Finite Group Databasebeta

← #29579731

Subgroups — #29579731

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839991C211yesmaximal, characteristic
2#21445857C211yesmaximal, characteristic, complemented
3#21444338C211yesmaximal, characteristic, complemented
4#21444334C211yesmaximal, characteristic
5#21444182C211yesmaximal, characteristic, complemented
6#21444178C211yesmaximal, characteristic
7C61×#1752C211yesmaximal, characteristic, complemented
8C22×#11758190C411yescharacteristic, complemented
9C2×#14519756C2211yescharacteristic
10C2×#14519751C2211yescharacteristic
11#16839991—21no—
12#16839991—21no—
13#16839991—21no—
14#16839991—21no—
15C2×#14519713—21no—
16C2×#14519709C411yescharacteristic, complemented
17C2×#14519702C411yescharacteristic, complemented
18C2×#14519699C411yescharacteristic, complemented
19#16839726C2211yescharacteristic
20#16839726C2211yescharacteristic
21#16839527—21no—
22#16839498—22no—
23#16839495—22no—
24#16839487—22no—
25#16839485—22no—
26C23×C8×C61C2211yescharacteristic
27C2×C22⋊C8×C61C2211yescharacteristic
28C2×#97×C61—21no—
29C4×C22⋊C4×C61C2211yescharacteristic
30C61×#336—21no—
31C22×Dic122C2×C411yescharacteristic
32C22×D244C2×C411yescharacteristic
33C22×C22⋊D61—21no—
34C2×#11758190—21no—
35C2×#11758190—21no—
36C2×#11758190—42no—
37C2×#11758194—21no—
38C2×#11758194—21no—
39#14519756—21no—
40#14519756—21no—
41#14519756—21no—
42#14519756—21no—
43#14519751—21no—
44#14519751—21no—
45#14519751—21no—
46#14519751—21no—
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×C4⋊Dic61C2×C411yescharacteristic
49C2×#11758196—21no—
50C2×C22⋊Dic61—21no—
51C2×#11758198—21no—
52#14519713—42no—
53#14519709—21no—
54#14519709—21no—
55#14519702—21no—
56#14519702—21no—
57#14519699—21no—
58#14519699—21no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63C23×C4×C61C2311yescharacteristic
64C22×C8×C61—21no—
65C22×C8×C61—42no—
66C22×C8×C61C2×C411yescharacteristic
67C22×C8×C61C2×C411yescharacteristic
68C22×C8×C61D411yescharacteristic
69C22×C8×C61D411yescharacteristic
70C22×C8×C61D411yescharacteristic
71C22×C8×C61D411yescharacteristic
72C2×C22⋊C4×C61—21no—
73C2×C42×C61—21no—
74C22⋊C8×C61—21no—
75C22⋊C8×C61D412yes—
76#97×C61—42no—
77#95×C61D412yes—
78C24×C61#4111yescharacteristic
79C23×D61—21no—
80C22×C4×C61—22no—
81C22×C4×C61—41no—
82C22×C4×C61—41no—
83C22×C4×C61C22×C411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C2×C22⋊D61—41no—
90C2×C22⋊D61—41no—
91C2×C22⋊D61—82no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—41no—
98C2×D244—41no—
99C2×Dic122—21no—
100C2×Dic122—21no—
101C2×Dic122—41no—
102C2×Dic122—41no—
103C2×C8×C61—22no—
104C2×C8×C61—22no—
105C2×C8×C61—22no—
106C2×C8×C61—22no—
107C2×C8×C61—41no—
108C2×C8×C61—41no—
109C2×C8×C61—41no—
110C2×C8×C61—41no—
111C2×C8×C61—41no—
112C2×C8×C61C22⋊C411yescharacteristic
113C2×C8×C61C22⋊C411yescharacteristic
114C2×C8×C61C22⋊C411yescharacteristic
115C2×C8×C61C22⋊C411yescharacteristic
116C22⋊C4×C61—42no—
117C42×C61—41no—
118C42×C61—41no—
119C22⋊Dic61—41no—
120C22⋊Dic61—41no—
121#11758198—41no—
122#11758198—41no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126C4⋊Dic61—21no—
127#11758196—41no—
128#11758196—41no—
129#11758194—41no—
130#11758194—41no—
131#11758194—41no—
132#11758194—41no—
133C2×#92418—41no—
134C2×#92418—41no—
135#11758190—42no—
136#11758190—42no—
137C23×C61—22no—
138C23×C61—22no—
139C23×C61—41no—
140C23×C61C4×D411yescharacteristic
141C23×C61#12111yescharacteristic
142C23×C61#12411yescharacteristic
143C22×D61—41no—
144C22×D61—42no—
145C22×D61—81no—
146C22×D61—81no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61C2×C22⋊C411yescharacteristic
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12711yescharacteristic
163C22⋊D61—82no—
164C22⋊D61—82no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—42no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171D244—41no—
172D244—41no—
173D244—81no—
174Dic122—41no—
175Dic122—41no—
176Dic122—81no—
177C8×C61—21no—
178C8×C61—21no—
179C8×C61—21no—
180C8×C61—42no—
181C8×C61#9912yes—
182#92418—81no—
183#92418—81no—
184#9816—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—44no—
193C22×C61#39011yescharacteristic
194C22×C61#44412yes—
195C22×C61#47112yes—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—162no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—82no—
204C4×C61C2×#9911yescharacteristic
205C4×C61#41011yescharacteristic
206Dic61—81no—
207Dic61—81no—
208Dic61—81no—
209C2×#465—611no—
210#1810—611no—
211#1752D6111yescharacteristic, complemented
212#1508—611no—
213#1507—611no—
214#1472—611no—
215#1471—611no—
216C2×C61—22no—
217C2×C61—22no—
218C2×C61—41no—
