Finite Group Databasebeta

← #29834927

Subgroups — #29834927

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455477C211yesmaximal, characteristic
2#21454427C211yesmaximal, characteristic, complemented
3#21448037C214yesmaximal, complemented
4#21448032C214yesmaximal, complemented
5C2×#16839586C211yesmaximal, characteristic, complemented
6C2×#16839585C211yesmaximal, characteristic, complemented
7#21447568C211yesmaximal, characteristic, complemented
8#21447439C211yesmaximal, characteristic, complemented
9C61×#2148C211yesmaximal, characteristic, complemented
10#16840528—21no—
11C2×#14519783C2211yescharacteristic
12#16840340—21no—
13C2×#14519717C2211yescharacteristic
14#16840017C2212yes—
15#16840012C2212yes—
16#16839988—22no—
17#16839810—21no—
18#16839809C2214yes—
19C2×#14519713C2211yescharacteristic
20#108×D61—22no—
21C2×#14519709C2211yescharacteristic
22#16839763—21no—
23#16839586C2214yes—
24#16839585C2214yes—
25#16839578C2212yes—
26#16839575C2212yes—
27#16839543C2214yes—
28C2×#135×C61C2211yescharacteristic
29C2×M32×C61C2211yescharacteristic
30C2×#108×C61C2211yescharacteristic
31C61×#362C2212yes—
32C61×#361C2212yes—
33C2×#11758222D411yescharacteristic
34#14519785—42no—
35#14519783—21no—
36#14519783—21no—
37#14519783—21no—
38#14519783—21no—
39#14519783—42no—
40C2×#11758194—21no—
41C2×#11758194D411yescharacteristic
42C2×#11758189C2312yes—
43M16×D61—24no—
44C2×C8×D61D412yes—
45#14519756—41no—
46#14519753—22no—
47#14519751—41no—
48#14519749—42no—
49#14519748—22no—
50C2×C4⋊Dic61—21no—
51#14519717—21no—
52#14519717C2312yes—
53C2×#11758185—21no—
54#14519713—21no—
55#14519713C2312yes—
56#14519712—24no—
57#14519709—21no—
58#14519709C2312yes—
59#14519701—22no—
60#14519698—22no—
61C2×#41×C61—21no—
62#135×C61—21no—
63#135×C61—21no—
64#135×C61—21no—
65#135×C61—21no—
66C2×Q16×C61C2311yescharacteristic
67C2×SD16×C61—21no—
68C2×D8×C61C2311yescharacteristic
69C2×M16×C61—21no—
70C22×C8×C61C2311yescharacteristic
71M32×C61—21no—
72M32×C61C2312yes—
73C2×C16×C61—21no—
74#108×C61—21no—
75#108×C61C2312yes—
76#41×C61—41no—
77#41×C61—41no—
78#41×C61—41no—
79#41×C61—41no—
80C2×Q8×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—41no—
83C22×C4×C61—41no—
84C22×C4×C61C2×D411yescharacteristic
85C2×C22⋊D61—42no—
86C2×C22⋊D61SD1612yes—
87C22×Dic61—41no—
88#11758222—21no—
89#11758222—21no—
90#11758222—21no—
91#11758222—21no—
92C2×D244SD1611yescharacteristic
93C2×C4×D61C2×D412yes—
94C2×Dic122SD1611yescharacteristic
95Q16×C61—21no—
96Q16×C61—21no—
97SD16×C61—41no—
98SD16×C61—41no—
99D8×C61—21no—
100D8×C61—21no—
101M16×C61—22no—
102M16×C61—41no—
103C2×C8×C61—41no—
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61#4111yescharacteristic
106C2×C8×C61#4111yescharacteristic
107C2×C8×C61#4111yescharacteristic
108C2×C8×C61#4111yescharacteristic
109C2×C8×C61C2411yescharacteristic
110C22⋊Dic61—42no—
111#11758198—42no—
112C4⋊Dic61—22no—
113C4⋊Dic61—41no—
114#11758196—42no—
115#11758194—21no—
116#11758194—21no—
117#11758194—22no—
