Finite Group Databasebeta

← #11798454

Subgroups — #11798454

Order 1152.

477 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 80 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
1C3×#5757C211yesmaximal, characteristic
2C2×#63654C211yesmaximal, characteristic, complemented
3#10590932C211yesmaximal, characteristic, complemented
4#10590638C211yesmaximal, characteristic, complemented
5C3×#4563C211yesmaximal, characteristic, complemented
6#10589120C211yesmaximal, characteristic, complemented
7#10589111C211yesmaximal, characteristic, complemented
8#72123—31nomaximal
9#69259—31nomaximal
10C3×#1004—21no—
11C2×C32×D8C2211yescharacteristic
12C2×#3746C2211yescharacteristic, complemented
13C3×#932—21no—
14C3×#912—21no—
15C3×#911—22no—
16C3×D3×D8—21no—
17C3×#904C2211yescharacteristic
18C3×#901C2211yescharacteristic
19#63831—21no—
20#63654—22no—
21C2×C3⋊(C3⋊C16)C2211yescharacteristic, complemented
22C3×#819C2211yescharacteristic, complemented
23C3×#812C2211yescharacteristic, complemented
24#63561—21no—
25#63551—21no—
26#63548—21no—
27C3×#576—31no—
28#5757D311yescharacteristic
29#5195—31no—
30C2×#827—31no—
31C2×#827—31no—
32C2×#827—61no—
33#5143—31no—
34#4909—31no—
35#4896—31no—
36C3×#361—31no—
37#4567—31no—
38#4563D311yescharacteristic, complemented
39#4561—31no—
40#4515—31no—
41#4497—31no—
42C2×C32×D4—21no—
43C2×#634—21no—
44C2×C3×#67—42no—
45C3×#237—22no—
46C3×#237—41no—
47C3×D3×D4—22no—
48C3×D3×D4—41no—
49C3×#235—21no—
50C3×#235D411yescharacteristic
51C32×D8—21no—
52C32×D8D412yes—
53C2×C32×C8C2311yescharacteristic
54#3746—21no—
55#3746—21no—
56C3×#214—22no—
57C3×#213—22no—
58C3×#208—21no—
59C3×#208D411yescharacteristic
60C3×Dic12D411yescharacteristic
61C3×D24D411yescharacteristic
62C3×#204—21no—
63C3×ΓL(1,𝔽25)—21no—
64C3×D3×C8—21no—
65#3719—41no—
66#3716—41no—
67C3⋊(C3⋊C16)—21no—
68C3×C3⋊C16—21no—
69#2194—91noSylow
70C3×H(5,𝔽2)—61no—
71#1004—21no—
72C3×AGL(1,ℤ/8)—61no—
73C3×AGL(1,ℤ/8)—62no—
74C3×#135—31no—
75C3×#135—61no—
76C2×C3×D8—21no—
77C2×C3×D8—61no—
78C2×C3×D8C2×D311yescharacteristic
79C2×C3×D8C2×D311yescharacteristic
80C3×#131—31no—
81#951—61no—
82#932—21no—
83#912—21no—
84#911—22no—
85D3×D8—21no—
86#909—61no—
87C2×Dic12—31no—
88C2×Dic12—31no—
89C2×Dic12—61no—
90#904C2×D311yescharacteristic
91#901C2×D311yescharacteristic
92C3×M32—31no—
93C3×#108—31no—
94#829—61no—
95#827—61no—
96#827—64no—
97#827—62no—
98#827—62no—
99#824—61no—
100#824—61no—
101#819C2×D311yescharacteristic, complemented
102#812C2×D311yescharacteristic, complemented
103C2×C3⋊C16—31no—
104C2×C3⋊C16—31no—
105C2×C3⋊C16—61no—
106Dic24—61no—
107#800—61no—
108C23×C32—41no—
109C22×C3×D3—42no—
110C32×D4—22no—
111C32×D4—41no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C3⋊Dic3—41no—
114#634—41no—
115#634—41no—
116C3×#67—21no—
117C3×#67—42no—
118C3×#67—42no—
119C3×#67—41no—
120C2×C3×Dic3—42no—
121C3×D12—21no—
122C3×D12D811yescharacteristic
123C3×C4×D3—21no—
124C3×C4×D3—41no—
