Finite Group Databasebeta

← #29598733

Subgroups — #29598733

Order 15616.

532 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16840203C211yesmaximal, characteristic, complemented
2C2×#16840034C211yesmaximal, characteristic, complemented
3#21455491C211yesmaximal, characteristic
4C2×#16839570C211yesmaximal, characteristic, complemented
5C2×#16839545C211yesmaximal, characteristic, complemented
6#21443927C211yesmaximal, characteristic, complemented
7C2×C61×#358C211yesmaximal, characteristic, complemented
8C2×#14519808—21no—
9C2×#14519799C2211yescharacteristic
10C2×#14519783C2211yescharacteristic
11#16840343—21no—
12#16840203—22no—
13#16840203—22no—
14C2×#14519723—21no—
15C2×#14519720—21no—
16C2×#14519717C2211yescharacteristic
17#16840034—22no—
18#16840034—22no—
19#16840009—22no—
20#16840004—21no—
21#16839999—21no—
22C2×#14519713C2211yescharacteristic
23#16839761—21no—
24C2×#14519696—21no—
25#16839570—22no—
26#16839545—22no—
27C22×Q16×C61C2211yescharacteristic
28C22×C16×C61C2211yescharacteristic
29C2×#107×C61C2211yescharacteristic
30C61×#358—22no—
31C2×#11758222—21no—
32C2×#11758222D411yescharacteristic
33#14519808—24no—
34#14519808—42no—
35#14519799—21no—
36#14519799—21no—
37#14519799—21no—
38#14519799—21no—
39C2×Dic244D411yescharacteristic
40C2×#11758202—41no—
41C2×#11758190—21no—
42C2×#11758201—41no—
43C2×D488D411yescharacteristic
44#14519783—21no—
45#14519783—21no—
46#14519783—21no—
47#14519783—21no—
48C2×#11758194—21no—
49C2×#11758194D411yescharacteristic
50C2×#11758189—21no—
51C2×C8×D61—21no—
52#14519757—22no—
53#14519757—41no—
54#14519752—22no—
55#14519752—41no—
56#14519751—42no—
57#14519723—24no—
58#14519720—24no—
59#14519717—21no—
60#14519717D412yescomplemented
61C2×#11758185—21no—
62#14519713—21no—
63#14519713D412yescomplemented
64#14519700—22no—
65#14519697—22no—
66#14519696—24no—
67C22×Q8×C61—21no—
68C2×Q16×C61—21no—
69C2×Q16×C61—21no—
70C2×Q16×C61—21no—
71C2×Q16×C61—21no—
72C2×Q16×C61—21no—
73C2×Q16×C61D412yescomplemented
74C22×C8×C61C2311yescharacteristic
75C2×C4⋊C4×C61—21no—
76C2×C16×C61—21no—
77C2×C16×C61—21no—
78C2×C16×C61—21no—
79#107×C61—21no—
80#107×C61D412yes—
81C2×Q8×C61—22no—
82C2×Q8×C61—41no—
83C2×Q8×C61—41no—
84C2×Q8×C61—41no—
85C2×Q8×C61—41no—
86C2×Q8×C61—41no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61C2×D411yescharacteristic
90C2×C22⋊D61—21no—
91C2×C22⋊D61—41no—
92#11758222—21no—
93#11758222—21no—
94#11758222—21no—
95#11758222—21no—
96#11758222—41no—
97#11758222—41no—
98#11758222—41no—
99#11758222—41no—
100C2×D244—21no—
101C2×D244—42no—
102C2×D244D811yescharacteristic
103C2×C4×D61—21no—
104C2×C4×D61—41no—
105C2×Dic122—21no—
106C2×Dic122D811yescharacteristic
107Q16×C61—22no—
108Q16×C61—22no—
109Q16×C61—41no—
110Q16×C61—41no—
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61#4111yescharacteristic
117C4⋊C4×C61—22no—
118C4⋊C4×C61—41no—
119#11758202—44no—
120#11758201—44no—
121#11758198—42no—
122#11758198—42no—
123C4⋊Dic61—42no—
124#11758194—22no—
125#11758194—21no—
126#11758194—21no—
127#11758194—41no—
128C2×#92418—21no—
129C2×#92418—41no—
130Dic244—21no—
131Dic244—21no—
132D488—21no—
133D488—21no—
134#11758190—41no—
135#11758190—41no—
