Finite Group Databasebeta

← #11786380

Subgroups — #11786380

Order 1152.

972 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
1C22×#3743C211yesmaximal, characteristic
2C2×C32×C4⋊C8C211yesmaximal, characteristic, complemented
3C2×#63647C211yesmaximal, characteristic
4C2×#63641C211yesmaximal, characteristic, complemented
5C4×#3752C211yesmaximal, characteristic
6#10589416C211yesmaximal, characteristic, complemented
7#10589390C211yesmaximal, characteristic, complemented
8#70478—34nomaximal
9C22×C4×C3⋊D3C2211yescharacteristic
10C22×C32×C8C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C2×C32×C42C2211yescharacteristic
13C2×#3752C411yescharacteristic
14C2×#3752C411yescharacteristic
15C2×C4×C3⋊Dic3C2211yescharacteristic
16C22×C3⋊(C3⋊C8)C2211yescharacteristic
17C22×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×#3743—22no—
19C2×#3743—22no—
20C2×#3743—22no—
21C32×C4⋊C8—22no—
22#63658C411yescharacteristic
23#63658C411yescharacteristic
24#63647—22no—
25#63641—22no—
26C22×ΓL(1,𝔽25)—34no—
27C2×C3×C4⋊C8D314yescomplemented
28C2×(C2×D3)⋊C8—34no—
29C2×Dic3⋊C8—34no—
30C4×(C2×D3)⋊C4—34no—
31#4550—34no—
32#4524—34no—
33C23×C3⋊D3C2×C411yescharacteristic
34C22×C32×C4C2×C411yescharacteristic
35C22×C32×C4C2×C411yescharacteristic
36C22×C32×C4C2311yescharacteristic
37C22×C3⋊Dic3C2×C411yescharacteristic
38C22×C3⋊Dic3C2×C411yescharacteristic
39C22×C3⋊Dic3C2×C411yescharacteristic
40C2×C4×C3⋊D3—21no—
41C2×C4×C3⋊D3—21no—
42C2×C4×C3⋊D3—21no—
43C2×C4×C3⋊D3—21no—
44C2×C4×C3⋊D3D412yes—
45C2×C4×C3⋊D3Q812yes—
46C2×C32×C8—21no—
47C2×C32×C8—21no—
48C2×C32×C8—21no—
49C2×C32×C8—21no—
50C2×C32×C8—21no—
51C2×C32×C8D412yes—
52C32×C42—21no—
53C32×C42—21no—
54#3752—22no—
55#3752—22no—
56C4×C3⋊Dic3—21no—
57C4×C3⋊Dic3—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C2×C3⋊(C3⋊C8)—21no—
60C2×C3⋊(C3⋊C8)—21no—
61C2×C3⋊(C3⋊C8)—21no—
62C2×C3⋊(C3⋊C8)—21no—
63C2×C3⋊(C3⋊C8)D412yes—
64#3743—28no—
65#1851—91noSylow
66C22×C4×D3—34no—
67C22×C3×C8C2×D314yes—
68C22×C3×C8C2×D314yes—
69C2×C3×C42C2×D314yes—
70C2×(C2×D3)⋊C4—34no—
71C2×(C2×D3)⋊C4—34no—
72C2×C4×Dic3—34no—
73C22×C3⋊C8—34no—
74C22×C3⋊C8—34no—
75C2×ΓL(1,𝔽25)—68no—
76C2×ΓL(1,𝔽25)—68no—
77C2×ΓL(1,𝔽25)—68no—
78C3×C4⋊C8—28no—
79C3⋊#95—34no—
80C3⋊#95—34no—
81(C2×D3)⋊C8—68no—
82Dic3⋊C8—68no—
83C23×C32C22×C411yescharacteristic
84C22×C3⋊D3—21no—
85C22×C3⋊D3—21no—
86C22×C3⋊D3—21no—
87C22×C3⋊D3—21no—
88C22×C3⋊D3—21no—
89C22×C3⋊D3—21no—
90C22×C3⋊D3C4⋊C412yes—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4M1612yescomplemented
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×D411yescharacteristic
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×Q811yescharacteristic
101C2×C32×C4#4111yescharacteristic
102C2×C32×C4#4111yescharacteristic
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—21no—
107C2×C3⋊Dic3—21no—
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3C4⋊C412yes—
111C2×C3⋊Dic3M1612yescomplemented
112C4×C3⋊D3—22no—
113C4×C3⋊D3—22no—
114C4×C3⋊D3—22no—
115C4×C3⋊D3—22no—
116C32×C8—22no—
117C32×C8—41no—
