Finite Group Databasebeta

← #29572676

Subgroups — #29572676

Order 15616.

898 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×Dic61C211yesmaximal, characteristic
2#21445797C212yesmaximal, complemented
3C2×#16839486C211yesmaximal, characteristic
4#21443661C212yesmaximal, complemented
5C2×C61×#323C211yesmaximal, characteristic
6C24×Dic61C411yescharacteristic
7C22×C22⋊Dic61C411yescharacteristic
8C22×C4×Dic61C411yescharacteristic
9C22×C4×Dic61C2211yescharacteristic
10C2×#14519706—22no—
11C2×C22⋊#92418C2212yes—
12C22⋊C4×Dic61—22no—
13C22⋊C4×Dic61—22no—
14C2×#14519684C411yescharacteristic
15C2×#14519684C2211yescharacteristic
16#16839516—22no—
17#16839486—22no—
18C22×C22⋊C4×C61C2211yescharacteristic
19C2×C22⋊C8×C61C2212yes—
20C61×#323—22no—
21C23×Dic61—22no—
22C23×Dic61—21no—
23C23×Dic61—21no—
24C23×Dic61—22no—
25C23×Dic61—42no—
26C23×Dic61—42no—
27C23×Dic61C2×C411yescharacteristic
28C23×Dic61C2×C411yescharacteristic
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2×C411yescharacteristic
31C22×#92418—22no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—22no—
34C2×C22⋊Dic61C814yescomplemented
35C2×C4×Dic61—21no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61—42no—
38C2×C4×Dic61—42no—
39C2×C4×Dic61D412yes—
40C2×C4×Dic61D412yes—
41#14519706—44no—
42C22⋊#92418—22no—
43C22⋊#92418—22no—
44#14519684—22no—
45#14519684—22no—
46C25×C61C2×C411yescharacteristic
47C23×C4×C61C2×C411yescharacteristic
48C23×C4×C61C2311yescharacteristic
49C22×C8×C61—22no—
50C2×C22⋊C4×C61—22no—
51C2×C22⋊C4×C61—22no—
52C2×C22⋊C4×C61D412yes—
53C2×C22⋊C4×C61Q812yes—
54C22⋊C8×C61—22no—
55C22⋊C8×C61—22no—
56C24×C61—21no—
57C24×C61—21no—
58C24×C61—21no—
59C24×C61—21no—
60C24×C61—41no—
61C24×C61—41no—
62C24×C61—41no—
63C24×C61—41no—
64C24×C61C4⋊C412yes—
65C24×C61C2×C812yes—
66C24×C61M1612yes—
67C24×C61C22×C411yescharacteristic
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—41no—
71C22×C4×C61—41no—
72C22×C4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61C4⋊C412yes—
75C22×C4×C61C2×D412yes—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×Q811yescharacteristic
78C22×C4×C61#4112yes—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—22no—
82C22×Dic61—22no—
83C22×Dic61—24no—
84C22×Dic61—24no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—41no—
90C22×Dic61—41no—
91C22×Dic61—48no—
92C22×Dic61—48no—
93C22×Dic61—42no—
94C22×Dic61—42no—
95C22×Dic61—42no—
96C22×Dic61—42no—
97C22×Dic61—41no—
98C22×Dic61—41no—
99C22×Dic61—41no—
100C22×Dic61—41no—
101C22×Dic61C22⋊C412yes—
102C22×Dic61C22⋊C411yescharacteristic
103C22×Dic61C22⋊C411yescharacteristic
104C22×Dic61C2×C812yes—
105C22×Dic61C2×C812yes—
106C22×Dic61M1612yes—
107C22×Dic61M1612yes—
108C2×C8×C61—42no—
109C2×C8×C61—42no—
110C2×C8×C61—42no—
111C22⋊C4×C61—24no—
112C22⋊C4×C61—24no—
113C22⋊Dic61—28no—
114C4×Dic61—24no—
115C4×Dic61—42no—
116C4×Dic61—42no—
117C4×Dic61—42no—
118C2×#92418—42no—
119C2×#92418—42no—
120C2×#92418—42no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—21no—
124C23×C61—21no—
125C23×C61—22no—
126C23×C61—22no—
127C23×C61—22no—
128C23×C61—22no—
129C23×C61—22no—
130C23×C61—22no—
131C23×C61—42no—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61—42no—
135C23×C61—42no—
136C23×C61—42no—
137C23×C61—42no—
138C23×C61—42no—
