no—
468C4—4361no—
469C4—4361no—
470C4—4361no—
471C4—2182no—
472C2—42no—
473C2—4362no—
474C2—4362no—
475C2—4362no—
476C2#2029549811yescharacteristic
477C2#2029549811yescharacteristic
478C2#2029550011yescharacteristic
479C2#2029576411yescharacteristic
480C2#2029576611yescharacteristic
481C2#2029576711yescharacteristic
482C2#2029576811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D109
commutatorC22×C109C23×C4
Fitting#388×C109C2
socle, cosocleC23×C109C24
prime radicalC2×#16812669C2
hypercenter#388D109
hypocenterC109#2375
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #20294928C2semidirect
3. #20294927C2semidirect
4. #20294926C2semidirect
5. #20294919C2semidirect
6. #20294918C2semidirect
7. #20294917C2semidirect
8. #20294916C2semidirect
9. #20294915C2semidirect
10. #20294904C2semidirect
11. #20294904C2semidirect
12. #20294902C2semidirect
13. #20294881C2semidirect
14. #20294881C2semidirect
15. #388×C109C2semidirect
20. C22×C4×D109C22semidirect
21. #16812669C4semidirect
22. C2×C4⋊Dic109C22semidirect
23. C2×#13182008C22semidirect
36. C2×#13182006C22semidirect
37. C2×#13182006C22semidirect
38. C2×#13182006C22semidirect
39. C2×C4×Dic109C22semidirect
40. C2×C4×Dic109C22semidirect
41. #16812606C22semidirect
42. #16812606C22semidirect
43. #16812606C22semidirect
44. #16812606C22semidirect
45. C2×C4⋊C4×C109C22semidirect
49. C2×C42×C109C22semidirect
50. #95×C109C22semidirect
51. #95×C109C22semidirect
58. C23×D109C2×C4semidirect
94. #13182006C2×C4semidirect
95. #13182006C2×C4semidirect
108. C22⋊C4×C109C2×C4semidirect
242. C109#2375semidirect
252. #388D109semidirect
266. C2×C4⋊C4C2×D109semidirect
287. C2×C42C2×D109semidirect
292. #95C2×D109semidirect
293. #95C2×D109semidirect
348. C22⋊C4C4×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126692
#202949282
#202949272
#202949262
#202949192
#202949182
#202949172
#202949162
#202949152
#202949042
#202949022
#202948812
#388×C1092
#2375109