Finite Group Databasebeta

← #29572517

Subgroups — #29572517

Order 15616.

754 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16840385C211yesmaximal, characteristic
2#21445900C211yesmaximal, characteristic, complemented
3#21445894C211yesmaximal, characteristic, complemented
4#21444352C211yesmaximal, characteristic, complemented
5#21444349C211yesmaximal, characteristic
6#21444347C211yesmaximal, characteristic
7C61×#1511C211yesmaximal, characteristic, complemented
8#16840385—22no—
9#16840385—22no—
10#16840385—22no—
11#16840385—22no—
12C2×#14519776C411yescharacteristic, complemented
13C2×#14519772—21no—
14C2×#14519766C411yescharacteristic, complemented
15C2×#14519763—21no—
16C2×#14519759—21no—
17C2×#14519757—21no—
18C2×#14519751C2211yescharacteristic
19C2×Q8×Dic61C411yescharacteristic, complemented
20C2×#14519739C411yescharacteristic, complemented
21C2×#14519736C2211yescharacteristic
22#16839726C2211yescharacteristic
23#16839625C2211yescharacteristic
24#16839530—21no—
25#16839530—21no—
26#16839528—21no—
27#16839528—21no—
28#16839499—22no—
29#16839497—22no—
30#16839495—22no—
31C2×C22⋊Q8×C61C2211yescharacteristic
32C61×#407C2211yescharacteristic
33C61×#381C2211yescharacteristic
34C61×#328—22no—
35C22×Dic122—21no—
36C22×D244C2×C411yescharacteristic
37C22×C22⋊D61—21no—
38C22×C4×D61—21no—
39C2×#11758194—21no—
40#14519776—22no—
41#14519776—22no—
42#14519772—24no—
43#14519772—24no—
44#14519766—22no—
45#14519766—22no—
46#14519763—24no—
47#14519763—24no—
48#14519759—24no—
49#14519759—24no—
50#14519757—24no—
51#14519757—24no—
52#14519751—21no—
53#14519751—21no—
54#14519751D412yes—
55#14519751D412yes—
56Q8×Dic61—22no—
57Q8×Dic61—22no—
58#14519739—22no—
59#14519739—22no—
60#14519736—21no—
61#14519736—21no—
62#14519736—21no—
63#14519736—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61D411yescharacteristic
68C2×#11758196—21no—
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61D411yescharacteristic
71C2×C22⋊Dic61D411yescharacteristic
72C2×#11758198—21no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61C2×C411yescharacteristic
78C2×C4×Dic61C2×C411yescharacteristic
79C22⋊#92418—21no—
80C22⋊#92418—21no—
81#14519684—21no—
82#14519684—21no—
83C22×Q8×C61C2×C411yescharacteristic
84C23×C4×C61C2311yescharacteristic
85C2×M16×C61—21no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61D412yes—
89C22⋊Q8×C61D412yes—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2×C411yescharacteristic
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C22⋊C8×C61—21no—
95C22⋊C8×C61—21no—
96#95×C61—21no—
97#95×C61—21no—
98C24×C61C2×D411yescharacteristic
99C23×D61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—42no—
102C2×Q8×C61C22⋊C412yes—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—42no—
106C22×C4×C61—41no—
107C22×C4×C61C22×C411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C2×C22⋊D61—22no—
114C2×C22⋊D61—22no—
115C2×C22⋊D61—82no—
116C22×Dic61—21no—
117C22×Dic61—41no—
118C22×Dic61C22⋊C411yescharacteristic
119C22×Dic61C22⋊C411yescharacteristic
120C22×Dic61C22⋊C411yescharacteristic
121C22×Dic61C22⋊C411yescharacteristic
122C2×D244—21no—
123C2×D244—42no—
124C2×D244C22⋊C412yes—
125C2×C4×D61—24no—
126C2×C4×D61—82no—
127C2×Dic122—24no—
128C2×Dic122—82no—
129M16×C61—42no—
130C2×C8×C61—41no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61C22⋊C412yes—
135C22⋊C4×C61—22no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—42no—
