Finite Group Databasebeta

← #11798456

Subgroups — #11798456

Order 1152.

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

normal subgroups only

This group has 463 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×C3×#827C211yesmaximal, characteristic, complemented
3#10590930C211yesmaximal, characteristic, complemented
4#10590636C211yesmaximal, characteristic, complemented
5#10589429C211yesmaximal, characteristic, complemented
6#10589121C211yesmaximal, characteristic, complemented
7#10589112C211yesmaximal, characteristic, complemented
8#72123—31nomaximal
9#71490—31nomaximal
10C3×#1004—21no—
11C2×C32×D8C2211yescharacteristic
12C3×#932—21no—
13C3×#912—21no—
14C3×#911—22no—
15C3×D3×D8—21no—
16C2×C3×Dic12C2211yescharacteristic, complemented
17C3×#904C2211yescharacteristic
18C3×#901C2211yescharacteristic
19#63827—21no—
20#63646C2211yescharacteristic, complemented
21#63639C2211yescharacteristic, complemented
22C3×#827—22no—
23C2×C3×C3⋊C16C2211yescharacteristic, complemented
24#63560—21no—
25#63550—21no—
26#63547—21no—
27C3×#576—31no—
28#5757D311yescharacteristic
29C2×C3×SD32—31no—
30#5195—31no—
31#5192—31no—
32#5179—31no—
33C2×#827D311yescharacteristic, complemented
34#5143—31no—
35#4567—31no—
36#4563—31no—
37#4563—31no—
38#4563—61no—
39#4561—31no—
40#4510—31no—
41#4508—31no—
42C2×C32×D4—21no—
43C2×C3×#67—42no—
44C3×#237—22no—
45C3×#237—41no—
46C3×D3×D4—22no—
47C3×D3×D4—41no—
48C3×#235—21no—
49C3×#235D411yescharacteristic
50C2×C3×Dic6—21no—
51C32×D8—21no—
52C32×D8D412yes—
53C2×C32×C8C2311yescharacteristic
54#3748—21no—
55C3×#214—22no—
56C3×#213—22no—
57C3×#208D411yescharacteristic
58C3×Dic12—21no—
59C3×Dic12—21no—
60C3×Dic12D411yescharacteristic
61C3×D24D411yescharacteristic
62C3×#204—21no—
63C3×ΓL(1,𝔽25)—21no—
64C3×D3×C8—21no—
65#3718—41no—
66#3715—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×Q16—31no—
77C2×C3×D8—21no—
78C2×C3×D8—61no—
79C2×C3×D8C2×D311yescharacteristic
80C2×C3×D8C2×D311yescharacteristic
81C3×#131—31no—
82#951—61no—
83#942—61no—
84#932—21no—
85#912—21no—
86#911—22no—
87D3×D8—21no—
88C2×Dic12C2×D311yescharacteristic, complemented
89#904C2×D311yescharacteristic
90#901C2×D311yescharacteristic
91C3×SD32—62no—
92C2×C3×C16—31no—
93#836—61no—
94#829—61no—
95#829—61no—
96#828—61no—
97#827—22no—
98#827—61no—
99#824—61no—
100#819—31no—
101#819—31no—
102#819—61no—
103#812—31no—
104#812—31no—
105#812—61no—
106C2×C3⋊C16C2×D311yescharacteristic, complemented
107C23×C32—41no—
108C22×C3×D3—42no—
109C32×D4—22no—
110C32×D4—41no—
111C2×C32×C4C2×D411yescharacteristic
112C3×#67—21no—
113C3×#67—42no—
114C3×#67—42no—
115C3×#67—41no—
116C2×C3×Dic3—42no—
117C2×C3×Dic3—41no—
118C3×D12—21no—
119C3×D12D811yescharacteristic
120C3×C4×D3—21no—
121C3×C4×D3—41no—
122C3×Dic6—21no—
123C3×Dic6—41no—
124C3×Dic6—41no—
125C3×Dic6D811yescharacteristic
126C32×C8C2×D411yescharacteristic
127C32×C8C2×D411yescharacteristic
128C3⋊(C3⋊C8)—41no—
129C3×C3⋊C8—21no—
130#576—31no—
131#510—91no—
132C2×SD32—31no—
133#473—91no—
134#361—91no—
135#359—91no—
136#350—91no—
137C3×#41—31no—
138C3×#41—62no—
139C3×#41—61no—
140C3×#41—121no—
141C2×C3×Q8—61no—
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—21no—
156C3×Q16—31no—
157C3×Q16—61no—
158C3×Q16—61no—
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×D8#6712yes—
168C3×D8#6712yes—
169C3×M16—31no—
170C3×M16—61no—
171C2×C3×C8—21no—
172C2×C3×C8—61no—
173C2×C3×C8C22×D311yescharacteristic
174C2×C3×C8C22×D311yescharacteristic
175#216—121no—
176#216—121no—
177#215—121no—
178#214—22no—
179#214—121no—
180#213—22no—
181#208—61no—
182#208—61no—
183#208—121no—
184#208#6711yescharacteristic
185Dic12—21no—
186Dic12—21no—
187Dic12#6711yescharacteristic
188D24#6711yescharacteristic
189#204—21no—
190ΓL(1,𝔽25)—21no—
191D3×C8—21no—
192C3×C16—61no—
193C3⋊C16—21no—
194C3⋊C16—61no—
195C3⋊C16—61no—
196C3⋊C16—121no—
197C22×C32—42no—
198C22×C32—161no—
199C22×C32AGL(1,ℤ/8)11yescharacteristic
200C2×C3×D3—21no—
201C2×C3×D3—41no—
202C2×C3×D3—162no—
