Finite Group Databasebeta

← #11787166

Subgroups — #11787166

Order 1152.

1224 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3743C211yesmaximal, characteristic, complemented
2C2×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C2×#63647C211yesmaximal, characteristic, complemented
4C2×#63647C211yesmaximal, characteristic, complemented
5C2×#63647C211yesmaximal, characteristic, complemented
6C32×#336C211yesmaximal, characteristic, complemented
7#10589390C211yesmaximal, characteristic, complemented
8#68669—34nomaximal
9C22×C4⋊(C3⋊D3)C411yescharacteristic
10C22×C4×C3⋊D3C2211yescharacteristic
11C22×C4×C3⋊D3C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C2×C32×C42C2211yescharacteristic
15C2×#3752C411yescharacteristic
16C2×#3752C411yescharacteristic
17C2×C4⋊(C3⋊Dic3)C411yescharacteristic
18C22×C3⋊(C3⋊C8)C2211yescharacteristic
19C22×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×#3743—22no—
21C2×#3743—22no—
22C2×#3743—22no—
23C4×C4⋊(C3⋊D3)—22no—
24C4×C4⋊(C3⋊D3)—22no—
25#63647—22no—
26#63647—22no—
27#63647—22no—
28C22×ΓL(1,𝔽25)—34no—
29C2×C4×D12—34no—
30C2×(C2×D3)⋊C8—34no—
31C2×(C2×D3)⋊C8—34no—
32C2×(C2×D3)⋊C8—34no—
33C3×#336D314yescomplemented
34#4524—34no—
35C23×C3⋊D3C2×C411yescharacteristic
36C23×C3⋊D3C2×C411yescharacteristic
37C22×C32×C4C2×C411yescharacteristic
38C22×C32×C4C2×C411yescharacteristic
39C22×C32×C4C2311yescharacteristic
40C22×C3⋊Dic3C2×C411yescharacteristic
41C22×C3⋊Dic3C2×C411yescharacteristic
42C2×C4⋊(C3⋊D3)—22no—
43C2×C4⋊(C3⋊D3)—22no—
44C2×C4⋊(C3⋊D3)—22no—
45C2×C4×C3⋊D3—21no—
46C2×C4×C3⋊D3—21no—
47C2×C4×C3⋊D3—21no—
48C2×C4×C3⋊D3—21no—
49C2×C4×C3⋊D3—21no—
50C2×C4×C3⋊D3—21no—
51C2×C4×C3⋊D3—21no—
52C2×C4×C3⋊D3—21no—
53C2×C4×C3⋊D3—21no—
54C2×C4×C3⋊D3—21no—
55C2×C4×C3⋊D3D412yes—
56C2×C4×C3⋊D3D412yes—
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C2×C32×C8—21no—
61C2×C32×C8—21no—
62C2×C32×C8D412yescomplemented
63C32×C42—21no—
64C32×C42—21no—
65#3752—22no—
66#3752—22no—
67C4⋊(C3⋊Dic3)—22no—
68C2×C3⋊(C3⋊C8)—21no—
69C2×C3⋊(C3⋊C8)—21no—
70C2×C3⋊(C3⋊C8)—21no—
71C2×C3⋊(C3⋊C8)—21no—
72C2×C3⋊(C3⋊C8)—21no—
73C2×C3⋊(C3⋊C8)D412yescomplemented
74#3743—28no—
75#1870—91noSylow
76C22×D12—34no—
77C22×C4×D3—34no—
78C22×C4×D3—34no—
79C22×C3×C8C2×D314yes—
80C22×C3×C8C2×D314yes—
81C2×C3×C42C2×D314yes—
82C2×(C2×D3)⋊C4—34no—
83C2×(C2×D3)⋊C4—34no—
84C2×C4⋊Dic3—34no—
85C22×C3⋊C8—34no—
86C22×C3⋊C8—34no—
87C2×ΓL(1,𝔽25)—68no—
88C2×ΓL(1,𝔽25)—68no—
89C2×ΓL(1,𝔽25)—68no—
90C4×D12—68no—
91C4×D12—68no—
92(C2×D3)⋊C8—68no—
93(C2×D3)⋊C8—68no—
94(C2×D3)⋊C8—68no—
95C23×C32C22×C411yescharacteristic
96C22×C3⋊D3—21no—
