Finite Group Databasebeta

← #30603113

Subgroups — #30603113

Order 15616.

1634 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1634 conjugacy classes of subgroups — too many to draw. Only its 172 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
1C22×#14519835C212yesmaximal
2C2×#16840705C212yesmaximal, complemented
3C2×#16840640C212yesmaximal, complemented
4C22×#14519759C211yesmaximal, characteristic
5C22×C22⋊#6923C2214yes—
6C2×#14519835—28no—
7C2×#14519835—24no—
8#16840705—28no—
9C2×(C2×#6923)⋊C4—24no—
10#16840640—24no—
11C23×C22⋊D61C2211yescharacteristic
12C23×C4×D61C2211yescharacteristic
13C2×#14519759—24no—
14C2×#14519759—24no—
15C2×#14519759—24no—
16C22×C4⋊Dic61C411yescharacteristic, complemented
17C22×#11758196C411yescharacteristic, complemented
18C22×C22⋊Dic61C411yescharacteristic, complemented
19C22×#11758198C2211yescharacteristic
20C22×C22⋊C4×C61C411yescharacteristic, complemented
21C23×#6923—24no—
22C2×C22⋊#6923—24no—
23C2×C22⋊#6923—24no—
24C2×C22⋊#6923—24no—
25C2×C22⋊#6923—24no—
26C2×C22⋊#6923—24no—
27C2×C22⋊#6923D418yes—
28#14519835—216no—
29(C2×#6923)⋊C4—216no—
30C24×D61C2311yescharacteristic
31C22×C22⋊D61—24no—
32C22×C22⋊D61—22no—
33C22×C22⋊D61—22no—
34C22×C22⋊D61—24no—
35C22×C22⋊D61D414yes—
36C23×Dic61C2×C411yescharacteristic
37C23×Dic61C2×C411yescharacteristic
38C23×Dic61C2×C411yescharacteristic
39C22×C4×D61—24no—
40C22×C4×D61—22no—
41C22×C4×D61—22no—
42C22×C4×D61—24no—
43C22×C4×D61D414yes—
44#14519759—232no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—22no—
47C2×C4⋊Dic61—22no—
48C2×#11758196—22no—
49C2×#11758196—22no—
50C2×#11758196—22no—
51C2×C22⋊Dic61—22no—
52C2×C22⋊Dic61—22no—
53C2×C22⋊Dic61—22no—
54C2×#11758198—22no—
55C2×#11758198—22no—
56C2×#11758198D414yes—
57C25×C61C2×C411yescharacteristic
58C23×C4×C61C2×C411yescharacteristic
59C23×C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61—22no—
61C2×C22⋊C4×C61—22no—
62C2×C22⋊C4×C61—22no—
63C24×C61—21no—
64C24×C61—21no—
65C24×C61—24no—
66C24×C61—42no—
67C24×C61—42no—
68C24×C61C22⋊C412yes—
69C24×C61C22×C411yescharacteristic
70C23×D61—42no—
71C23×D61—42no—
72C23×D61C2×D412yes—
73C23×D61C2×D412yes—
74C23×D61C2×D414yes—
75C23×D61C2×D412yes—
76C23×D61#4114yes—
77C22×#6923—28no—
78C22×#6923—48no—
79C22×#6923—48no—
80C22×#6923—44no—
81C22×#6923—44no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—24no—
87C22×C4×C61—42no—
88C22×C4×C61C22⋊C412yes—
89C22×C4×C61C22⋊C412yes—
90C2×C22⋊D61—24no—
91C2×C22⋊D61—28no—
92C2×C22⋊D61—28no—
93C2×C22⋊D61—24no—
94C2×C22⋊D61—42no—
95C2×C22⋊D61—44no—
96C2×C22⋊D61—44no—
97C2×C22⋊D61—42no—
98C2×C22⋊D61—42no—
99C2×C22⋊D61—42no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—24no—
107C22×Dic61—24no—
108C22×Dic61—42no—
109C22×Dic61C22⋊C412yes—
110C22×Dic61C22⋊C412yes—
111C22×Dic61C22⋊C412yes—
112C2×C4×D61—24no—
113C2×C4×D61—28no—
114C2×C4×D61—28no—
115C2×C4×D61—24no—
116C2×C4×D61—42no—
117C2×C4×D61—44no—
118C2×C4×D61—44no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×C4×D61—42no—
122C22⋊#6923—28no—
123C22⋊#6923—28no—
124C22⋊#6923—44no—
125C22⋊#6923—44no—
126C22⋊C4×C61—28no—
127C22⋊Dic61—28no—
128#11758198—28no—
129C4⋊Dic61—28no—
130#11758196—28no—