136#11758189—41no—
137#11758189—41no—
138C8×D61—41no—
139C8×D61—41no—
140C16×C61—41no—
141#11758185—22no—
142C23×C61AGL(1,ℤ/8)11yescharacteristic
143C22×D61—21no—
144C22×D61—41no—
145Q8×C61—42no—
146Q8×C61—42no—
147Q8×C61—81no—
148Q8×C61—81no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—81no—
152C2×C4×C61—81no—
153C2×C4×C61—161no—
154C2×C4×C61#12011yescharacteristic
155C2×C4×C61#12011yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61#13511yescharacteristic
159C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
160C22⋊D61—41no—
161C22⋊D61—41no—
162C22⋊D61—81no—
163C22⋊D61—81no—
164C2×Dic61—21no—
165C2×Dic61—41no—
166D244—21no—
167D244—21no—
168D244—41no—
169D244—41no—
170D244—162no—
171C4×D61—41no—
172C4×D61—41no—
173C4×D61—81no—
174C4×D61—81no—
175Dic122—21no—
176Dic122—21no—
177Dic122—41no—
178Dic122—41no—
179C8×C61#12011yescharacteristic
180C8×C61#12111yescharacteristic
181C8×C61#12111yescharacteristic
182C8×C61#12711yescharacteristic
183#92418—41no—
184#92418—42no—
185#13308—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61#44711yescharacteristic
189C22×C61#44911yescharacteristic
190C22×C61#46911yescharacteristic
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—41no—
194C2×D61—81no—
195C2×D61—81no—
196C2×D61—81no—
197C4×C61—82no—
198C4×C61—82no—
199C4×C61#39211yescharacteristic
200C4×C61#44711yescharacteristic
201C4×C61#44911yescharacteristic
202C4×C61#46611yescharacteristic
203Dic61—41no—
204Dic61—81no—
205C2×#510—611no—
206C2×#473—611no—
207#3061—611no—
208C2×#359—611no—
209C2×#358D6111yescharacteristic, complemented
210C2×#350—611no—
211#1387—611no—
212C2×C61—21no—
213C2×C61—21no—
214C2×C61#199911yescharacteristic
215C2×C61#219411yescharacteristic
216C2×C61#219511yescharacteristic
217D61—81no—
218D61—161no—
219C2×#137—1221no—
220C2×#135—611no—
221C22×Q16C2×D6111yescharacteristic
222C2×#131—611no—
223#537—1221no—
224#510—1222no—
225#510—1222no—
226C2×Q32—1221no—
227C2×SD32—1221no—
228C2×M32—611no—
229C22×C16C2×D6111yescharacteristic
230#473—1222no—
231#473—1222no—
232#468—1222no—
233#467—1221no—
234#466—1221no—
235C2×#108—611no—
236C2×#107C2×D6111yescharacteristic
237#417—1221no—
238C2×#101—1221no—
239#359—1222no—
240#358—22no—
241#350—1222no—
242C61#1330811yescharacteristic, complemented, Sylow
243C2×#41—611no—
244C2×#41—1221no—
245C22×Q8—21no—
246#137—1224no—
247#137—2442no—
248#135—1221no—
249#135—1221no—
250#135—1221no—
251#135—1221no—
252C2×Q16—21no—
253C2×Q16—21no—
254C2×Q16—21no—
255C2×Q16—21no—
256C2×Q16—21no—
257C2×Q16—611no—
258C2×Q16—2441no—
259C2×Q16D24412yescomplemented
260C2×SD16—1221no—
261C2×SD16—2441no—
262C2×D8—611no—
263#131—1221no—
264#131—1221no—
265#131—1221no—
266#131—1221no—
267C2×M16—611no—
268C2×M16—1221no—
269C2×M16—1221no—
270C22×C8—1221no—
271C22×C8C22×D6111yescharacteristic
272C22⋊Q8—1222no—
273C22⋊Q8—2441no—
274#121—1222no—
275#121—2441no—
276#121—2442no—
277C2×C4⋊C4—21no—
278Q32—1224no—
279SD32—1224no—
280M32—612no—
281M32—1221no—
282C2×C16—21no—
283C2×C16—21no—
284C2×C16—21no—
285C2×C16—1221no—
286#108—612no—
287#108—1221no—
288#107—21no—
289#107C22⋊D6112yes—
290#103—1222no—
291#102—1222no—
292#101—1224no—