97C22×C3⋊D3—21no—
98C22×C3⋊D3—21no—
99C22×C3⋊D3—21no—
100C22×C3⋊D3—21no—
101C22×C3⋊D3—21no—
102C22×C3⋊D3—21no—
103C22×C3⋊D3—21no—
104C22×C3⋊D3—21no—
105C22×C3⋊D3—21no—
106C22×C3⋊D3—21no—
107C22×C3⋊D3—21no—
108C22×C3⋊D3C22⋊C412yes—
109C22×C3⋊D3M1612yescomplemented
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4M1612yescomplemented
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4#4111yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3C22⋊C412yes—
128C4⋊(C3⋊D3)—28no—
129C4×C3⋊D3—22no—
130C4×C3⋊D3—22no—
131C4×C3⋊D3—22no—
132C4×C3⋊D3—22no—
133C4×C3⋊D3—41no—
134C4×C3⋊D3—41no—
135C4×C3⋊D3—41no—
136C4×C3⋊D3—41no—
137C32×C8—22no—
138C32×C8—41no—
139C3⋊(C3⋊C8)—22no—
140C3⋊(C3⋊C8)—41no—
141C22×M16—91no—
142C2×C4×D4—91no—
143C2×C22⋊C8—91no—
144C2×C22⋊C8—91no—
145C2×C22⋊C8—91no—
146#336—91no—
147#336C3⋊D311yescharacteristic, complemented
148C23×D3—34no—
149C23×D3—34no—
150C22×C3×C4C4×D314yes—
151C22×C3×C4C4×D314yes—
152C22×C3×C4C22×D314yes—
153C22×Dic3—34no—
154C22×Dic3—34no—
155C2×D12—68no—
156C2×D12—68no—
157C2×D12—68no—
158C2×C4×D3—38no—
159C2×C4×D3—38no—
160C2×C4×D3—64no—
161C2×C4×D3—64no—
162C2×C4×D3—64no—
163C2×C4×D3—64no—
164C2×C4×D3—64no—
165C2×C4×D3—64no—
166C2×C4×D3—64no—
167C2×C4×D3—64no—
168C2×C4×D3—64no—
169C2×C4×D3—64no—
170C2×C3×C8—24no—
171C2×C3×C8—24no—
172C2×C3×C8—24no—
173C2×C3×C8—24no—
174C2×C3×C8—24no—
175C2×C3×C8D1218yescomplemented
176C3×C42—24no—
177C3×C42—24no—
178(C2×D3)⋊C4—68no—
179(C2×D3)⋊C4—68no—
180C4⋊Dic3—68no—
181C2×C3⋊C8—38no—
182C2×C3⋊C8—64no—
183C2×C3⋊C8—64no—
184C2×C3⋊C8—64no—
185C2×C3⋊C8—64no—
186C2×C3⋊C8—64no—
187ΓL(1,𝔽25)—632no—
188C22×C32C2×C22⋊C411yescharacteristic
189C22×C32C4×D411yescharacteristic
190C22×C32C4×D411yescharacteristic
191C22×C32C2×M1611yescharacteristic
192C22×C32C2×M1611yescharacteristic
193C22×C32#13111yescharacteristic
194C22×C32#13111yescharacteristic
195C2×C3⋊D3—22no—
196C2×C3⋊D3—22no—
197C2×C3⋊D3—22no—
198C2×C3⋊D3—22no—
199C2×C3⋊D3—22no—
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C32×C4—22no—
210C32×C4—41no—
211C32×C4#12011yescharacteristic
212C32×C4#12111yescharacteristic
213C32×C4#12111yescharacteristic
214C32×C4#12311yescharacteristic
215C3⋊Dic3—22no—
216C3⋊Dic3—41no—
217C22×D4—91no—
218C23×C4—91no—
219C23×C4—91no—
220C2×M16—182no—
221C2×M16—182no—
222C2×M16—182no—
223C22×C8—91no—
224C22×C8—91no—
225C22×C8C2×C3⋊D311yescharacteristic
226C22×C8C2×C3⋊D311yescharacteristic
227C4×D4—182no—
228C4×D4—182no—
229C2×C4⋊C4—91no—
230C2×C22⋊C4—91no—
231C2×C22⋊C4—91no—
232C2×C42C2×C3⋊D311yescharacteristic
233C22⋊C8—182no—
234C22⋊C8—182no—
235C22⋊C8—182no—