131C23×C61—22no—
132C23×C61—22no—
133C23×C61—22no—
134C23×C61—24no—
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—88no—
144C23×C61C2×C22⋊C411yescharacteristic
145C23×C61C2×C22⋊C411yescharacteristic
146C23×C61C2×C22⋊C411yescharacteristic
147C23×C61C4×D414yes—
148C22×D61—24no—
149C22×D61—24no—
150C22×D61—48no—
151C22×D61—42no—
152C22×D61—42no—
153C22×D61—42no—
154C22×D61—42no—
155C22×D61—48no—
156C22×D61#12012yes—
157C22×D61#12014yes—
158C22×D61#12012yes—
159C22×D61#12114yes—
160C22×D61#12114yes—
161C22×D61#12114yes—
162C22×D61#12312yes—
163C22×D61#12312yes—
164C22×D61#12412yes—
165C22×D61#12712yes—
166C2×#6923—416no—
167C2×#6923—48no—
168C2×#6923—88no—
169C2×#6923—84no—
170C2×#6923—84no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—24no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C22⋊D61—48no—
180C22⋊D61—48no—
181C2×Dic61—22no—
182C2×Dic61—22no—
183C2×Dic61—22no—
184C2×Dic61—24no—
185C2×Dic61—24no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C4×D61—48no—
194C4×D61—48no—
195#10966—611nomaximal, Sylow
196C22×C61—28no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—42no—
202C22×C61—42no—
203C22×C61—42no—
204C22×C61—84no—
205C22×C61—84no—
206C22×C61—84no—
207C22×C61#37911yescharacteristic
208C22×C61#38612yes—
209C22×C61#38612yes—
210C22×C61#39012yes—
211C22×C61#41012yes—
212C22×C61#41012yes—
213C2×D61—24no—
214C2×D61—28no—
215C2×D61—24no—
216C2×D61—24no—
217C2×D61—24no—
218C2×D61—42no—
219C2×D61—44no—
220C2×D61—44no—
221C2×D61—42no—
222C2×D61—42no—
223C2×D61—42no—
224C2×D61#39214yes—
225C2×D61#39414yes—
226#6923—88no—
227C4×C61—42no—
228C4×C61—42no—
229Dic61—42no—
230Dic61—42no—
231Dic61—42no—
232C22×#123—611no—
233C22×#99—612no—
234C2×#381—612no—
235C2×#342—612no—
236C2×C61—22no—
237C2×C61—22no—
238C2×C61—82no—
239C2×C61#143811yescharacteristic
240C2×C61#178211yescharacteristic
241C2×C61#178211yescharacteristic
242C2×C61#189914yes—
243D61—24no—
244D61—24no—
245C23×D4—611no—
246C24×C4—611no—
247C2×#123—1224no—
248C2×#123—1224no—
249C2×#123—1224no—
250C22×C4⋊C4—611no—
251C22×C4⋊C4—611no—
252C22×C22⋊C4—611no—
253C22×C22⋊C4—614no—
254C22×C22⋊C4—611no—
255C22×C22⋊C4#692311yescharacteristic, complemented
256C2×#99—1228no—
257C2×#99—1224no—
258#381—1228no—
259C2×#95—1224no—
260#342—1224no—
261C61#1096611yescharacteristic, complemented, Sylow
262C25—611no—
263C25C2×#692311yescharacteristic
264C22×D4—614no—
265C22×D4—1222no—
266C22×D4—1222no—
267C22×D4—1224no—
268C22×D4—1224no—
269C23×C4—611no—
270C23×C4—611no—
271C23×C4—611no—
272C23×C4—614no—
273C23×C4—1222no—
274C23×C4—1222no—
275C23×C4—1224no—
276C23×C4—1224no—
277C23×C4—1224no—
278C23×C4C2×#692311yescharacteristic
279C23×C4C2×#692311yescharacteristic
280#123—12232no—
281C2×C4⋊C4—1222no—
282C2×C4⋊C4—1222no—
283C2×C4⋊C4—1222no—
284C2×C4⋊C4—1222no—
285C2×C4⋊C4—1222no—
286C2×C4⋊C4—1222no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—614no—
291C2×C22⋊C4—618no—
292C2×C22⋊C4—1222no—
293C2×C22⋊C4—1222no—
294C2×C22⋊C4—1222no—
295C2×C22⋊C4—1222no—
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—1224no—
298C2×C22⋊C4—1224no—
299C2×C22⋊C4—1224no—
300C2×C22⋊C4—1224no—
301C2×C22⋊C4—1224no—
302#99—12216no—