139C23×C61—42no—
140C23×C61—42no—
141C23×C61—42no—
142C23×C61—42no—
143C23×C61—41no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61C4⋊C814yescomplemented
148C23×C61C2×C22⋊C411yescharacteristic
149C23×C61C2×C4⋊C411yescharacteristic
150C23×C61#11711yescharacteristic
151C23×C61C22×C811yescharacteristic
152C23×C61C2×M1611yescharacteristic
153C23×C61C2×M1611yescharacteristic
154C23×C61C2×M1611yescharacteristic
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61#12112yes—
166C2×C4×C61C22⋊Q812yes—
167C2×Dic61—24no—
168C2×Dic61—24no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—22no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—22no—
175C2×Dic61—22no—
176C2×Dic61—22no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—48no—
180C2×Dic61—48no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61C22⋊C814yes—
190C2×Dic61#9912yes—
191C2×Dic61#10012yes—
192C8×C61—82no—
193#92418—82no—
194#8559—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—24no—
208C22×C61—44no—
209C22×C61—44no—
210C22×C61—44no—
211C22×C61—44no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—42no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61#38011yescharacteristic
219C22×C61C2×C22⋊C811yescharacteristic
220C22×C61#40711yescharacteristic
221C22×C61C2×#9911yescharacteristic
222C22×C61#41011yescharacteristic
223C22×C61C2×#10011yescharacteristic
224C22×C61#41311yescharacteristic
225C22×C61C2×C4⋊C811yescharacteristic
226C22×C61#42311yescharacteristic
227C22×C61#43111yescharacteristic
228C22×C61#43211yescharacteristic
229C4×C61—22no—
230C4×C61—22no—
231C4×C61—42no—
232Dic61—24no—
233Dic61—22no—
234Dic61—22no—
235Dic61—42no—
236Dic61—42no—
237C2×C4×C22⋊C4—611no—
238#1797—612no—
239C2×#323—611no—
240C2×#323D6111yescharacteristic
241#1280—612no—
242C2×C61—21no—
243C2×C61—21no—
244C2×C61—21no—
245C2×C61—21no—
246C2×C61—44no—
247C2×C61#145911yescharacteristic
248C2×C61#146011yescharacteristic
249C2×C61#146211yescharacteristic
250C2×C61#146411yescharacteristic
251C2×C61#179711yescharacteristic
252C2×C61#179911yescharacteristic
253C2×C61#180011yescharacteristic
254C24×C4—611no—
255C22×C22⋊C4—611no—
256C22×C22⋊C4C2×D6111yescharacteristic
257C22×C42—611no—
258C22×C42—611no—
259C2×C4⋊C8—1222no—
260C2×C22⋊C8—612no—
261C2×C22⋊C8C2×D6112yes—
262C4×C22⋊C4—1222no—
263C4×C22⋊C4—1222no—
264C2×#95—611no—
265C2×#95—611no—
266#336—1222no—
267#323—22no—
268#323—1222no—
269C61#855911yescharacteristic, complemented, Sylow
270C25C4×D6111yescharacteristic
271C23×C4—611no—
272C23×C4—611no—
273C23×C4—611no—
274C23×C4—611no—
275C23×C4—1222no—
276C23×C4—1221no—
277C23×C4—1221no—
278C23×C4—1222no—
279C23×C4—2442no—
280C23×C4—2442no—
281C23×C4C4×D6111yescharacteristic
282C23×C4C22×D6111yescharacteristic
283C22×C8—22no—
284C22×C8—1222no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—614no—
288C2×C22⋊C4—1222no—
289C2×C22⋊C4—1222no—
290C2×C22⋊C4Dic12212yes—
291C2×C22⋊C4C22⋊D6112yes—
292C2×C42—612no—
293C2×C42—612no—
294C2×C42—1221no—
295C2×C42—1221no—
296C2×C42—2442no—
297C2×C42—2442no—
298C4⋊C8—2444no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—1222no—
302C22⋊C8—1222no—
303#95—1222no—
304#95—1222no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24C8×D6112yes—
314C24#1175818912yes—
