Finite Group Databasebeta

← #30600204

Subgroups — #30600204

Order 15616.

712 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840783C211yesmaximal, characteristic, complemented
2#21461710C211yesmaximal, characteristic, complemented
3C2×#16840606C212yesmaximal
4#21460657C212yesmaximal, complemented
5C2×#16839652C211yesmaximal, characteristic
6#16840783—24no—
7C2×C4⋊#92420—21no—
8C2×C22⋊#92420C2212yes—
9C2×C22⋊#92420C2212yes—
10C2×#14519832—21no—
11#16840628—21no—
12#16840628—21no—
13#16840606—24no—
14C22×#11758198C411yescharacteristic
15C22×#11758198C411yescharacteristic
16C22×#11758198C2211yescharacteristic
17C22×C4×Dic61C2211yescharacteristic
18#16839652—21no—
19#16839652—21no—
20#16839652—21no—
21#16839652—21no—
22C2×#14519684C411yescharacteristic
23C2×#14519684C2211yescharacteristic
24C22×C4⋊C4×C61C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27C4⋊#92420—24no—
28C22⋊#92420—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C22⋊#92420D414yes—
32#14519832—24no—
33C24×D61C2×C411yescharacteristic
34C23×Dic61C2×C411yescharacteristic
35C23×Dic61C2×C411yescharacteristic
36C23×Dic61C2311yescharacteristic
37C2×#11758198—22no—
38C2×#11758198—22no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—22no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—22no—
47C2×#11758198—42no—
48C2×C4×Dic61—22no—
49C2×C4×Dic61—22no—
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58C23×C4×C61C2×C411yescharacteristic
59C23×C4×C61C2×C411yescharacteristic
60C23×C4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C4⋊C4×C61—42no—
64C24×C61C22×C411yescharacteristic
65C23×D61—21no—
66C23×D61—21no—
67C23×D61—22no—
68C23×D61—22no—
69C23×D61—41no—
70C23×D61—41no—
71C23×D61—41no—
72C23×D61—41no—
73C23×D61M1612yes—
74C22×C4×C61—21no—
75C22×C4×C61—22no—
76C22×C4×C61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—21no—
79C22×C4×C61—22no—
80C22×C4×C61—41no—
81C22×C4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61C22⋊C411yescharacteristic
86C22×C4×C61C22⋊C411yescharacteristic
87C22×C4×C61M1612yes—
88C22×Dic61—22no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—22no—
92C22×Dic61—21no—
93C22×Dic61—22no—
94C22×Dic61—41no—
95C22×Dic61—41no—
96C22×Dic61—41no—
97C22×Dic61—41no—
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C2×D411yescharacteristic
101C22×Dic61C2×D411yescharacteristic
102C22×Dic61C2×D412yes—
103C22×Dic61#4112yes—
104C2×#92420—24no—
105C2×#92420—42no—
106C2×#92420—42no—
107C2×#92420—42no—
108C2×#92420—42no—
109C2×#92420—42no—
110C4⋊C4×C61—42no—
111C4⋊C4×C61—42no—
112#11758198—24no—
113#11758198—24no—
114#11758198—42no—
115#11758198—42no—
116#11758198—41no—
117#11758198—42no—
118#11758198—41no—
119C4×Dic61—21no—
120C4×Dic61—21no—
121C4×Dic61—22no—
122C4×Dic61—22no—
123C4×Dic61—21no—
124C4×Dic61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—21no—
129C23×C61C2×C22⋊C411yescharacteristic
130C23×C61C4×D412yes—
131C23×C61C2×M1611yescharacteristic
132C23×C61C2×M1611yescharacteristic
133C23×C61#13112yes—
134C22×D61—22no—
135C22×D61—24no—
136C22×D61—42no—
137C22×D61—42no—
138C22×D61—41no—
139C22×D61—41no—
140C22×D61—41no—
141C22×D61—41no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—42no—
145C22×D61—42no—
146C22×D61—42no—
147C22×D61—82no—
148C22×D61—82no—
149C22×D61—82no—
150C22×D61—82no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—22no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—44no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—82no—
166C2×C4×C61#10012yes—
167C2×Dic61—21no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—82no—
185C2×Dic61#12011yescharacteristic
186C2×Dic61#12112yes—
187C2×Dic61#12311yescharacteristic
188#92420—44no—
189#92420—82no—
190#9084—611nomaximal, Sylow
191C22×C61—22no—
192C22×C61—22no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—22no—
198C22×C61—22no—
199C22×C61#38611yescharacteristic
200C22×C61#40711yescharacteristic
201C22×C61#40811yescharacteristic
202C22×C61#41011yescharacteristic
203C22×C61#41011yescharacteristic
204C22×C61C2×#10011yescharacteristic
205C22×C61#41311yescharacteristic
206C22×C61#43512yes—
207C22×C61#43612yes—
208C2×D61—42no—
209C2×D61—42no—
210C2×D61—42no—
211C2×D61—84no—
212C2×D61—81no—
213C2×D61—81no—
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—84no—
217C4×C61—22no—
218C4×C61—41no—
219C4×C61—42no—
220C4×C61—81no—
221Dic61—21no—
222Dic61—22no—
223Dic61—21no—
224Dic61—41no—
225Dic61—41no—
226Dic61—81no—
227C2×#432—611no—
228C2×#388—611no—
229#1817—611no—
230C2×#323—612no—
231#1280—612no—
232C2×C61—22no—
233C2×C61—22no—
234C2×C61#146411yescharacteristic
