Finite Group Databasebeta

← #11898346

Subgroups — #11898346

Order 1152.

1080 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596203C211yesmaximal, characteristic, complemented
2#10596013C211yesmaximal, characteristic, complemented
3C32×#461C211yesmaximal, characteristic, complemented
4#10592867C212yesmaximal
5#10592865C212yesmaximal, complemented
6#10592791C211yesmaximal, characteristic, complemented
7C4×#3753C211yesmaximal, characteristic
8#10592540C211yesmaximal, characteristic, complemented
9#10592529C211yesmaximal, characteristic, complemented
10#10592497C212yesmaximal, complemented
11#10592489C212yesmaximal, complemented
12#80421—34nomaximal
13C2×C32×SD16C2212yes—
14C32×C4⋊Q8C2211yescharacteristic
15C32×C4×D4C2211yescharacteristic
16#64156C2212yes—
17Q8×C3⋊Dic3C2212yes—
18C2×#3756C2212yes—
19C2×#3756C2212yes—
20#64145C2212yes—
21C2×#3753C2211yescharacteristic
22#64135—21no—
23#64133—21no—
24C2×#3744C2212yes—
25#64107—22no—
26#64106—21no—
27#64101—21no—
28#64096C2212yes—
29#64088—22no—
30C4×C4⋊(C3⋊D3)C2211yescharacteristic
31#64078C2211yescharacteristic
32C4×#634C2211yescharacteristic
33C32×C4⋊C8C2211yescharacteristic
34C32×#103C2212yes—
35#63659C2211yescharacteristic
36#63643C2212yes—
37#63641C2212yes—
38C8×C3⋊Dic3C2212yes—
39#63636C2211yescharacteristic
40#63634C2211yescharacteristic
41C4×C3⋊(C3⋊C8)C2211yescharacteristic
42#5732—34no—
43#5542—34no—
44C3×#461D314yescomplemented
45#5168—38no—
46#5166—38no—
47#5092—34no—
48C4×#213—34no—
49#4841—34no—
50#4830—34no—
51#4798—38no—
52#4790—38no—
53C2×C32×Q8C2312yes—
54C2×C32×D4C2311yescharacteristic
55C22×C32×C4—21no—
56C2×C4⋊(C3⋊D3)C2311yescharacteristic
57C2×C4×C3⋊D3—21no—
58C2×#634D411yescharacteristic
59C2×#634C2312yes—
60C32×SD16—42no—
61C2×C32×C8C2312yes—
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2311yescharacteristic
65C32×C22⋊C4—21no—
66C32×C42C2311yescharacteristic
67#3757—22no—
68#3757—22no—
69#3756—24no—
70#3756—42no—
71#3753—22no—
72#3752—21no—
73#3752—22no—
74#3752—22no—
75C4⋊(C3⋊Dic3)—21no—
76C4⋊(C3⋊Dic3)—21no—
77C4⋊(C3⋊Dic3)—22no—
78C4⋊(C3⋊Dic3)D411yescharacteristic
79C4⋊(C3⋊Dic3)C2311yescharacteristic
80#3750—22no—
81#3750D412yes—
82C4×C3⋊Dic3—22no—
83C4×C3⋊Dic3—22no—
84C4×C3⋊Dic3C2312yes—
85C2×C3⋊(C3⋊C8)C2311yescharacteristic
86C2×C3⋊(C3⋊C8)C2311yescharacteristic
87#3744—42no—
88#3393—91noSylow
89C2×C3×SD16C2×D318yes—
90C3×C4⋊Q8C2×D314yes—
91C3×C4×D4C2×D314yes—
92#947—38no—
93Q8×Dic3—38no—
94C2×#216—38no—
95C2×#216—38no—
96#936—38no—
97C2×#213—34no—
98#926—64no—
99#924—64no—
100C2×#204—38no—
101#898—68no—
102#897—64no—
103#892—64no—
104#887—38no—
105#879—68no—
106C4×D12—34no—
107#869—34no—
108C4×Dic6—34no—
109C3×C4⋊C8C2×D314yes—
110C3×#103C2×D318yes—
111#832—34no—
112#816—38no—
113Dic3⋊C8—38no—