203C32×C4#12011yescharacteristic
204C32×C4C2×D811yescharacteristic
205C3×Dic3—21no—
206C3×Dic3—41no—
207C3×Dic3—81no—
208H(5,𝔽2)—61no—
209#137—181no—
210AGL(1,ℤ/8)—61no—
211AGL(1,ℤ/8)—62no—
212#135—31no—
213#135—61no—
214C2×Q16—31no—
215C2×D8—61no—
216C2×D8D3211yescharacteristic
217#131—31no—
218Q32—181no—
219SD32—62no—
220SD32—181no—
221M32—91no—
222C2×C16—31no—
223#108—91no—
224#101—181no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—81no—
228C23×C3—122no—
229C22×D3—42no—
230C3×Q8—31no—
231C3×Q8—61no—
232C3×Q8—121no—
233C3×Q8—121no—
234C3×D4—22no—
235C3×D4—22no—
236C3×D4—31no—
237C3×D4—42no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—61no—
241C3×D4—61no—
242C3×D4—81no—
243C3×D4—122no—
244C3×D4—122no—
245C3×D4—121no—
246C2×C3×C4—21no—
247C2×C3×C4—61no—
248C2×C3×C4—122no—
249C2×C3×C4—121no—
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—41no—
259D12—21no—
260D12#21311yescharacteristic
261C4×D3—21no—
262C4×D3—41no—
263Dic6—21no—
264Dic6—41no—
265Dic6—41no—
266Dic6#21311yescharacteristic
267C3×C8—21no—
268C3×C8—21no—
269C3×C8—61no—
270C3×C8D3×D411yescharacteristic
271C3×C8C2×#6711yescharacteristic
272C3×C8C2×#6711yescharacteristic
273C3×C8C2×#6711yescharacteristic
274C3⋊C8—21no—
275C3⋊C8—121no—
276C3⋊C8—121no—
277C3⋊C8—241no—
278C2×C32—21no—
279C2×C32—82no—
280C2×C32#44711yescharacteristic
281C3×D3—41no—
282C3×D3—81no—
283#41—31no—
284#41—62no—
285#41—61no—
286#41—121no—
287C2×Q8—61no—
288C2×D4—21no—
289C2×D4—62no—
290C2×D4—121no—
291C2×D4—122no—
292Q16—31no—
293Q16—61no—
294Q16—61no—
295Q16—361no—
296SD16—62no—
297SD16—61no—
298SD16—361no—
299D8—21no—
300D8—31no—
301D8—62no—
302D8#62512yes—
303M16—31no—
304M16—61no—
305M16—181no—
306C2×C8—61no—
307C2×C8C2×D3211yescharacteristic
308C16—61no—
309C16—181no—
310C22×C3—21no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—61no—
314C22×C3—82no—
315C22×C3—121no—
316C22×C3—161no—
317C22×C3—161no—
318C22×C3—162no—
319C22×C3—482no—
320C22×C3#91111yescharacteristic
321C22×C3#93211yescharacteristic
322C2×D3—21no—
323C2×D3—41no—
324C2×D3—162no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—61no—
328C3×C4—121no—
329C3×C4—241no—
330C3×C4D3×D811yescharacteristic
331C3×C4C2×#21311yescharacteristic
332C3×C4#93711yescharacteristic
333C3×C4#95311yescharacteristic
334Dic3—21no—
335Dic3—41no—
336Dic3—81no—
337C32#219411yescharacteristic, complemented, Sylow
338C23—41no—
339C23—122no—
340Q8—31no—
341Q8—61no—
342Q8—121no—
343Q8—121no—
344D4—22no—
345D4—31no—
346D4—41no—
347D4—61no—
348D4—61no—
349D4—122no—
350D4—122no—
351D4—121no—
352C2×C4—61no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4D3×#6711yescharacteristic
357C8—61no—
358C8—361no—
359C8C2×#62511yescharacteristic
360C8D3×#6711yescharacteristic
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—41no—
365C2×C3—82no—
366C2×C3—82no—
367C2×C3—121no—
368C2×C3—162no—
369C2×C3—241no—
370C2×C3#515611yescharacteristic
371C2×C3#521711yescharacteristic
372D3—41no—
373D3—81no—
374C22—42no—
375C22—61no—
376C22—121no—
377C22—161no—
378C22—482no—
379C22#6393211yescharacteristic
380C4—61no—
381C4—121no—
382C4—241no—
383C4D3×#21311yescharacteristic
384C4#6397611yescharacteristic
385C3—21no—
386C3#7149011yescharacteristic, complemented
387C3#7212311yescharacteristic, complemented
388C2—21no—
389C2—82no—
390C2—121no—
391C2—241no—
392C2#1059144711yescharacteristic
subgroupG/H
centerC2#10591447
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×C3×#827C2semidirect
3. #10590930C2semidirect
4. #10590636C2semidirect
5. #10589429C2semidirect
6. #10589121C2semidirect
7. #10589112C2semidirect
16. C2×C3×Dic12C22semidirect
20. #63646C22semidirect
21. #63639C22semidirect
23. C2×C3×C3⋊C16C22semidirect
33. C2×#827D3semidirect
88. C2×Dic12C2×D3semidirect
106. C2×C3⋊C16C2×D3semidirect
337. C32#2194semidirect
386. C3#71490semidirect
387. 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×C3×#8272
#105909302
#105906362
#105894292
#105891212
#105891122
#721233
#714903