138C22⋊Dic61—21no—
139C22⋊Dic61—21no—
140C22⋊Dic61—21no—
141C22⋊Dic61—21no—
142C22⋊Dic61—42no—
143#11758198—24no—
144#11758198—24no—
145#11758198—24no—
146#11758198—22no—
147#11758198—22no—
148C4⋊Dic61—24no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61C22⋊C412yes—
155#11758196—24no—
156C4×Dic61—24no—
157C4×Dic61—21no—
158C4×Dic61—21no—
159C4×Dic61—21no—
160C4×Dic61—21no—
161#11758194—42no—
162C2×#92418—41no—
163C23×C61—22no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61C2×C22⋊C411yescharacteristic
167C23×C61#12011yescharacteristic
168C23×C61#12011yescharacteristic
169C22×D61—22no—
170C22×D61—42no—
171C22×D61—42no—
172C22×D61—42no—
173Q8×C61—42no—
174Q8×C61—42no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—81no—
186C2×C4×C61—81no—
187C2×C4×C61—81no—
188C2×C4×C61C2×C22⋊C411yescharacteristic
189C2×C4×C61C2×C22⋊C411yescharacteristic
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61#12011yescharacteristic
192C22⋊D61—82no—
193C22⋊D61—82no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—22no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—42no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—81no—
205C2×Dic61—81no—
206C2×Dic61—81no—
207D244—42no—
208D244—42no—
209C4×D61—84no—
210Dic122—84no—
211C8×C61—81no—
212#92418—81no—
213#8519—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—82no—
223C22×C61#37911yescharacteristic
224C22×C61#41012yes—
225C2×D61—42no—
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—84no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—42no—
234C4×C61—42no—
235C4×C61—161no—
236Dic61—41no—
237Dic61—42no—
238Dic61—41no—
239Dic61—41no—
240Dic61—41no—
241Dic61—161no—
242C2×#542—611no—
243#1822—611no—
244#1820—611no—
245#1511—611no—
246#1511D6111yescharacteristic, complemented
247#1510—611no—
248#1510—611no—
249C2×C61—22no—
250C2×C61—41no—
251C2×C61—41no—
252C2×C61#178211yescharacteristic
253C2×C61#179511yescharacteristic
254C2×C61#179611yescharacteristic
255D61—82no—
256#542—1222no—
257#542—1222no—
258#542—1222no—
259#542—1222no—
260C2×C4⋊Q8—611no—
261C2×#126—1221no—
262C2×#124—611no—
263C2×#124—1221no—
264C2×#123—1221no—
265C2×C22⋊Q8—1221no—
266C2×C22⋊Q8C2×D6111yescharacteristic
267C2×#121—611no—
268C2×C4×Q8—611no—
269C2×C4×D4—611no—
270C2×#117—611no—
271#407—611no—
272#407C2×D6111yescharacteristic
273#381—611no—
274#381C2×D6111yescharacteristic
275#343—1221no—
276#343—1221no—
277#342—1221no—
278#342—1221no—
279#328—22no—
280#328—1222no—
281#327—1222no—
282#327—1222no—
283C61#851911yescharacteristic, complemented, Sylow
284C22×Q8—1221no—
285C22×Q8C4×D6111yescharacteristic
286C22×D4—611no—
287C22×D4—1221no—
288C23×C4—1221no—
289C23×C4C22×D6111yescharacteristic
290C2×M16—21no—
291C2×M16—1221no—
292C4⋊Q8—1222no—
293C4⋊Q8—1222no—
294#126—1224no—
295#126—1224no—
296#124—1224no—
297#124—1224no—
298#124—1222no—
299#124—1222no—
300#123—1224no—
301#123—1224no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304C22⋊Q8—1224no—
305C22⋊Q8—1224no—
306C22⋊Q8D24412yes—
307C22⋊Q8C22⋊D6112yes—
308#121—612no—
309#121—612no—
310#121—1221no—
311#121—1221no—
312C4×Q8—1222no—
313C4×Q8—1222no—
314C4×D4—1222no—
315C4×D4—1222no—
316#117—1221no—
317#117—1221no—
318#117—1221no—
319#117—1221no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—611no—
322C2×C4⋊C4—611no—
323C2×C4⋊C4—611no—
324C2×C4⋊C4—1221no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4C4×D6111yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—611no—