236C23×C3C2×C4×D314yes—
237C22×D3—38no—
238C22×D3—38no—
239C22×D3—64no—
240C22×D3—64no—
241C22×D3—64no—
242C22×D3—64no—
243C22×D3—64no—
244C22×D3—64no—
245C22×D3—64no—
246C22×D3—64no—
247C22×D3—64no—
248C22×D3—64no—
249C22×D3—64no—
250C22×D3—64no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4ΓL(1,𝔽25)18yescomplemented
257C2×C3×C4C2×D1214yes—
258C2×C3×C4#23514yes—
259C2×C3×C4D3×D414yes—
260C2×C3×C4D3×D414yes—
261C2×C3×C4D3×D414yes—
262C2×C3×C4#23914yes—
263C2×Dic3—38no—
264C2×Dic3—64no—
265C2×Dic3—64no—
266C2×Dic3—64no—
267C2×Dic3—64no—
268C2×Dic3—64no—
269D12—632no—
270C4×D3—68no—
271C4×D3—68no—
272C4×D3—68no—
273C4×D3—68no—
274C4×D3—124no—
275C4×D3—124no—
276C4×D3—124no—
277C4×D3—124no—
278C3×C8—28no—
279C3×C8—44no—
280C3⋊C8—68no—
281C3⋊C8—124no—
282C2×C32#38611yescharacteristic
283C2×C32#40711yescharacteristic
284C2×C32#40811yescharacteristic
285C2×C32#43511yescharacteristic
286C2×C32#43511yescharacteristic
287C2×C32#43611yescharacteristic
288C2×C32#43611yescharacteristic
289C3⋊D3—42no—
290C3⋊D3—42no—
291C24—91no—
292C24—91no—
293C2×D4—182no—
294C2×D4—182no—
295C2×D4—182no—
296C22×C4—91no—
297C22×C4—92no—
298C22×C4—92no—
299C22×C4—91no—
300C22×C4—181no—
301C22×C4—181no—
302C22×C4—181no—
303C22×C4—181no—
304C22×C4—181no—
305C22×C4—181no—
306C22×C4—181no—
307C22×C4—181no—
308C22×C4—181no—
309C22×C4—181no—
310C22×C4C4×C3⋊D311yescharacteristic
311C22×C4C4×C3⋊D311yescharacteristic
312C22×C4C22×C3⋊D311yescharacteristic
313M16—188no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—92no—
320C2×C8—181no—
321C2×C8—181no—
322C2×C8—181no—
323C2×C8—181no—
324C2×C8—181no—
325C2×C8C4⋊(C3⋊D3)12yescomplemented
326C4⋊C4—182no—
327C22⋊C4—182no—
328C22⋊C4—182no—
329C42—21no—
330C42—21no—
331C22×C3C4×D1214yes—
332C22×C3D3×C22⋊C414yes—
333C22×C3#89314yes—
334C22×C3C2×ΓL(1,𝔽25)14yes—
335C22×C3#90114yes—
336C22×C3D3×M1614yes—
337C22×C3#90714yes—
338C2×D3—68no—
339C2×D3—68no—
340C2×D3—68no—
341C2×D3—68no—
342C2×D3—68no—
343C2×D3—68no—
344C2×D3—124no—
345C2×D3—124no—
346C2×D3—124no—
347C2×D3—124no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—124no—
351C2×D3—124no—
352C3×C4—28no—
353C3×C4—44no—
354C3×C4#88214yes—
355C3×C4#88414yes—
356C3×C4#89414yes—
357C3×C4#89514yes—
358Dic3—68no—
359Dic3—124no—
360C32#187011yescharacteristic, complemented, Sylow
361C23—92no—
362C23—92no—
363C23—181no—
364C23—181no—
365C23—181no—
366C23—181no—
367C23—181no—
368C23—181no—
369C23—181no—
370C23—181no—
371C23—181no—
372C23—181no—
373C23—181no—
374C23—181no—
375C23C2×C4×C3⋊D311yescharacteristic
376D4—188no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—92no—