315C24#1175819612yes—
316C24C2×C4×D6111yescharacteristic
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—612no—
324C22×C4—611no—
325C22×C4—611no—
326C22×C4—612no—
327C22×C4—612no—
328C22×C4—612no—
329C22×C4—612no—
330C22×C4—1222no—
331C22×C4—1222no—
332C22×C4—1221no—
333C22×C4—1221no—
334C22×C4—1224no—
335C22×C4—1224no—
336C22×C4—1221no—
337C22×C4—1221no—
338C22×C4—1221no—
339C22×C4—1221no—
340C22×C4—2442no—
341C22×C4—2442no—
342C22×C4—2441no—
343C22×C4—2441no—
344C22×C4—2448no—
345C22×C4—2448no—
346C22×C4—2442no—
347C22×C4—2442no—
348C22×C4—2441no—
349C22×C4—2441no—
350C22×C4—2441no—
351C22×C4—2441no—
352C22×C4#1175819612yes—
353C22×C4C2×Dic12211yescharacteristic
354C22×C4D4×D6112yes—
355C22×C4#1175822412yes—
356C22×C4C2×C22⋊D6111yescharacteristic
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—2442no—
361C2×C8—2442no—
362C2×C8—2442no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—1228no—
366C42—1224no—
367C42—2442no—
368C42—2442no—
369C42—2442no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23#1451970614yescomplemented
397C23C22⋊C4×D6111yescharacteristic
398C23C2×#1175819611yescharacteristic
399C23#1451973311yescharacteristic
400C23C2×C8×D6111yescharacteristic
401C23M16×D6111yescharacteristic
402C23M16×D6111yescharacteristic
403C23C2×#1175818911yescharacteristic
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—614no—
415C2×C4—612no—
416C2×C4—612no—
417C2×C4—1224no—
418C2×C4—1224no—
419C2×C4—1221no—
420C2×C4—1221no—
421C2×C4—1222no—
422C2×C4—1221no—
423C2×C4—1221no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—1222no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2448no—
430C2×C4—2448no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2441no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4#1451974912yes—
440C2×C4#1451975412yes—
441C8—82no—
442C8—4882no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—24no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22#1683962011yescharacteristic
467C22C22⋊C8×D6111yescharacteristic
468C22#1683972411yescharacteristic
469C22#99×D6111yescharacteristic
470C22#1683973411yescharacteristic
471C22#100×D6111yescharacteristic
472C22#1683974311yescharacteristic
473C22C2×#1451970611yescharacteristic
474C22#1683978511yescharacteristic
475C22#1683981211yescharacteristic
476C22#1683981611yescharacteristic
477C4—22no—
478C4—22no—
479C4—42no—
480C4—1224no—
481C4—1222no—
482C4—1222no—
483C4—2442no—
484C4—2442no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—44no—
490C2#2144413411yescharacteristic
491C2#2144413811yescharacteristic
492C2#2144414411yescharacteristic
493C2#2144415011yescharacteristic
494C2#2144579711yescharacteristic
495C2#2144580511yescharacteristic
496C2#2144580911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC23×C61C22×C8
FrattiniC23×C4C22×D61
FittingC2×C61×#323C2
socleC23×C61C2×C22⋊C4
prime radicalC25×C61C2×C4
cosocleC23×C4×C61C23
hypercenterC2×#323D61
hypocenterC61#8559
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #21445797C2semidirect
4. #21443661C2semidirect
34. C2×C22⋊Dic61C8semidirect
147. C23×C61C4⋊C8semidirect
269. C61#8559semidirect
396. C23#14519706semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×Dic612
#214457972
C2×#168394862
#214436612
C2×C61×#3232
#855961