125C3×Dic6—21no—
126C3×Dic6D811yescharacteristic
127C32×C8C2×D411yescharacteristic
128C32×C8C2×D411yescharacteristic
129C3×C3⋊C8—21no—
130C3×C3⋊C8—41no—
131#576—31no—
132#510—91no—
133C2×SD32—91no—
134#473—91no—
135#361—31no—
136#359—91no—
137#350—91no—
138C3×#41—31no—
139C3×#41—62no—
140C3×#41—61no—
141C3×#41—121no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—41no—
145C2×C3×D4—62no—
146C2×C3×D4—121no—
147C2×C3×D4—122no—
148C2×#67—42no—
149#237—22no—
150#237—41no—
151D3×D4—22no—
152D3×D4—41no—
153#235—21no—
154#235#6711yescharacteristic
155C2×Dic6—61no—
156C2×Dic6—61no—
157C2×Dic6—121no—
158C3×Q16—31no—
159C3×SD16—62no—
160C3×SD16—61no—
161C3×D8—21no—
162C3×D8—21no—
163C3×D8—22no—
164C3×D8—31no—
165C3×D8—41no—
166C3×D8—62no—
167C3×D8D1212yes—
168C3×D8#6712yes—
169C3×M16—31no—
170C3×M16—61no—
171C3×M16—61no—
172C2×C3×C8—21no—
173C2×C3×C8—61no—
174C2×C3×C8C22×D311yescharacteristic
175C2×C3×C8C22×D311yescharacteristic
176#216—121no—
177#214—22no—
178#214—121no—
179#213—22no—
180#208—21no—
181#208#6711yescharacteristic
182Dic12—62no—
183Dic12—61no—
184Dic12—61no—
185Dic12—61no—
186Dic12—61no—
187Dic12—121no—
188Dic12—121no—
189Dic12#6711yescharacteristic
190D24#6711yescharacteristic
191#204—21no—
192#204—121no—
193ΓL(1,𝔽25)—21no—
194D3×C8—21no—
195C3×C16—61no—
196C3⋊C16—21no—
197C3⋊C16—62no—
198C3⋊C16—61no—
199C3⋊C16—61no—
200C22×C32—42no—
201C22×C32—161no—
202C22×C32AGL(1,ℤ/8)11yescharacteristic
203C2×C3×D3—21no—
204C2×C3×D3—41no—
205C2×C3×D3—162no—
206C32×C4#12011yescharacteristic
207C32×C4C2×D811yescharacteristic
208C3⋊Dic3—81no—
209C3×Dic3—21no—
210C3×Dic3—41no—
211H(5,𝔽2)—61no—
212#137—181no—
213AGL(1,ℤ/8)—61no—
214AGL(1,ℤ/8)—62no—
215#135—31no—
216#135—61no—
217C2×Q16—91no—
218C2×D8—61no—
219C2×D8D3211yescharacteristic
220#131—31no—
221Q32—181no—
222SD32—181no—
223SD32—182no—
224M32—31no—
225C2×C16—91no—
226#108—31no—
227#101—181no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—81no—
231C23×C3—122no—
232C22×D3—42no—
233C3×Q8—31no—
234C3×Q8—61no—
235C3×D4—22no—
236C3×D4—22no—
237C3×D4—31no—
238C3×D4—42no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—61no—
242C3×D4—61no—
243C3×D4—81no—
244C3×D4—122no—
245C3×D4—122no—
246C3×D4—121no—
247C2×C3×C4—21no—
248C2×C3×C4—61no—
249C2×C3×C4—122no—
250C2×C3×C4—121no—
251C2×C3×C4D3×D411yescharacteristic
252C2×C3×C4C2×#6711yescharacteristic
253#67—21no—
254#67—42no—
255#67—42no—
256#67—41no—
257C2×Dic3—42no—
258C2×Dic3—121no—
259C2×Dic3—121no—
260C2×Dic3—241no—
261D12—21no—
262D12#21311yescharacteristic
263C4×D3—21no—
264C4×D3—41no—
265Dic6—21no—
266Dic6—122no—
267Dic6—121no—
268Dic6—121no—
269Dic6—121no—
270Dic6—121no—
271Dic6—241no—
272Dic6#21311yescharacteristic
273C3×C8—21no—
274C3×C8—21no—
275C3×C8—61no—
276C3×C8—121no—
277C3×C8C2×D1211yescharacteristic
278C3×C8D3×D411yescharacteristic
279C3×C8C2×#6711yescharacteristic
280C3×C8C2×#6711yescharacteristic
281C3⋊C8—21no—
282C3⋊C8—41no—
