Finite Group Databasebeta

← #29595528

Subgroups — #29595528

Order 15616.

962 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16840336C211yesmaximal, characteristic
2C2×#16839560C212yesmaximal, complemented
3C2×#16839558C212yesmaximal, complemented
4#21445743C211yesmaximal, characteristic, complemented
5C61×#1784C211yesmaximal, characteristic, complemented
6C22×#11758222C411yescharacteristic
7#16840336—24no—
8#16840336—24no—
9C2×#14519775C411yescharacteristic
10C2×#14519775C2211yescharacteristic
11C2×#14519766C411yescharacteristic
12C2×#14519766C2211yescharacteristic
13C2×#14519760—21no—
14C2×#14519760—21no—
15C2×#14519753—21no—
16C2×#14519753—21no—
17C2×#14519736C411yescharacteristic
18C2×#14519696—21no—
19C2×#14519696C2212yes—
20#16839726—21no—
21#16839560—24no—
22#16839558—24no—
23C23×Q8×C61C2211yescharacteristic
24C2×#101×C61—21no—
25C2×#101×C61C2212yes—
26C61×#407—21no—
27C2×#11758222—22no—
28C2×#11758222—22no—
29C2×#11758222—42no—
30C2×#11758222—42no—
31C22×Dic122C2×C411yescharacteristic
32C22×D244C2×C411yescharacteristic
33C22×C22⋊D61—21no—
34C22×C4×D61—21no—
35C2×#11758194—22no—
36#14519775—22no—
37#14519775—22no—
38#14519775—22no—
39#14519775D414yes—
40#14519766—22no—
41#14519766—22no—
42#14519766—22no—
43#14519766D414yes—
44#14519760—28no—
45#14519760—28no—
46#14519753—28no—
47#14519753—28no—
48#14519736—22no—
49#14519736—22no—
50C2×C4⋊Dic61—21no—
51C2×#11758196—21no—
52C2×#11758196—21no—
53C2×#11758196—21no—
54C2×#11758196—21no—
55C2×C22⋊Dic61—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×C4×Dic61C2×C411yescharacteristic
61C2×C4×Dic61C2×C411yescharacteristic
62#14519696—22no—
63#14519696—22no—
64#14519696—42no—
65C22⋊#92418—42no—
66C22×Q8×C61—22no—
67C22×Q8×C61—22no—
68C22×Q8×C61—42no—
69C22×Q8×C61—42no—
70C22×Q8×C61C2×C411yescharacteristic
71C22×Q8×C61D411yescharacteristic
72C22×Q8×C61D411yescharacteristic
73C22×Q8×C61C2311yescharacteristic
74C23×C4×C61C2×C411yescharacteristic
75C23×C4×C61D411yescharacteristic
76C23×C4×C61D411yescharacteristic
77C2×M16×C61—22no—
78#101×C61—22no—
79#101×C61—22no—
80#101×C61—42no—
81C22⋊C8×C61—42no—
82C24×C61C22⋊C411yescharacteristic
83C23×D61—21no—
84C2×Q8×C61—22no—
85C2×Q8×C61—22no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×Q8×C61—22no—
89C2×Q8×C61—22no—
90C2×Q8×C61—41no—
91C2×Q8×C61—41no—
92C2×Q8×C61—41no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C2×Q8×C61—41no—
96C2×Q8×C61—48no—
97C2×Q8×C61—48no—
98C2×Q8×C61C2×D412yes—
99C2×Q8×C61C2×D412yes—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—41no—
109C22×C4×C61—41no—
110C22×C4×C61C22⋊C411yescharacteristic
111C22×C4×C61C22⋊C411yescharacteristic
112C22×C4×C61C22⋊C411yescharacteristic
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C2×C22⋊D61—24no—
117C2×C22⋊D61—42no—
118C2×C22⋊D61—42no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122#11758222—48no—
123#11758222—48no—
124C2×D244—21no—
125C2×D244—41no—
126C2×D244—41no—
127C2×D244C22⋊C412yes—
128C2×C4×D61—24no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×Dic122—21no—