114C8×Dic3—38no—
115#809—34no—
116#807—34no—
117C4×C3⋊C8—34no—
118C23×C32—21no—
119C22×C3⋊D3—21no—
120C32×Q8—22no—
121C32×Q8—22no—
122C32×D4—21no—
123C32×D4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—42no—
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2411yescharacteristic
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3C2×D412yes—
133C2×C3⋊Dic3C2×D412yes—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136C4×C3⋊D3—42no—
137#634—22no—
138#634—22no—
139#634—22no—
140C32×C8—22no—
141C3⋊(C3⋊C8)—21no—
142C3⋊(C3⋊C8)#4112yes—
143#556—91no—
144#556—91no—
145#495—92no—
146#494—91no—
147#464—92no—
148#464—92no—
149#461—91no—
150#461C3⋊D311yescharacteristic, complemented
151C8×Q8—91no—
152C4×Q16—92no—
153C4×D8—91no—
154C2×C3×Q8C22×D318yes—
155C2×C3×D4C22×D314yes—
156C22×C3×C4—24no—
157C2×D12—34no—
158C2×C4×D3—64no—
159C2×Dic6—34no—
160C2×Dic6—38no—
161C3×SD16—48no—
162C2×C3×C8C22×D318yes—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4C22×D314yes—
166C3×C22⋊C4—24no—
167C3×C42C22×D314yes—
168#217—68no—
169#217—68no—
170#216—616no—
171#216—128no—
172#213—68no—
173(C2×D3)⋊C4—68no—
174(C2×D3)⋊C4—68no—
175(C2×D3)⋊C4—64no—
176C4⋊Dic3—34no—
177C4⋊Dic3—34no—
178C4⋊Dic3—68no—
179C4⋊Dic3—64no—
180C4⋊Dic3—64no—
181Dic3⋊C4—38no—
182Dic3⋊C4—68no—
183C4×Dic3—38no—
184C4×Dic3—68no—
185C4×Dic3—68no—
186C2×C3⋊C8—34no—
187C2×C3⋊C8—34no—
188#204—128no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32C22×D411yescharacteristic
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C32×C4—21no—
197C32×C4—42no—
198C32×C4—41no—
199C32×C4#13512yes—
200C32×C4C2×#4111yescharacteristic
201C32×C4H(5,𝔽2)11yescharacteristic
202C3⋊Dic3—22no—
203C3⋊Dic3—22no—
204C3⋊Dic3—42no—
205C3⋊Dic3—41no—
206C2×Q16—92no—
207C2×Q16—92no—
208C2×SD16—92no—
209C2×SD16C2×C3⋊D312yes—
210C2×D8—91no—
211C4⋊Q8—91no—
212C4⋊Q8C2×C3⋊D311yescharacteristic
213#126—182no—
214#126—182no—
215#124—92no—
216#124—92no—
217C22⋊Q8—181no—
218C22⋊Q8—181no—
219C4×Q8—91no—
220C4×Q8—92no—
221C4×Q8—92no—
222C4×D4—91no—
223C4×D4C2×C3⋊D311yescharacteristic
224#117—181no—
225#117—181no—
226#107—91no—
227C4⋊C8—92no—
228C4⋊C8C2×C3⋊D311yescharacteristic
229#103—92no—
230#103C2×C3⋊D312yes—
231#102—91no—
232#102—91no—
233C4×C8—92no—
234C4×C8—91no—
235C23×C3—24no—
236C22×D3—64no—
237C3×Q8—28no—
238C3×Q8—28no—
239C3×D4—24no—
240C3×D4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—28no—
243C2×C3×C4—48no—
244C2×C3×C4D3×D414yes—
245C2×C3×C4D3×D414yes—
246C2×C3×C4C23×D314yes—
247C2×Dic3—38no—
248C2×Dic3—38no—
249C2×Dic3—64no—
250C2×Dic3—68no—
251D12—64no—
252D12—64no—
253C4×D3—128no—
254Dic6—68no—
255Dic6—68no—
256Dic6—68no—
257C3×C8—28no—
258C3⋊C8—38no—
259C3⋊C8—64no—
260C2×C32—81no—
261C2×C32#55011yescharacteristic
