Finite Group Databasebeta

← #11787160

Subgroups — #11787160

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
1C2×C4×#638C211yesmaximal, characteristic
2C22×#3743C211yesmaximal, characteristic, complemented
3C2×C32×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×#63657C211yesmaximal, characteristic, complemented
5C2×#63647C211yesmaximal, characteristic, complemented
6#10589416C211yesmaximal, characteristic, complemented
7#10589416C211yesmaximal, characteristic, complemented
8#69958—34nomaximal
9C23×C32×C4C2211yescharacteristic
10C22×#638C411yescharacteristic
11C22×C4×C3⋊D3C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C2×#3757C411yescharacteristic
15C4×#638—22no—
16C4×#638—22no—
17C2×#3752C411yescharacteristic
18C2×#3750C411yescharacteristic
19C2×C4×C3⋊Dic3C2211yescharacteristic
20C22×C3⋊(C3⋊C8)C2211yescharacteristic
21C22×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×#3743—22no—
23C2×#3743—22no—
24C2×#3743—22no—
25C32×C22⋊C8—22no—
26#63657—22no—
27#63647—22no—
28C2×C4×#67—34no—
29C22×ΓL(1,𝔽25)—34no—
30C2×C3×C22⋊C8D314yescomplemented
31C2×#830—34no—
32C2×(C2×D3)⋊C8—34no—
33#4550—34no—
34#4550—34no—
35C24×C32C2×C411yescharacteristic
36C23×C3⋊D3C2×C411yescharacteristic
37C22×C32×C4—21no—
38C22×C32×C4—21no—
39C22×C32×C4—21no—
40C22×C32×C4—21no—
41C22×C32×C4—21no—
42C22×C32×C4—21no—
43C22×C32×C4C2×C411yescharacteristic
44C22×C32×C4C2311yescharacteristic
45C2×#638—22no—
46C2×#638—22no—
47C2×#638—22no—
48C22×C3⋊Dic3C2×C411yescharacteristic
49C22×C3⋊Dic3C2×C411yescharacteristic
50C22×C3⋊Dic3C2×C411yescharacteristic
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
63#3757—22no—
64#3752—22no—
65#3750—22no—
66C4×C3⋊Dic3—21no—
67C4×C3⋊Dic3—21no—
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
76C23×C3×C4C2×D314yes—
77C22×#67—34no—
78C22×C4×D3—34no—
79C22×C3×C8C2×D314yes—
80C22×C3×C8C2×D314yes—
81C2×#217—34no—
82C4×#67—68no—
83C4×#67—68no—
84C2×(C2×D3)⋊C4—34no—
85C2×Dic3⋊C4—34no—
86C2×C4×Dic3—34no—
87C22×C3⋊C8—34no—
88C22×C3⋊C8—34no—
89C2×ΓL(1,𝔽25)—68no—
90C2×ΓL(1,𝔽25)—68no—
91C2×ΓL(1,𝔽25)—68no—
92C3×C22⋊C8—28no—
93#830—68no—
94(C2×D3)⋊C8—68no—
95C23×C32—21no—
96C23×C32—21no—
97C23×C32—21no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32—21no—
101C23×C32M1612yescomplemented
102C23×C32C22×C411yescharacteristic
103C22×C3⋊D3—21no—
104C22×C3⋊D3—21no—
105C22×C3⋊D3—21no—
106C22×C3⋊D3—21no—
107C22×C3⋊D3—21no—
108C22×C3⋊D3—21no—
109C22×C3⋊D3C22⋊C412yes—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123#638—28no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3C22⋊C412yes—
132C2×C3⋊Dic3M1612yescomplemented
133C4×C3⋊D3—22no—
134C4×C3⋊D3—22no—
135C4×C3⋊D3—22no—
136C4×C3⋊D3—22no—
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⋊C8C3⋊D311yescharacteristic, complemented
146#336—91no—
147#336—91no—
148C24×C3C4×D314yes—
149C23×D3—34no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4C4×D314yes—
157C22×C3×C4C22×D314yes—