132C2×Dic122—41no—
133C2×Dic122—41no—
134C2×Dic122C22⋊C412yes—
135M16×C61—42no—
136M16×C61—42no—
137C2×C8×C61—42no—
138C22⋊Dic61—24no—
139#11758198—24no—
140#11758198—24no—
141#11758198—24no—
142#11758198—24no—
143C4⋊Dic61—24no—
144#11758196—24no—
145#11758196—24no—
146#11758196—24no—
147#11758196—24no—
148C4×Dic61—21no—
149C4×Dic61—21no—
150C4×Dic61C22⋊C412yes—
151C4×Dic61C22⋊C412yes—
152#11758194—42no—
153#11758194—42no—
154C2×#92418—42no—
155C23×C61—21no—
156C23×C61—21no—
157C23×C61—42no—
158C23×C61#9912yes—
159C23×C61C2×C22⋊C411yescharacteristic
160C22×D61—22no—
161C22×D61—44no—
162C22×D61—41no—
163C22×D61—41no—
164Q8×C61—42no—
165Q8×C61—42no—
166Q8×C61—44no—
167Q8×C61—44no—
168Q8×C61—42no—
169Q8×C61—42no—
170C2×C4×C61—24no—
171C2×C4×C61—24no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—44no—
187C2×C4×C61—44no—
188C2×C4×C61#9912yes—
189C2×C4×C61C2×C22⋊C411yescharacteristic
190C2×C4×C61C2×C22⋊C411yescharacteristic
191C2×C4×C61#12012yes—
192C2×C4×C61#12012yes—
193C22⋊D61—48no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201D244—42no—
202D244—42no—
203C4×D61—48no—
204Dic122—42no—
205Dic122—42no—
206C8×C61—82no—
207#92418—82no—
208#12702—611nomaximal, Sylow
209C22×C61—22no—
210C22×C61—24no—
211C22×C61—22no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61#37911yescharacteristic
215C22×C61C2×#9911yescharacteristic
216C22×C61C2×#9911yescharacteristic
217C2×D61—44no—
218C2×D61—42no—
219C2×D61—82no—
220C2×D61—82no—
221C4×C61—22no—
222C4×C61—22no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—22no—
226C4×C61—41no—
227C4×C61—41no—
228Dic61—42no—
229Dic61—42no—
230Dic61—42no—
231C2×#536—611no—
232C2×#356—612no—
233C2×#355—612no—
234#1784—611no—
235#1784D6111yescharacteristic, complemented
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—42no—
239C2×C61#178111yescharacteristic
240C2×C61#213312yes—
241D61—82no—
242C22×#41—611no—
243C23×Q8C2×D6111yescharacteristic
244#536—1224no—
245#536—1224no—
246C2×C4⋊Q8—611no—
247C2×C4⋊Q8—611no—
248C2×#124—611no—
249C2×#124—611no—
250C2×#123—1221no—
251C2×#123—1221no—
252C2×C22⋊Q8—1221no—
253C2×C22⋊Q8—1221no—
254C2×#117—611no—
255C2×#101—21no—
256C2×#101—612no—
257C2×#101—1221no—
258C2×#101C2×D6112yes—
259#407—21no—
260#407—1221no—
261#356—1224no—
262#355—1224no—
263C61#1270211yescharacteristic, complemented, Sylow
264C2×#41—1222no—
265C2×#41—1222no—
266C2×#41—2442no—
267C2×#41—2442no—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8—42no—
271C22×Q8—42no—
272C22×Q8—611no—
273C22×Q8C4×D6111yescharacteristic
274C22×Q8D24411yescharacteristic
275C22×Q8C22⋊D6111yescharacteristic
276C22×Q8C22×D6111yescharacteristic
277C22×D4—611no—
278C22×D4—1221no—
279C23×C4—1221no—
280C23×C4C4×D6111yescharacteristic
281C23×C4D24411yescharacteristic
282C23×C4C22⋊D6111yescharacteristic
283C2×M16—22no—
284C2×M16—1222no—
285C4⋊Q8—614no—
286C4⋊Q8—1222no—
287C4⋊Q8—1222no—
288C4⋊Q8—1222no—
289#124—614no—
290#124—1222no—
291#124—1222no—
292#124—1222no—
293#123—1228no—
294#123—1228no—
295C22⋊Q8—1228no—
296C22⋊Q8—1228no—
