Finite Group Databasebeta

← #29572678

Subgroups — #29572678

Order 15616.

898 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
1#21445799C212yesmaximal, complemented
2C2×#16839486C212yesmaximal
3#21443662C212yesmaximal, complemented
4C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
5C2×C4⋊#92418—22no—
6C2×C22⋊#92418C2214yes—
7#16839517—22no—
8#16839486—24no—
9C24×C4×C61C411yescharacteristic
10C22×C22⋊C4×C61C411yescharacteristic
11C22×C22⋊C4×C61C2211yescharacteristic
12C22×C42×C61C411yescharacteristic
13C22×C42×C61C2211yescharacteristic
14C4×C22⋊C4×C61—22no—
15C4×C22⋊C4×C61—22no—
16C2×#95×C61C411yescharacteristic
17C2×#95×C61C2211yescharacteristic
18C22×#92418—24no—
19C4⋊#92418—44no—
20C22⋊#92418—24no—
21C22⋊#92418—24no—
22C25×C61C2×C411yescharacteristic
23C23×C4×C61—22no—
24C23×C4×C61—21no—
25C23×C4×C61—21no—
26C23×C4×C61—22no—
27C23×C4×C61—42no—
28C23×C4×C61—42no—
29C23×C4×C61C2×C411yescharacteristic
30C23×C4×C61C2×C411yescharacteristic
31C23×C4×C61C2×C411yescharacteristic
32C23×C4×C61C2×C411yescharacteristic
33C23×C4×C61C2×C411yescharacteristic
34C23×C4×C61C2311yescharacteristic
35C2×C22⋊C4×C61—22no—
36C2×C22⋊C4×C61—22no—
37C2×C22⋊C4×C61—22no—
38C2×C22⋊C4×C61—22no—
39C2×C22⋊C4×C61C814yescomplemented
40C2×C22⋊C4×C61D412yes—
41C2×C22⋊C4×C61Q812yes—
42C2×C42×C61—21no—
43C2×C42×C61—21no—
44C2×C42×C61—42no—
45C2×C42×C61—42no—
46C2×C42×C61D412yes—
47C2×C42×C61D412yes—
48#95×C61—22no—
49#95×C61—22no—
50C24×C61—21no—
51C24×C61—21no—
52C24×C61—21no—
53C24×C61—21no—
54C24×C61—41no—
55C24×C61—41no—
56C24×C61—41no—
57C24×C61—41no—
58C24×C61C4⋊C412yes—
59C24×C61C2×C812yes—
60C24×C61M1612yes—
61C24×C61C22×C411yescharacteristic
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—24no—
67C22×C4×C61—24no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61—41no—
77C22×C4×C61—41no—
78C22×C4×C61—48no—
79C22×C4×C61—48no—
80C22×C4×C61—42no—
81C22×C4×C61—42no—
82C22×C4×C61—42no—
83C22×C4×C61—42no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61C22⋊C412yes—
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22⋊C411yescharacteristic
93C22×C4×C61C4⋊C412yes—
94C22×C4×C61C2×C812yes—
95C22×C4×C61C2×C812yes—
96C22×C4×C61M1612yes—
97C22×C4×C61M1612yes—
98C22×C4×C61C2×D412yes—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×Q811yescharacteristic
101C22×C4×C61#4112yes—
102C22⋊C4×C61—28no—
103C22⋊C4×C61—24no—
104C22⋊C4×C61—24no—
105C42×C61—24no—
106C42×C61—42no—
107C42×C61—42no—
108C42×C61—42no—
109C2×#92418—44no—
110C2×#92418—44no—
111C2×#92418—44no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—22no—
117C23×C61—22no—
118C23×C61—22no—
119C23×C61—22no—
120C23×C61—22no—
121C23×C61—22no—
122C23×C61—42no—
123C23×C61—42no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61—42no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61—42no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61C4⋊C814yescomplemented
139C23×C61C2×C22⋊C411yescharacteristic
140C23×C61C2×C4⋊C411yescharacteristic
141C23×C61#11711yescharacteristic
142C23×C61C22×C811yescharacteristic
143C23×C61C2×M1611yescharacteristic
144C23×C61C2×M1611yescharacteristic
145C23×C61C2×M1611yescharacteristic
146C2×C4×C61—24no—
147C2×C4×C61—24no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61—21no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—48no—
165C2×C4×C61—48no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61C22⋊C814yes—
179C2×C4×C61#9912yes—
180C2×C4×C61#10012yes—
181C2×C4×C61#12112yes—
182C2×C4×C61C22⋊Q812yes—
183#92418—84no—
184#8559—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—24no—
198C22×C61—44no—
199C22×C61—44no—
200C22×C61—44no—
201C22×C61—44no—
202C22×C61—42no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61#38011yescharacteristic
209C22×C61C2×C22⋊C811yescharacteristic
210C22×C61#40711yescharacteristic
211C22×C61C2×#9911yescharacteristic
212C22×C61#41011yescharacteristic
213C22×C61C2×#10011yescharacteristic
214C22×C61#41311yescharacteristic
215C22×C61C2×C4⋊C811yescharacteristic
216C22×C61#42311yescharacteristic
217C22×C61#43111yescharacteristic
218C22×C61#43211yescharacteristic
219C4×C61—22no—
220C4×C61—22no—
221C4×C61—24no—
222C4×C61—22no—
223C4×C61—22no—
224C4×C61—42no—
225C4×C61—42no—
226C4×C61—42no—
227C2×C4×C22⋊C4D6111yescharacteristic