158C2×#67—68no—
159C2×#67—68no—
160C2×#67—68no—
161C22×Dic3—34no—
162C22×Dic3—34no—
163C22×Dic3—34no—
164C2×C4×D3—38no—
165C2×C4×D3—38no—
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×C8#6718yescomplemented
176#217—68no—
177(C2×D3)⋊C4—68no—
178Dic3⋊C4—68no—
179C4×Dic3—64no—
180C4×Dic3—64no—
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×C32—22no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—41no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32C2×C22⋊C411yescharacteristic
196C22×C32C4×D411yescharacteristic
197C22×C32C4×D411yescharacteristic
198C22×C32C2×M1611yescharacteristic
199C22×C32C2×M1611yescharacteristic
200C22×C32#13111yescharacteristic
201C22×C32#13111yescharacteristic
202C2×C3⋊D3—22no—
203C2×C3⋊D3—22no—
204C2×C3⋊D3—22no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C32×C4—41no—
210C32×C4#12011yescharacteristic
211C32×C4#12111yescharacteristic
212C32×C4#12111yescharacteristic
213C32×C4#12311yescharacteristic
214C3⋊Dic3—22no—
215C3⋊Dic3—22no—
216C3⋊Dic3—41no—
217C22×D4—91no—
218C23×C4—91no—
219C23×C4C2×C3⋊D311yescharacteristic
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×C42—91no—
233C22⋊C8—22no—
234C22⋊C8—182no—
235C22⋊C8—182no—
236C23×C3—24no—
237C23×C3—24no—
238C23×C3—24no—
239C23×C3—24no—
240C23×C3—24no—
241C23×C3—24no—
242C23×C3ΓL(1,𝔽25)18yescomplemented
243C23×C3C2×C4×D314yes—
244C22×D3—38no—
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—44no—
255C2×C3×C4—44no—
256C2×C3×C4—44no—
257C2×C3×C4—44no—
258C2×C3×C4#23514yes—
259C2×C3×C4D3×D414yes—
260C2×C3×C4D3×D414yes—
261C2×C3×C4D3×D414yes—
262C2×C3×C4#23714yes—
263C2×C3×C4C2×#6714yes—
264#67—632no—
265C2×Dic3—38no—
266C2×Dic3—38no—
267C2×Dic3—64no—
268C2×Dic3—64no—
269C2×Dic3—64no—
270C2×Dic3—64no—
271C2×Dic3—64no—
272C2×Dic3—64no—
273C2×Dic3—64no—
274C4×D3—68no—
275C4×D3—68no—
276C4×D3—68no—
277C4×D3—68no—
278C3×C8—28no—
279C3×C8—44no—
280C3⋊C8—68no—
281C3⋊C8—124no—
282C2×C32—42no—
283C2×C32#38611yescharacteristic
284C2×C32#40711yescharacteristic
285C2×C32#40811yescharacteristic
286C2×C32#43511yescharacteristic
287C2×C32#43511yescharacteristic
288C2×C32#43611yescharacteristic
289C2×C32#43611yescharacteristic
290C3⋊D3—42no—
291C24—91no—
292C24C4×C3⋊D311yescharacteristic
293C2×D4—182no—
294C2×D4—182no—
295C2×D4—182no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—91no—
303C22×C4—91no—
304C22×C4—92no—
305C22×C4—92no—
306C22×C4—91no—
307C22×C4—181no—
308C22×C4—181no—
309C22×C4—181no—
310C22×C4—181no—
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×C8#63812yescomplemented
326C4⋊C4—182no—
327C22⋊C4—182no—
328C22⋊C4—182no—
329C42—181no—
330C42—181no—
331C22×C3—28no—
332C22×C3—28no—
333C22×C3—28no—
334C22×C3—44no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—44no—
338C22×C3D3×C22⋊C414yes—
339C22×C3#67⋊C414yes—
340C22×C3C2×ΓL(1,𝔽25)14yes—
341C22×C3#90114yes—
342C22×C3D3×M1614yes—
343C22×C3#90714yes—
344C22×C3C4×#6714yes—