219C2×C61#196411yescharacteristic
220C2×C61#197511yescharacteristic
221C2×C61#197711yescharacteristic
222D61—161no—
223C22×SD16—611no—
224C23×C8C2×D6111yescharacteristic
225C2×C22⋊Q8—611no—
226C2×#121—611no—
227#465—1221no—
228#465—1221no—
229#465—1221no—
230#465—1221no—
231C2×#108—1221no—
232C2×#106—611no—
233C2×#103—611no—
234C2×#102—611no—
235#407—611no—
236#407—611no—
237C2×C22⋊C8C2×D6111yescharacteristic
238C2×#97—21no—
239C4×C22⋊C4C2×D6111yescharacteristic
240#341—1221no—
241#336—21no—
242#328—1222no—
243#327—1222no—
244#324—1222no—
245#323—1222no—
246C61#981611yescharacteristic, complemented, Sylow
247C22×Q8—611no—
248C22×D4—611no—
249C22×D4—1221no—
250C23×C4C22×D6111yescharacteristic
251C2×SD16—1221no—
252C2×SD16—1221no—
253C2×SD16—2442no—
254C2×M16—1221no—
255C2×M16—1221no—
256C22×C8—21no—
257C22×C8—42no—
258C22×C8C4×D6111yescharacteristic
259C22×C8C4×D6111yescharacteristic
260C22×C8D24411yescharacteristic
261C22×C8D24411yescharacteristic
262C22×C8C22⋊D6111yescharacteristic
263C22×C8C22⋊D6111yescharacteristic
264C22⋊Q8—1221no—
265C22⋊Q8—1221no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268#121—1221no—
269#121—1221no—
270#121—1221no—
271#121—1221no—
272C2×C4⋊C4—611no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—1221no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—1221no—
277C2×C22⋊C4—1221no—
278C2×C42—21no—
279#108—2442no—
280#106—1221no—
281#106—1221no—
282#103—1221no—
283#103—1221no—
284#102—1221no—
285#102—1221no—
286C22⋊C8—21no—
287C22⋊C8—1221no—
288C22⋊C8—1221no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291C22⋊C8D24412yes—
292#97—42no—
293#95D24412yes—
294C24—1221no—
295C24#1175822211yescharacteristic
296C2×Q8—1221no—
297C2×Q8—1221no—
298C2×Q8—2441no—
299C2×Q8—2441no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—4882no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2441no—
306C2×D4—2441no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4C2×C4×D6111yescharacteristic
314C22×C4C2×D24411yescharacteristic
315C22×C4C2×D24411yescharacteristic
316C22×C4C2×D24411yescharacteristic
317C22×C4#1175822211yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319SD16—2442no—
320SD16—2442no—
321M16—2441no—
322M16—2441no—
323M16—2441no—
324M16—2441no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—2441no—
335C2×C8—2441no—
336C2×C8#1175819811yescharacteristic
337C2×C8#1175819811yescharacteristic
338C2×C8#1175819811yescharacteristic
339C2×C8#1175819811yescharacteristic
340C4⋊C4—1221no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C4⋊C4—1221no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C22⋊C4—42no—
347C22⋊C4—2441no—
348C22⋊C4—2441no—
349C22⋊C4—2441no—
350C22⋊C4—2441no—
351C42—41no—
352C42—41no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—4881no—
357C23—4881no—
358C23—2442no—
359C23—2441no—
360C23C4×D24411yescharacteristic
361C23#1451975111yescharacteristic
362C23#1451976511yescharacteristic
363Q8—4881no—
364Q8—2441no—
365Q8—2441no—
366D4—4881no—
367D4—4882no—
368D4—4882no—
369D4—2441no—
370D4—2441no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—4881no—
384C2×C4—4881no—
385C2×C4—2442no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4C2×#1175819811yescharacteristic
390C2×C4C4×D24411yescharacteristic
391C2×C4#1451975111yescharacteristic
392C2×C4#1451977411yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—42no—
397C8—4881no—
398C8—4881no—
399C8#1451969112yes—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—9762no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22#1683966411yescharacteristic
413C22#1683988512yes—
414C22#1684002512yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—82no—
419C4—4881no—
420C4—4881no—
421C4—4881no—
422C4C2×#1451969111yescharacteristic
423C4#1683973511yescharacteristic
424C2—22no—
425C2—22no—
426C2—41no—
427C2—9761no—
428C2#2144665511yescharacteristic
429C2#2144671211yescharacteristic
430C2#2144672211yescharacteristic
subgroupG/H
centerC22#16839664
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1752C2
socleC22×C61#390
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1752D61
hypocenterC61#9816
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
2. #21445857C2semidirect
3. #21444338C2semidirect
5. #21444182C2semidirect
7. C61×#1752C2semidirect
8. C22×#11758190C4semidirect
16. C2×#14519709C4semidirect
17. C2×#14519702C4semidirect
18. C2×#14519699C4semidirect
211. #1752D61semidirect
246. C61#9816semidirect

Maximal subgroups

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

subgroupindex
C2×#168399912
#214458572
#214443382
#214443342
#214441822
#214441782
C61×#17522
#981661