118C3⋊(C3⋊C8)—22no—
119C3⋊(C3⋊C8)—41no—
120C22×M16—91no—
121C2×C4⋊C8—91no—
122C2×C4⋊C8C3⋊D311yescharacteristic, complemented
123C2×C22⋊C8—91no—
124C4×C22⋊C4—91no—
125#336—91no—
126#336—91no—
127C23×D3—34no—
128C22×C3×C4C4×D314yes—
129C22×C3×C4C4×D314yes—
130C22×C3×C4C22×D314yes—
131C22×Dic3—34no—
132C22×Dic3—34no—
133C22×Dic3—34no—
134C2×C4×D3—38no—
135C2×C4×D3—38no—
136C2×C4×D3—64no—
137C2×C4×D3—64no—
138C2×C4×D3—64no—
139C2×C4×D3—64no—
140C2×C3×C8—24no—
141C2×C3×C8—24no—
142C2×C3×C8—24no—
143C2×C3×C8—24no—
144C2×C3×C8—24no—
145C2×C3×C8#6718yes—
146C3×C42—24no—
147C3×C42—24no—
148(C2×D3)⋊C4—68no—
149(C2×D3)⋊C4—68no—
150C4×Dic3—64no—
151C4×Dic3—64no—
152C2×C3⋊C8—38no—
153C2×C3⋊C8—64no—
154C2×C3⋊C8—64no—
155C2×C3⋊C8—64no—
156C2×C3⋊C8—64no—
157C2×C3⋊C8—64no—
158ΓL(1,𝔽25)—632no—
159C22×C32C2×C4⋊C411yescharacteristic
160C22×C32C4×D411yescharacteristic
161C22×C32C4×D411yescharacteristic
162C22×C32C2×M1611yescharacteristic
163C22×C32C2×M1611yescharacteristic
164C22×C32#13111yescharacteristic
165C22×C32#13111yescharacteristic
166C2×C3⋊D3—22no—
167C2×C3⋊D3—22no—
168C2×C3⋊D3—22no—
169C2×C3⋊D3—41no—
170C2×C3⋊D3—41no—
171C2×C3⋊D3—41no—
172C2×C3⋊D3—41no—
173C32×C4—22no—
174C32×C4—41no—
175C32×C4#12011yescharacteristic
176C32×C4C22⋊Q811yescharacteristic
177C32×C4C22⋊Q811yescharacteristic
178C32×C4#12311yescharacteristic
179C3⋊Dic3—22no—
180C3⋊Dic3—22no—
181C3⋊Dic3—41no—
182C23×C4—91no—
183C2×M16—182no—
184C2×M16—182no—
185C2×M16—182no—
186C22×C8—91no—
187C22×C8—91no—
188C22×C8C2×C3⋊D311yescharacteristic
189C22×C8C2×C3⋊D311yescharacteristic
190C2×C22⋊C4—91no—
191C2×C22⋊C4—91no—
192C2×C42—91no—
193C2×C42C2×C3⋊D311yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—182no—
196C22⋊C8—182no—
197#95—91no—
198#95—91no—
199C23×C3C2×C4×D314yes—
200C22×D3—38no—
201C22×D3—64no—
202C22×D3—64no—
203C22×D3—64no—
204C22×D3—64no—
205C22×D3—64no—
206C22×D3—64no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—24no—
211C2×C3×C4—24no—
212C2×C3×C4ΓL(1,𝔽25)18yescomplemented
213C2×C3×C4#23514yes—
214C2×C3×C4D3×D414yes—
215C2×C3×C4D3×D414yes—
216C2×C3×C4D3×Q814yes—
217C2×C3×C4#23914yes—
218C2×C3×C4C2×#6714yes—
219C2×Dic3—38no—
220C2×Dic3—38no—
221C2×Dic3—64no—
222C2×Dic3—64no—
223C2×Dic3—64no—
224C2×Dic3—64no—
225C2×Dic3—64no—
226C2×Dic3—64no—
227C2×Dic3—64no—
228C4×D3—68no—
229C4×D3—68no—
230C4×D3—68no—
231C4×D3—68no—
232C3×C8—28no—
233C3×C8—44no—
234C3⋊C8—68no—
235C3⋊C8—124no—
236C2×C32#38511yescharacteristic
237C2×C32#42311yescharacteristic
238C2×C32#42411yescharacteristic
239C2×C32#43511yescharacteristic
240C2×C32#43511yescharacteristic
241C2×C32#43611yescharacteristic
242C2×C32#43611yescharacteristic
243C3⋊D3—42no—
244C24—91no—
245C22×C4—91no—
246C22×C4—91no—
247C22×C4—92no—
248C22×C4—92no—
249C22×C4—91no—
250C22×C4—181no—
251C22×C4—181no—
252C22×C4—181no—
253C22×C4—181no—
254C22×C4C4×C3⋊D311yescharacteristic
255C22×C4C4×C3⋊D311yescharacteristic
256C22×C4C22×C3⋊D311yescharacteristic
257M16—188no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—92no—