118#11758194—41no—
119C2×#92418—41no—
120C2×#92418C2×D412yes—
121#11758189—22no—
122#11758189—22no—
123#11758189—44no—
124C8×D61—22no—
125C8×D61—22no—
126C8×D61—44no—
127C16×C61—22no—
128#11758185—22no—
129C23×C61—41no—
130C23×C61C2×SD1611yescharacteristic
131C22×D61C2×SD1612yes—
132Q8×C61—41no—
133Q8×C61—41no—
134D4×C61—41no—
135D4×C61—41no—
136D4×C61—81no—
137D4×C61—81no—
138C2×C4×C61—41no—
139C2×C4×C61—81no—
140C2×C4×C61—81no—
141C2×C4×C61#12311yescharacteristic
142C2×C4×C61#12311yescharacteristic
143C2×C4×C61C2×SD1611yescharacteristic
144C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C61C22×D411yescharacteristic
147C22⋊D61—22no—
148C22⋊D61—22no—
149C22⋊D61—162no—
150C2×Dic61—42no—
151C2×Dic61—161no—
152C2×Dic61C2×SD1612yes—
153D244—21no—
154D244—21no—
155C4×D61—22no—
156C4×D61—22no—
157Dic122—21no—
158Dic122—21no—
159C8×C61—42no—
160C8×C61#12311yescharacteristic
161C8×C61#12311yescharacteristic
162C8×C61C2×#4111yescharacteristic
163C8×C61C2×#4111yescharacteristic
164#92418—22no—
165#92418—42no—
166#33659—611nomaximal, Sylow
167C22×C61—41no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61#48111yescharacteristic
172C22×C61#48111yescharacteristic
173C22×C61C22×SD1611yescharacteristic
174C2×D61—22no—
175C2×D61—42no—
176C4×C61—81no—
177C4×C61#48111yescharacteristic
178C4×C61#48111yescharacteristic
179C4×C61C2×#12311yescharacteristic
180C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
181Dic61—22no—
182Dic61—82no—
183#3060—611no—
184#2916—611no—
185#2239—614no—
186#2238—614no—
187C2×#365—611no—
188C2×#365—611no—
189#2148—611no—
190#2148D6111yescharacteristic, complemented
191#2115—611no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—21no—
195C2×C61—161no—
196C2×C61C2×#48111yescharacteristic
197D61—42no—
198C2×#135C2×D6111yescharacteristic
199#568—1221no—
200C2×#131—611no—
201#537—1221no—
202C2×M32—611no—
203C2×M32C2×D6111yescharacteristic
204#470—612no—
205#469—612no—
206#465—1222no—
207#430—614no—
208#430—1221no—
209C2×#108—611no—
210C2×#108—1222no—
211C2×#108C2×D6111yescharacteristic
212C2×#106—611no—
213#417—1221no—
214#365—614no—
215#365—614no—
216#362—612no—
217#362C2×D6112yes—
218#361—612no—
219#361C2×D6112yes—
220#349—614no—
221C61#3365911yescharacteristic, complemented, Sylow
222C2×#41—21no—
223C2×#41—611no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135—21no—
228C2×Q16C22×D6111yescharacteristic
229C2×SD16—21no—
230C2×D8C22×D6111yescharacteristic
231#131—1221no—
232#131—1221no—
233#131—1221no—
234#131—1221no—
235#131—2442no—
236#131—2442no—
237C2×M16—21no—
238C2×M16—611no—
239C2×M16—612no—
240C2×M16—1224no—
241C2×M16—1221no—
242C22×C8—612no—
243C22×C8C22×D6111yescharacteristic
244C22⋊Q8—1222no—
245C22⋊Q8—2441no—
246#121—1222no—
247#121—2441no—
248#121—2442no—
249C2×C4⋊C4—1221no—
250M32—21no—
251M32—612no—
252M32—1221no—
253M32C22×D6112yes—
254C2×C16—21no—
255C2×C16—1221no—
256#108—21no—
257#108—612no—
258#108—1224no—
259#108—1221no—
260#108C22×D6112yes—