235C2×C61#146611yescharacteristic
236C2×C61#146911yescharacteristic
237C2×C61#147311yescharacteristic
238C2×C61#187011yescharacteristic
239C2×C61#189911yescharacteristic
240C2×C61#190011yescharacteristic
241D61—82no—
242C22×C4⋊C4#692311yescharacteristic
243C22×C22⋊C4—611no—
244C22×C22⋊C4—611no—
245C22×C22⋊C4—611no—
246C22×C42—611no—
247#432—1224no—
248C2×C4⋊C8—1221no—
249C2×C22⋊C8—612no—
250C2×C22⋊C8—612no—
251C2×#97—1221no—
252#388—1221no—
253#388—1221no—
254#388—1221no—
255#388—1221no—
256C2×#95—611no—
257C2×#95—611no—
258#336—1221no—
259#336—1221no—
260#323—1224no—
261C61#908411yescharacteristic, complemented, Sylow
262C25—611no—
263C23×C4—611no—
264C23×C4—611no—
265C23×C4—611no—
266C23×C4C2×#692311yescharacteristic
267C23×C4C2×#692311yescharacteristic
268C23×C4C2×#692311yescharacteristic
269C22×C8—1222no—
270C22×C8—1222no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—42no—
274C2×C22⋊C4—1221no—
275C2×C22⋊C4—1221no—
276C2×C22⋊C4—1222no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1222no—
279C2×C22⋊C4—1221no—
280C2×C22⋊C4—1221no—
281C2×C22⋊C4—1222no—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4—2442no—
285C2×C42—611no—
286C2×C42—611no—
287C2×C42—611no—
288C2×C42—611no—
289C2×C42—1222no—
290C2×C42—1222no—
291C4⋊C8—1224no—
292C22⋊C8—614no—
293C22⋊C8—1222no—
294C22⋊C8—1222no—
295C22⋊C8—1222no—
296#97—1224no—
297#95—1221no—
298#95—1221no—
299#95—1221no—
300#95—1221no—
301C24—612no—
302C24—1222no—
303C24—1222no—
304C24—1221no—
305C24—1221no—
306C24—2441no—
307C24—2441no—
308C24—2441no—
309C24—2441no—
310C24C22×#692311yescharacteristic
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—611no—
323C22×C4—611no—
324C22×C4—612no—
325C22×C4—612no—
326C22×C4—611no—
327C22×C4—611no—
328C22×C4—1222no—
329C22×C4—1221no—
330C22×C4—1221no—
331C22×C4—1222no—
332C22×C4—1222no—
333C22×C4—1221no—
334C22×C4—2441no—
335C22×C4—2441no—
336C22×C4—2441no—
337C22×C4—2441no—
338C22×C4#1175821312yes—
339C22×C4C22⋊#692311yescharacteristic
340C22×C4C22⋊#692311yescharacteristic
341C2×C8—1224no—
342C2×C8—2442no—
343C2×C8—2442no—
344C2×C8—2442no—
345C2×C8—2442no—
346C2×C8—2442no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C22⋊C4—1224no—
350C22⋊C4—1224no—
351C22⋊C4—2442no—
352C22⋊C4—2441no—
353C22⋊C4—2442no—
354C22⋊C4—2441no—
355C22⋊C4—2442no—
356C42—1221no—
357C42—1221no—
358C42—1222no—
359C42—1221no—
360C42—1222no—
361C42—1221no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—1224no—
367C23—1222no—
368C23—4882no—
369C23—4882no—
370C23—4882no—
371C23—4882no—
372C23—2442no—
373C23—2442no—
374C23—2442no—
375C23—2442no—
376C23—2441no—
377C23—2441no—
378C23—2441no—
379C23—2441no—
380C23—2441no—
381C23—2441no—
382C23—2442no—
383C23C2×C22⋊#692311yescharacteristic
384C23D4×#692312yes—
385C23D61⋊M1611yescharacteristic
386C23C2×#1175821311yescharacteristic
387C23#1451986512yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—82no—
403C2×C4—611no—
404C2×C4—612no—
405C2×C4—611no—
406C2×C4—1221no—
407C2×C4—1221no—
408C2×C4—1221no—
409C2×C4—1221no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—1221no—
416C2×C4—4882no—
417C2×C4—2442no—
418C2×C4—2442no—
419C2×C4—2441no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4#1451983712yes—
425C8—4882no—
426C8—2444no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—22no—
435C22—4884no—
436C22—4884no—
437C22—4881no—
438C22—4881no—
439C22—4881no—
440C22—4881no—
441C22—2442no—
442C22—2442no—
443C22—2442no—
444C22#1684071011yescharacteristic
445C22#1684073311yescharacteristic
446C22#1684073711yescharacteristic
447C22#1684074111yescharacteristic
448C22#1684074311yescharacteristic
449C22C2×#1451983711yescharacteristic
450C22#1684074911yescharacteristic
451C22#1684079212yes—
452C22#1684079612yes—
453C4—22no—
454C4—41no—
455C4—42no—
456C4—81no—
457C4—1222no—
458C4—1221no—
459C4—1221no—
460C4—4881no—
461C4—2441no—
462C4—2441no—
463C2—22no—
464C2—22no—
465C2—4882no—
466C2#2146107311yescharacteristic
467C2#2146107811yescharacteristic
468C2#2146109011yescharacteristic
469C2#2146110211yescharacteristic
470C2#2146179911yescharacteristic
471C2#2146184311yescharacteristic
472C2#2146184411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61C2×C4
cosocleC23×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#9084
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16840783C2semidirect
2. #21461710C2semidirect
4. #21460657C2semidirect
261. C61#9084semidirect

Maximal subgroups

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

subgroupindex
C2×#168407832
#214617102
C2×#168406062
#214606572
C2×#168396522
#908461