330C2×C22⋊C4—611no—
331C2×C22⋊C4—1221no—
332C2×C22⋊C4—1221no—
333C2×C22⋊C4—1221no—
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1221no—
336C2×C42—611no—
337C2×C42—611no—
338C2×C42—1221no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—1221no—
342C22⋊C8—1221no—
343#95—21no—
344#95—21no—
345#95—1221no—
346#95—1221no—
347C24—1221no—
348C24D4×D6111yescharacteristic
349C2×Q8—21no—
350C2×Q8—42no—
351C2×Q8—1224no—
352C2×Q8—4882no—
353C2×Q8#1175819812yes—
354C2×D4—612no—
355C2×D4—1222no—
356C2×D4—1222no—
357C2×D4—1221no—
358C2×D4—4882no—
359C2×D4—2442no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—611no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—611no—
368C22×C4—1224no—
369C22×C4—1221no—
370C22×C4—4882no—
371C22×C4—2441no—
372C22×C4C2×C4×D6111yescharacteristic
373C22×C4C2×D24411yescharacteristic
374C22×C4D4×D6111yescharacteristic
375C22×C4D4×D6111yescharacteristic
376C22×C4D4×D6111yescharacteristic
377C22×C4C2×C22⋊D6111yescharacteristic
378M16—42no—
379M16—2442no—
380C2×C8—41no—
381C2×C8—2441no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—612no—
386C4⋊C4—1224no—
387C4⋊C4—1224no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4#1175819812yes—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—42no—
397C22⋊C4—1224no—
398C22⋊C4—1224no—
399C22⋊C4—1224no—
400C22⋊C4—1221no—
401C22⋊C4—1221no—
402C22⋊C4—1221no—
403C22⋊C4—1221no—
404C22⋊C4—1222no—
405C22⋊C4—1222no—
406C22⋊C4—2442no—
407C42—1224no—
408C42—1221no—
409C42—1221no—
410C42—1221no—
411C42—1221no—
412C23—22no—
413C23—41no—
414C23—41no—
415C23—1222no—
416C23—2442no—
417C23—2442no—
418C23—2442no—
419C23C22⋊C4×D6111yescharacteristic
420C23#1451974511yescharacteristic
421C23#1451974711yescharacteristic
422Q8—42no—
423Q8—42no—
424Q8—4884no—
425D4—4882no—
426D4—4882no—
427D4—2442no—
428D4—2442no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—1221no—
445C2×C4—1222no—
446C2×C4—1221no—
447C2×C4—4884no—
448C2×C4—4881no—
449C2×C4—4881no—
450C2×C4—4881no—
451C2×C4—2442no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4C22⋊C4×D6111yescharacteristic
457C2×C4C2×#1175819811yescharacteristic
458C2×C4#1451974511yescharacteristic
459C2×C4#1451974711yescharacteristic
460C8—81no—
461C8—4881no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—82no—
471C22—4884no—
472C22—2442no—
473C22—2442no—
474C22—2442no—
475C22#1683961711yescharacteristic
476C22#1683973412yes—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—42no—
482C4—42no—
483C4—161no—
484C4—9761no—
485C4—2441no—
486C4—2442no—
487C4—2441no—
488C4—2441no—
489C4—2441no—
490C2—22no—
491C2—41no—
492C2—41no—
493C2—4882no—
494C2#2144573511yescharacteristic
495C2#2144578811yescharacteristic
496C2#2144579111yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1511C2
socleC22×C61#379
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1511D61
hypocenterC61#8519
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
2. #21445900C2semidirect
3. #21445894C2semidirect
4. #21444352C2semidirect
7. C61×#1511C2semidirect
12. C2×#14519776C4semidirect
14. C2×#14519766C4semidirect
19. C2×Q8×Dic61C4semidirect
20. C2×#14519739C4semidirect
246. #1511D61semidirect
283. C61#8519semidirect

Maximal subgroups

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

subgroupindex
C2×#168403852
#214459002
#214458942
#214443522
#214443492
#214443472
C61×#15112
#851961