228#1797—612no—
229C2×#323—612no—
230#1280—612no—
231C2×C61—21no—
232C2×C61—21no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—44no—
236C2×C61#145911yescharacteristic
237C2×C61#146011yescharacteristic
238C2×C61#146211yescharacteristic
239C2×C61#146411yescharacteristic
240C2×C61#179711yescharacteristic
241C2×C61#179911yescharacteristic
242C2×C61#180011yescharacteristic
243C24×C4Dic6111yescharacteristic
244C22×C22⋊C4Dic6111yescharacteristic
245C22×C22⋊C4C2×D6111yescharacteristic
246C22×C42Dic6111yescharacteristic
247C22×C42C2×D6111yescharacteristic
248C2×C4⋊C8—1222no—
249C2×C22⋊C8—614no—
250C4×C22⋊C4—22no—
251C4×C22⋊C4—22no—
252C2×#95Dic6111yescharacteristic
253C2×#95C2×D6111yescharacteristic
254#336—1222no—
255#323—1224no—
256C61#855911yescharacteristic, complemented, Sylow
257C25C2×Dic6111yescharacteristic
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—42no—
263C23×C4—42no—
264C23×C4C2×Dic6111yescharacteristic
265C23×C4C2×Dic6111yescharacteristic
266C23×C4C2×Dic6111yescharacteristic
267C23×C4C2×Dic6111yescharacteristic
268C23×C4C2×Dic6111yescharacteristic
269C23×C4C22×D6111yescharacteristic
270C22×C8—1224no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4#9241814yescomplemented
276C2×C22⋊C4Dic12212yes—
277C2×C22⋊C4D24412yes—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42C22⋊D6112yes—
283C2×C42C22⋊D6112yes—
284C4⋊C8—2444no—
285C22⋊C8—1224no—
286C22⋊C8—1224no—
287#95—22no—
288#95—22no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24C2×#9241812yes—
298C24#1175819412yes—
299C24C4⋊Dic6112yes—
300C24C22×Dic6111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—48no—
318C22×C4—48no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4C2×#9241812yes—
330C22×C4C2×#9241812yes—
331C22×C4#1175819412yes—
332C22×C4#1175819412yes—
333C22×C4C4⋊Dic6112yes—
334C22×C4C22⋊Dic6112yes—
335C22×C4C22⋊Dic6111yescharacteristic
336C22×C4C22⋊Dic6111yescharacteristic
337C22×C4C2×Dic12211yescharacteristic
338C22×C4C2×D24411yescharacteristic
339C22×C4#1175822212yes—
340C22×C4C2×C22⋊D6112yes—
341C2×C8—2444no—
342C2×C8—2444no—
343C2×C8—2444no—
344C22⋊C4—28no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C42—24no—
348C42—42no—
349C42—42no—
350C42—42no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23C4⋊#9241814yescomplemented
378C23C2×C22⋊Dic6111yescharacteristic
379C23C2×C4⋊Dic6111yescharacteristic
380C23#1451973611yescharacteristic
381C23C22×#9241811yescharacteristic
382C23C2×#1175819411yescharacteristic
383C23C2×#1175819411yescharacteristic
384C23C2×#1175819411yescharacteristic
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4C22⋊#9241814yes—
418C2×C4#1451969212yes—
419C2×C4#1451969412yes—
420C2×C4#1451975112yes—
421C2×C4#1451975612yes—
422C8—4884no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—24no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22#1683962211yescharacteristic
447C22C2×C22⋊#9241811yescharacteristic
448C22#1683972611yescharacteristic
449C22C2×#1451969211yescharacteristic
450C22#1683973611yescharacteristic
451C22C2×#1451969411yescharacteristic
452C22#1683974611yescharacteristic
453C22C2×C4⋊#9241811yescharacteristic
454C22#1683978611yescharacteristic
455C22#1683981511yescharacteristic
456C22#1683981911yescharacteristic
457C4—24no—
458C4—22no—
459C4—22no—
460C4—22no—
461C4—22no—
462C4—42no—
463C4—42no—
464C4—42no—
465C2—21no—
466C2—21no—
467C2—21no—
468C2—21no—
469C2—44no—
470C2#2144413711yescharacteristic
471C2#2144414011yescharacteristic
472C2#2144414611yescharacteristic
473C2#2144415311yescharacteristic
474C2#2144579911yescharacteristic
475C2#2144580711yescharacteristic
476C2#2144581111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC23×C61C22×C8
FrattiniC23×C4C22×D61
FittingC2×C4×C22⋊C4×C61C2
socleC23×C61C2×C22⋊C4
prime radicalC25×C61C2×C4
cosocleC23×C4×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#8559
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #21445799C2semidirect
3. #21443662C2semidirect
39. C2×C22⋊C4×C61C8semidirect
138. C23×C61C4⋊C8semidirect
256. C61#8559semidirect
275. C2×C22⋊C4#92418semidirect
377. C23C4⋊#92418semidirect

Maximal subgroups

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

subgroupindex
#214457992
C2×#168394862
#214436622
C2×C4×C22⋊C4×C612
#855961