383C2×C4—181no—
384C2×C4—181no—
385C2×C4—182no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—182no—
389C2×C4—181no—
390C2×C4—181no—
391C2×C4—181no—
392C2×C4—361no—
393C2×C4—361no—
394C2×C4—361no—
395C2×C4—361no—
396C2×C4#374312yescomplemented
397C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
398C2×C4#382811yescharacteristic
399C2×C4D4×C3⋊D311yescharacteristic
400C2×C4D4×C3⋊D311yescharacteristic
401C2×C4D4×C3⋊D311yescharacteristic
402C2×C4#383211yescharacteristic
403C8—22no—
404C8—41no—
405C8—182no—
406C8—361no—
407C2×C3#466514yes—
408C2×C3#468814yes—
409C2×C3#470614yes—
410C2×C3#472614yes—
411C2×C3#472714yes—
412C2×C3#483614yes—
413C2×C3#483814yes—
414D3—128no—
415D3—128no—
416C22—182no—
417C22—182no—
418C22—182no—
419C22—182no—
420C22—182no—
421C22—182no—
422C22—361no—
423C22—361no—
424C22—361no—
425C22—361no—
426C22—361no—
427C22—361no—
428C22—361no—
429C22—361no—
430C22C4×C4⋊(C3⋊D3)11yescharacteristic
431C22C22⋊C4×C3⋊D311yescharacteristic
432C22#6410211yescharacteristic
433C22C2×#374311yescharacteristic
434C22#6411011yescharacteristic
435C22M16×C3⋊D311yescharacteristic
436C22#6411611yescharacteristic
437C4—22no—
438C4—41no—
439C4—182no—
440C4—361no—
441C4#6409111yescharacteristic
442C4#6409311yescharacteristic
443C4#6410311yescharacteristic
444C4#6410411yescharacteristic
445C3#6866914yescomplemented
446C2—362no—
447C2—362no—
448C2#1059236411yescharacteristic
449C2#1059238711yescharacteristic
450C2#1059240511yescharacteristic
451C2#1059242511yescharacteristic
452C2#1059242611yescharacteristic
453C2#1059253511yescharacteristic
454C2#1059253711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C22×C3⋊D3
commutator, socleC23×C32C22×C4
FittingC32×#336C2
prime radicalC22×C4⋊(C3⋊D3)C4
cosocleC22×C32×C4C23
hypercenter#336C3⋊D3
hypocenterC32#1870
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ #336

Complements

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

NH ≅ G/Ntype
1. C22×#3743C2semidirect
3. C2×#63647C2semidirect
4. C2×#63647C2semidirect
5. C2×#63647C2semidirect
6. C32×#336C2semidirect
7. #10589390C2semidirect
33. C3×#336D3semidirect
62. C2×C32×C8D4semidirect
73. C2×C3⋊(C3⋊C8)D4semidirect
109. C22×C3⋊D3M16semidirect
115. C2×C32×C4M16semidirect
147. #336C3⋊D3semidirect
175. C2×C3×C8D12semidirect
256. C2×C3×C4ΓL(1,𝔽25)semidirect
325. C2×C8C4⋊(C3⋊D3)semidirect
360. C32#1870semidirect
396. C2×C4#3743semidirect
445. C3#68669semidirect

Maximal subgroups

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

subgroupindex
C22×#37432
C2×C4×C4⋊(C3⋊D3)2
C2×#636472
C32×#3362
#105893902
#686693