283C2×C32—21no—
284C2×C32—82no—
285C2×C32#44711yescharacteristic
286C3×D3—41no—
287C3×D3—81no—
288#41—31no—
289#41—62no—
290#41—61no—
291#41—121no—
292C2×Q8—181no—
293C2×D4—21no—
294C2×D4—62no—
295C2×D4—121no—
296C2×D4—122no—
297Q16—31no—
298Q16—181no—
299Q16—181no—
300Q16—361no—
301SD16—62no—
302SD16—61no—
303SD16—361no—
304D8—21no—
305D8—31no—
306D8—62no—
307D8#62612yes—
308M16—31no—
309M16—61no—
310M16—61no—
311C2×C8—61no—
312C2×C8C2×D3211yescharacteristic
313C16—61no—
314C16—181no—
315C22×C3—21no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—61no—
319C22×C3—82no—
320C22×C3—121no—
321C22×C3—161no—
322C22×C3—161no—
323C22×C3—162no—
324C22×C3—482no—
325C22×C3#91411yescharacteristic
326C22×C3#93211yescharacteristic
327C2×D3—21no—
328C2×D3—41no—
329C2×D3—162no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—61no—
333C3×C4—121no—
334C3×C4#88211yescharacteristic
335C3×C4D3×D811yescharacteristic
336C3×C4C2×#21311yescharacteristic
337C3×C4#95311yescharacteristic
338Dic3—21no—
339Dic3—41no—
340Dic3—242no—
341Dic3—241no—
342Dic3—241no—
343C32#219411yescharacteristic, complemented, Sylow
344C23—41no—
345C23—122no—
346Q8—31no—
347Q8—61no—
348Q8—361no—
349Q8—361no—
350D4—22no—
351D4—31no—
352D4—41no—
353D4—61no—
354D4—61no—
355D4—122no—
356D4—122no—
357D4—121no—
358C2×C4—61no—
359C2×C4—122no—
360C2×C4—121no—
361C2×C4—361no—
362C2×C4D3×#6711yescharacteristic
363C8—61no—
364C8—121no—
365C8C2×#62611yescharacteristic
366C8D3×#6711yescharacteristic
367C2×C3—22no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—82no—
372C2×C3—82no—
373C2×C3—121no—
374C2×C3—162no—
375C2×C3—241no—
376C2×C3#477311yescharacteristic
377C2×C3#521711yescharacteristic
378D3—41no—
379D3—81no—
380C22—42no—
381C22—61no—
382C22—121no—
383C22—161no—
384C22—482no—
385C22#6393411yescharacteristic
386C4—61no—
387C4—121no—
388C4—721no—
389C4D3×#21311yescharacteristic
390C4#6397711yescharacteristic
391C3—21no—
392C3#6925911yescharacteristic, complemented
393C3#7212311yescharacteristic, complemented
394C2—21no—
395C2—82no—
396C2—121no—
397C2—241no—
398C2#1059144811yescharacteristic
subgroupG/H
centerC2#10591448
commutator, cosocleC2×C32×C8C23
FrattiniC2×C8C2×D32
FittingC2×C32×D8C22
socleC2×C32#447
prime radicalC3×#5757C2
hypercenterC2×D8D32
hypocenterC32#2194
LCSG ▷ C2×C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
2. C2×#63654C2semidirect
3. #10590932C2semidirect
4. #10590638C2semidirect
5. C3×#4563C2semidirect
6. #10589120C2semidirect
7. #10589111C2semidirect
12. C2×#3746C22semidirect
21. C2×C3⋊(C3⋊C16)C22semidirect
22. C3×#819C22semidirect
23. C3×#812C22semidirect
38. #4563D3semidirect
101. #819C2×D3semidirect
102. #812C2×D3semidirect
343. C32#2194semidirect
392. C3#69259semidirect
393. C3#72123semidirect

Maximal subgroups

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

subgroupindex
C3×#57572
C2×#636542
#105909322
#105906382
C3×#45632
#105891202
#105891112
#721233
#692593