262C2×C32C2×#13511yescharacteristic
263C2×C32#57811yescharacteristic
264C3⋊D3—81no—
265C2×Q8—91no—
266C2×Q8—92no—
267C2×Q8C22×C3⋊D312yes—
268C2×D4—91no—
269C2×D4C22×C3⋊D311yescharacteristic
270C22×C4—21no—
271C22×C4—181no—
272Q16—184no—
273Q16—362no—
274SD16—42no—
275SD16—362no—
276D8—182no—
277C2×C8—91no—
278C2×C8—91no—
279C2×C8C22×C3⋊D312yes—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—91no—
283C4⋊C4—92no—
284C4⋊C4—91no—
285C4⋊C4—182no—
286C4⋊C4—182no—
287C4⋊C4—181no—
288C4⋊C4—181no—
289C4⋊C4C22×C3⋊D311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—182no—
292C22⋊C4—182no—
293C22⋊C4—182no—
294C22⋊C4—182no—
295C22⋊C4—181no—
296C42—92no—
297C42—182no—
298C42—182no—
299C42C22×C3⋊D311yescharacteristic
300C22×C3—44no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3C2×D3×D414yes—
304C2×D3—124no—
305C2×D3—124no—
306C2×D3—124no—
307C3×C4—24no—
308C3×C4—44no—
309C3×C4—48no—
310C3×C4#91618yes—
311C3×C4#100414yes—
312C3×C4D3×#4114yes—
313Dic3—68no—
314Dic3—68no—
315Dic3—124no—
316Dic3—128no—
317C32#339311yescharacteristic, complemented, Sylow
318C23—21no—
319C23—181no—
320Q8—22no—
321Q8—22no—
322Q8—182no—
323Q8—182no—
324Q8—182no—
325D4—21no—
326D4—21no—
327D4—181no—
328D4—181no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—92no—
333C2×C4—92no—
334C2×C4—181no—
335C2×C4—182no—
336C2×C4—362no—
337C2×C4D4×C3⋊D311yescharacteristic
338C2×C4D4×C3⋊D311yescharacteristic
339C2×C4C23×C3⋊D311yescharacteristic
340C8—22no—
341C8—92no—
342C8—181no—
343C2×C3—84no—
344C2×C3#559914yes—
345C2×C3C2×#91614yes—
346C2×C3#577914yes—
347D3—244no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—361no—
352C22—361no—
353C22—361no—
354C22C2×D4×C3⋊D311yescharacteristic
355C4—21no—
356C4—41no—
357C4—42no—
358C4—182no—
359C4—182no—
360C4—361no—
361C4—362no—
362C4#6412512yes—
363C4#6436111yescharacteristic
364C4#41×C3⋊D311yescharacteristic
365C3#8042114yescomplemented
366C2—81no—
367C2—721no—
368C2#1059607011yescharacteristic
369C2C2×#6412511yescharacteristic
370C2#1059625011yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#461C2
socleC22×C32C22×D4
prime radicalC2×#3753C22
hypercenter#461C3⋊D3
hypocenterC32#3393
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #10596203C2semidirect
2. #10596013C2semidirect
3. C32×#461C2semidirect
5. #10592865C2semidirect
6. #10592791C2semidirect
8. #10592540C2semidirect
9. #10592529C2semidirect
10. #10592497C2semidirect
11. #10592489C2semidirect
44. C3×#461D3semidirect
150. #461C3⋊D3semidirect
317. C32#3393semidirect
365. C3#80421semidirect

Maximal subgroups

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

subgroupindex
#105962032
#105960132
C32×#4612
#105928672
#105928652
#105927912
C4×#37532
#105925402
#105925292
#105924972
#105924892
#804213