293#41—1221no—
294#41—1221no—
295#41—1221no—
296#41—1221no—
297#41—2441no—
298#41—2441no—
299#41—2441no—
300#41—2441no—
301C2×Q8—22no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—611no—
308C2×Q8—1221no—
309C2×D4—611no—
310C2×D4—1221no—
311C2×D4—1221no—
312C2×D4—2442no—
313C2×D4—2441no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—1221no—
317C22×C4—2441no—
318C22×C4D4×D6111yescharacteristic
319Q16—22no—
320Q16—22no—
321Q16—41no—
322Q16—41no—
323Q16—1221no—
324Q16—1221no—
325Q16—2444no—
326SD16—2444no—
327SD16—2441no—
328SD16—2441no—
329D8—1221no—
330D8—1221no—
331M16—1221no—
332M16—1222no—
333M16—1221no—
334M16—2441no—
335M16—2441no—
336M16—2441no—
337C2×C8—1221no—
338C2×C8—2441no—
339C2×C8—2441no—
340C2×C8—2441no—
341C2×C8C2×D24411yescharacteristic
342C2×C8#1175822211yescharacteristic
343C2×C8D4×D6111yescharacteristic
344C2×C8D4×D6111yescharacteristic
345C2×C8D4×D6111yescharacteristic
346C2×C8C2×C22⋊D6111yescharacteristic
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—2442no—
350C22⋊C4—2442no—
351C22⋊C4—2442no—
352C16—41no—
353C16—1222no—
354C23—1221no—
355C23—2441no—
356C23#1451980111yescharacteristic
357Q8—42no—
358Q8—42no—
359Q8—81no—
360Q8—81no—
361Q8—1221no—
362Q8—1221no—
363Q8—2441no—
364Q8—2441no—
365D4—1221no—
366D4—1221no—
367D4—9762no—
368D4—4881no—
369D4—4881no—
370D4—2441no—
371D4—2441no—
372D4—2441no—
373D4—2441no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—161no—
379C2×C4—1221no—
380C2×C4—4881no—
381C2×C4—4881no—
382C2×C4—2441no—
383C2×C4—2441no—
384C2×C4—2441no—
385C2×C4#1451974511yescharacteristic
386C2×C4#1451974711yescharacteristic
387C2×C4#1451975011yescharacteristic
388C2×C4D8×D6111yescharacteristic
389C2×C4#1451979811yescharacteristic
390C2×C4#1451980411yescharacteristic
391C8—2442no—
392C8—2441no—
393C8#1451974711yescharacteristic
394C8#1451975011yescharacteristic
395C8#1451975111yescharacteristic
396C8#1451977311yescharacteristic
397C22—21no—
398C22—21no—
399C22—4881no—
400C22—4881no—
401C22—4881no—
402C22—2441no—
403C22—2441no—
404C22—2441no—
405C22#1683989411yescharacteristic
406C22#1683990811yescharacteristic
407C22#1684001411yescharacteristic
408C4—82no—
409C4—82no—
410C4—4881no—
411C4—2441no—
412C4#1683967111yescharacteristic
413C4#1683989711yescharacteristic
414C4#1683990611yescharacteristic
415C4#1683999711yescharacteristic
416C2—21no—
417C2—21no—
418C2—9761no—
419C2—4881no—
420C2#2144681611yescharacteristic
421C2#2144778911yescharacteristic
422C2#2144779611yescharacteristic
subgroupG/H
centerC22#16839894
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC2×C61×#358C2
socleC22×C61#447
prime radical#21455491C2
hypercenterC2×#358D61
hypocenterC61#13308
LCSG ▷ C22×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. C2×#16840034C2semidirect
4. C2×#16839570C2semidirect
5. C2×#16839545C2semidirect
6. #21443927C2semidirect
7. C2×C61×#358C2semidirect
60. #14519717D4semidirect
63. #14519713D4semidirect
73. C2×Q16×C61D4semidirect
209. C2×#358D61semidirect
242. C61#13308semidirect
259. C2×Q16D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
C2×#168400342
#214554912
C2×#168395702
C2×#168395452
#214439272
C2×C61×#3582
#1330861