297#117—1222no—
298#117—1222no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1221no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1221no—
307C2×C22⋊C4—1221no—
308C2×C22⋊C4—1221no—
309C2×C42—611no—
310C2×C42—611no—
311#101—22no—
312#101—22no—
313#101—42no—
314#101—1222no—
315#101—1222no—
316#101—2442no—
317C22⋊C8—42no—
318C22⋊C8—2442no—
319C24—1221no—
320C24#1175819811yescharacteristic
321#41—2448no—
322#41—2448no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—612no—
338C2×Q8—1221no—
339C2×Q8—2441no—
340C2×Q8—2441no—
341C2×Q8D4×D6112yes—
342C2×Q8D4×D6112yes—
343C2×D4—612no—
344C2×D4—1224no—
345C2×D4—1221no—
346C2×D4—2441no—
347C2×D4—2441no—
348C2×D4—2442no—
349C2×D4—2442no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—1224no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—2442no—
365C22×C4—2442no—
366C22×C4#1175819811yescharacteristic
367C22×C4#1175819811yescharacteristic
368C22×C4#1175819811yescharacteristic
369C22×C4C2×C4×D6111yescharacteristic
370C22×C4C2×D24411yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372M16—42no—
373M16—42no—
374M16—2442no—
375M16—2442no—
376C2×C8—42no—
377C2×C8—2442no—
378C4⋊C4—1224no—
379C4⋊C4—1224no—
380C4⋊C4—1224no—
381C4⋊C4—1224no—
382C4⋊C4—1224no—
383C22⋊C4—1224no—
384C22⋊C4—1224no—
385C22⋊C4—1224no—
386C22⋊C4—1224no—
387C22⋊C4—1224no—
388C42—612no—
389C42—612no—
390C42—1221no—
391C42—1221no—
392C23—21no—
393C23—21no—
394C23—42no—
395C23—1222no—
396C23—2444no—
397C23—2441no—
398C23—2441no—
399C23#1451969112yes—
400C23C2×#1175819811yescharacteristic
401Q8—42no—
402Q8—42no—
403Q8—44no—
404Q8—44no—
405Q8—42no—
406Q8—42no—
407Q8—2442no—
408Q8—2442no—
409D4—2442no—
410D4—2442no—
411D4—2448no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—1222no—
431C2×C4—1222no—
432C2×C4—1222no—
433C2×C4—2448no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4#1451969112yes—
439C2×C4C22⋊C4×D6111yescharacteristic
440C2×C4C22⋊C4×D6111yescharacteristic
441C2×C4#1451974512yes—
442C2×C4#1451974712yes—
443C8—82no—
444C8—4882no—
445C22—22no—
446C22—24no—
447C22—22no—
448C22—42no—
449C22—42no—
450C22—4882no—
451C22—4882no—
452C22—2444no—
453C22—2442no—
454C22#1683961711yescharacteristic
455C22C2×#1451969111yescharacteristic
456C22C2×#1451969111yescharacteristic
457C4—22no—
458C4—22no—
459C4—22no—
460C4—21no—
461C4—21no—
462C4—41no—
463C4—41no—
464C4—2442no—
465C4—2442no—
466C4—2442no—
467C2—21no—
468C2—21no—
469C2—42no—
470C2—4882no—
471C2#2144573311yescharacteristic
472C2#2144750612yes—
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1784C2
socleC22×C61C2×#99
prime radicalC22×#11758222C4
cosocleC22×Q8×C61C23
hypercenter#1784D61
hypocenterC61#12702
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. C2×#16839560C2semidirect
3. C2×#16839558C2semidirect
4. #21445743C2semidirect
5. C61×#1784C2semidirect
235. #1784D61semidirect
263. C61#12702semidirect

Maximal subgroups

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

subgroupindex
C2×#168403362
C2×#168395602
C2×#168395582
#214457432
C61×#17842
#1270261