303#95—12216no—
304C24—21no—
305C24—21no—
306C24—24no—
307C24—42no—
308C24—42no—
309C24—612no—
310C24—612no—
311C24—612no—
312C24—614no—
313C24—614no—
314C24—2442no—
315C24—2442no—
316C24C22⋊#692312yes—
317C24C22×#692311yescharacteristic
318C2×D4—1224no—
319C2×D4—1228no—
320C2×D4—1228no—
321C2×D4—1224no—
322C2×D4—2442no—
323C2×D4—2444no—
324C2×D4—2442no—
325C2×D4—2444no—
326C2×D4—2442no—
327C2×D4—2442no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—42no—
334C22×C4—612no—
335C22×C4—612no—
336C22×C4—612no—
337C22×C4—1224no—
338C22×C4—1228no—
339C22×C4—1228no—
340C22×C4—1224no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4—1224no—
348C22×C4—1224no—
349C22×C4—1228no—
350C22×C4—2442no—
351C22×C4—2444no—
352C22×C4—2442no—
353C22×C4—2444no—
354C22×C4—2442no—
355C22×C4—2442no—
356C22×C4—2444no—
357C22×C4—2444no—
358C22×C4—2448no—
359C22×C4—2448no—
360C22×C4—2442no—
361C22×C4C22⋊#692312yes—
362C22×C4C22⋊#692312yes—
363C4⋊C4—1228no—
364C4⋊C4—1228no—
365C22⋊C4—28no—
366C22⋊C4—1228no—
367C22⋊C4—1228no—
368C22⋊C4—1228no—
369C22⋊C4—1228no—
370C22⋊C4—2444no—
371C22⋊C4—2444no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—88no—
385C23—612no—
386C23—614no—
387C23—612no—
388C23—614no—
389C23—612no—
390C23—614no—
391C23—612no—
392C23—614no—
393C23—612no—
394C23—612no—
395C23—1224no—
396C23—1224no—
397C23—2448no—
398C23—2448no—
399C23—2442no—
400C23—2442no—
401C23—2442no—
402C23—2442no—
403C23C2×C22⋊#692311yescharacteristic
404C23C2×C22⋊#692311yescharacteristic
405C23C2×C22⋊#692311yescharacteristic
406C23D4×#692314yes—
407D4—2448no—
408D4—2448no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—1224no—
418C2×C4—1224no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—4888no—
423C2×C4—4884no—
424C2×C4—4884no—
425C2×C4—2448no—
426C2×C4—2448no—
427C2×C4—2448no—
428C2×C4—24416no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2442no—
436C22—28no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—84no—
445C22—84no—
446C22—84no—
447C22—614no—
448C22—614no—
449C22—1228no—
450C22—1224no—
451C22—1224no—
452C22—1224no—
453C22—1224no—
454C22—2442no—
455C22—2444no—
456C22—2442no—
457C22—2442no—
458C22—2444no—
459C22—2442no—
460C22#1684070311yescharacteristic
461C22#1684071012yes—
462C22#1684071012yes—
463C22#1684071412yes—
464C22#1684074112yes—
465C22#1684074212yes—
466C4—42no—
467C4—42no—
468C4—4888no—
469C4—2442no—
470C4—2442no—
471C4—2442no—
472C2—22no—
473C2—22no—
474C2—82no—
475C2—1224no—
476C2—1224no—
477C2#2146100111yescharacteristic
478C2#2146164811yescharacteristic
479C2#2146164911yescharacteristic
480C2#2146184214yes—
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×C22⋊D61C22
cosocleC24×D61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#10966
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16840705C2semidirect
3. C2×#16840640C2semidirect
16. C22×C4⋊Dic61C4semidirect
17. C22×#11758196C4semidirect
18. C22×C22⋊Dic61C4semidirect
20. C22×C22⋊C4×C61C4semidirect
255. C22×C22⋊C4#6923semidirect
261. C61#10966semidirect

Maximal subgroups

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

subgroupindex
C22×#145198352
C2×#168407052
C2×#168406402
C22×#145197592
#1096661