261#106—612no—
262#106—1221no—
263#103—1222no—
264#102—1222no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—1221no—
270#41—1221no—
271#41—1221no—
272#41—1221no—
273C2×Q8—21no—
274C2×Q8—611no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4—612no—
278C2×D4—611no—
279C2×D4—2442no—
280C22×C4—41no—
281C22×C4—612no—
282C22×C4—2441no—
283C22×C4D4×D6111yescharacteristic
284Q16—21no—
285Q16—21no—
286SD16—41no—
287SD16—41no—
288D8—21no—
289D8—21no—
290M16—22no—
291M16—41no—
292M16—1221no—
293M16—1221no—
294M16—1222no—
295M16—1222no—
296M16—1222no—
297M16—2444no—
298M16—2441no—
299C2×C8—41no—
300C2×C8—612no—
301C2×C8—1222no—
302C2×C8—1222no—
303C2×C8—2444no—
304C2×C8—2441no—
305C2×C8#1175822211yescharacteristic
306C2×C8#1175822211yescharacteristic
307C2×C8D4×D6111yescharacteristic
308C2×C8#1175822411yescharacteristic
309C2×C8#1175822411yescharacteristic
310C2×C8C23×D6111yescharacteristic
311C4⋊C4—1222no—
312C4⋊C4—2442no—
313C4⋊C4—2441no—
314C22⋊C4—2442no—
315C22⋊C4—2442no—
316C16—22no—
317C16—1222no—
318C23—41no—
319C23—612no—
320C23SD16×D6111yescharacteristic
321Q8—41no—
322Q8—41no—
323Q8—1221no—
324Q8—1221no—
325D4—41no—
326D4—41no—
327D4—81no—
328D4—81no—
329D4—1222no—
330D4—1222no—
331D4—1221no—
332D4—1221no—
333D4—9762no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—612no—
338C2×C4—1222no—
339C2×C4—1222no—
340C2×C4—9761no—
341C2×C4—2442no—
342C2×C4#1451975911yescharacteristic
343C2×C4#1451975911yescharacteristic
344C2×C4SD16×D6111yescharacteristic
345C2×C4#1451980111yescharacteristic
346C2×C4#1451980111yescharacteristic
347C2×C4C2×D4×D6111yescharacteristic
348C8—42no—
349C8—1222no—
350C8—2442no—
351C8#1451975911yescharacteristic
352C8#1451975911yescharacteristic
353C8C2×#1175822411yescharacteristic
354C8C2×#1175822211yescharacteristic
355C22—41no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—1222no—
360C22—2442no—
361C22#1684007911yescharacteristic
362C22#1684007911yescharacteristic
363C22C2×SD16×D6111yescharacteristic
364C4—81no—
365C4—1222no—
366C4—4882no—
367C4#1684007911yescharacteristic
368C4#1684007911yescharacteristic
369C4C2×#1451975911yescharacteristic
370C4C2×#1451980111yescharacteristic
371C2—21no—
372C2—21no—
373C2—21no—
374C2—161no—
375C2—2442no—
376C2C2×#1684007911yescharacteristic
subgroupG/H
centerC2C2×#16840079
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2148C2
socleC2×C61C2×#481
prime radical#21455477C2
hypercenter#2148D61
hypocenterC61#33659
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. #21454427C2semidirect
3. #21448037C2semidirect
4. #21448032C2semidirect
5. C2×#16839586C2semidirect
6. C2×#16839585C2semidirect
7. #21447568C2semidirect
8. #21447439C2semidirect
9. C61×#2148C2semidirect
190. #2148D61semidirect
221. C61#33659semidirect

Maximal subgroups

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

subgroupindex
#214554772
#214544272
#214480372
#214480322
C2×#168395862
C2×#168395852
#214475682
#214474392
C61×#21482
#3365961