345C2×D3—68no—
346C2×D3—68no—
347C2×D3—68no—
348C2×D3—124no—
349C2×D3—124no—
350C2×D3—124no—
351C2×D3—124no—
352C3×C4—44no—
353C3×C4#88314yes—
354C3×C4#88414yes—
355C3×C4#93714yes—
356C3×C4#93814yes—
357Dic3—68no—
358Dic3—68no—
359Dic3—124no—
360C32#187011yescharacteristic, complemented, Sylow
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—92no—
368C23—181no—
369C23—181no—
370C23—181no—
371C23—181no—
372C23—181no—
373C23—181no—
374C23#374312yescomplemented
375C23C2×C4×C3⋊D311yescharacteristic
376D4—188no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—92no—
385C2×C4—92no—
386C2×C4—181no—
387C2×C4—181no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4—182no—
391C2×C4—182no—
392C2×C4—182no—
393C2×C4—182no—
394C2×C4—181no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4#382811yescharacteristic
398C2×C4D4×C3⋊D311yescharacteristic
399C2×C4D4×C3⋊D311yescharacteristic
400C2×C4D4×C3⋊D311yescharacteristic
401C2×C4#383011yescharacteristic
402C2×C4C2×#63811yescharacteristic
403C8—22no—
404C8—41no—
405C8—182no—
406C8—361no—
407C2×C3—48no—
408C2×C3#472614yes—
409C2×C3#472714yes—
410C2×C3#472914yes—
411C2×C3#473014yes—
412C2×C3#495614yes—
413C2×C3#511114yes—
414C2×C3#512814yes—
415D3—128no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—182no—
424C22—182no—
425C22—182no—
426C22—361no—
427C22—361no—
428C22—361no—
429C22—361no—
430C22C22⋊C4×C3⋊D311yescharacteristic
431C22#6409011yescharacteristic
432C22C2×#374311yescharacteristic
433C22#6411011yescharacteristic
434C22M16×C3⋊D311yescharacteristic
435C22#6411611yescharacteristic
436C22C4×#63811yescharacteristic
437C4—41no—
438C4—182no—
439C4—182no—
440C4—361no—
441C4#6409211yescharacteristic
442C4#6409311yescharacteristic
443C4#6414611yescharacteristic
444C4#6414711yescharacteristic
445C3#6995814yescomplemented
446C2—42no—
447C2—362no—
448C2#1059242511yescharacteristic
449C2#1059242611yescharacteristic
450C2#1059242811yescharacteristic
451C2#1059242911yescharacteristic
452C2#1059265511yescharacteristic
453C2#1059281011yescharacteristic
454C2#1059282711yescharacteristic
subgroupG/H
center, FrattiniC22×C4C22×C3⋊D3
commutator, socleC23×C32C22×C4
FittingC2×C32×C22⋊C8C2
prime radicalC22×#638C4
cosocleC22×C32×C4C23
hypercenterC2×C22⋊C8C3⋊D3
hypocenterC32#1870
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C22×#3743C2semidirect
3. C2×C32×C22⋊C8C2semidirect
4. C2×#63657C2semidirect
5. C2×#63647C2semidirect
6. #10589416C2semidirect
7. #10589416C2semidirect
30. C2×C3×C22⋊C8D3semidirect
62. C2×C32×C8D4semidirect
73. C2×C3⋊(C3⋊C8)D4semidirect
101. C23×C32M16semidirect
132. C2×C3⋊Dic3M16semidirect
145. C2×C22⋊C8C3⋊D3semidirect
175. C2×C3×C8#67semidirect
242. C23×C3ΓL(1,𝔽25)semidirect
325. C2×C8#638semidirect
360. C32#1870semidirect
374. C23#3743semidirect
445. C3#69958semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6382
C22×#37432
C2×C32×C22⋊C82
C2×#636572
C2×#636472
#105894162
#699583