264C2×C8—181no—
265C2×C8—181no—
266C2×C8—181no—
267C2×C8—181no—
268C2×C8—181no—
269C2×C8#63812yes—
270C22⋊C4—182no—
271C22⋊C4—182no—
272C42—21no—
273C42—21no—
274C42—181no—
275C42—181no—
276C22×C3D3×C4⋊C414yes—
277C22×C3#89314yes—
278C22×C3C2×ΓL(1,𝔽25)14yes—
279C22×C3#90114yes—
280C22×C3D3×M1614yes—
281C22×C3#90714yes—
282C22×C3C4×#6714yes—
283C2×D3—68no—
284C2×D3—68no—
285C2×D3—68no—
286C2×D3—124no—
287C2×D3—124no—
288C2×D3—124no—
289C2×D3—124no—
290C3×C4—28no—
291C3×C4—44no—
292C3×C4#89414yes—
293C3×C4#89614yes—
294C3×C4#93714yes—
295C3×C4#94614yes—
296Dic3—68no—
297Dic3—68no—
298Dic3—124no—
299C32#185111yescharacteristic, complemented, Sylow
300C23—92no—
301C23—181no—
302C23—181no—
303C23—181no—
304C23—181no—
305C23—181no—
306C23—181no—
307C23C2×C4×C3⋊D311yescharacteristic
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—92no—
314C2×C4—92no—
315C2×C4—181no—
316C2×C4—181no—
317C2×C4—181no—
318C2×C4—181no—
319C2×C4—182no—
320C2×C4—182no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4—181no—
326C2×C4#374312yescomplemented
327C2×C4#382811yescharacteristic
328C2×C4D4×C3⋊D311yescharacteristic
329C2×C4D4×C3⋊D311yescharacteristic
330C2×C4Q8×C3⋊D311yescharacteristic
331C2×C4#383211yescharacteristic
332C2×C4C2×#63811yescharacteristic
333C8—22no—
334C8—41no—
335C8—182no—
336C8—361no—
337C2×C3#483614yes—
338C2×C3#483714yes—
339C2×C3#483814yes—
340C2×C3#483914yes—
341C2×C3#498914yes—
342C2×C3#511114yes—
343C2×C3#512814yes—
344D3—128no—
345C22—182no—
346C22—182no—
347C22—182no—
348C22—361no—
349C22—361no—
350C22—361no—
351C22—361no—
352C22C4⋊C4×C3⋊D311yescharacteristic
353C22#6410211yescharacteristic
354C22C2×#374311yescharacteristic
355C22#6411011yescharacteristic
356C22M16×C3⋊D311yescharacteristic
357C22#6411611yescharacteristic
358C22C4×#63811yescharacteristic
359C4—22no—
360C4—41no—
361C4—182no—
362C4—182no—
363C4—361no—
364C4#6410311yescharacteristic
365C4#6410511yescharacteristic
366C4#6414611yescharacteristic
367C4#6415511yescharacteristic
368C3#7047814yescomplemented
369C2—362no—
370C2#1059253511yescharacteristic
371C2#1059253611yescharacteristic
372C2#1059253711yescharacteristic
373C2#1059253811yescharacteristic
374C2#1059268811yescharacteristic
375C2#1059281011yescharacteristic
376C2#1059282711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C22×C3⋊D3
commutator, socleC23×C32C22×C4
FittingC2×C32×C4⋊C8C2
prime radicalC23×C3⋊D3C2×C4
cosocleC22×C32×C4C23
hypercenterC2×C4⋊C8C3⋊D3
hypocenterC32#1851
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4⋊C8C2semidirect
4. C2×#63641C2semidirect
6. #10589416C2semidirect
7. #10589390C2semidirect
27. C2×C3×C4⋊C8D3semidirect
96. C2×C32×C4M16semidirect
111. C2×C3⋊Dic3M16semidirect
122. C2×C4⋊C8C3⋊D3semidirect
212. C2×C3×C4ΓL(1,𝔽25)semidirect
299. C32#1851semidirect
326. C2×C4#3743semidirect
368. C3#70478semidirect

Maximal subgroups

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

subgroupindex
C22×#37432
C2×C32×C4⋊C82
C2×#636472
C2×#636412
C4×#37522
#105894162
#105893902
#704783