Finite Group Databasebeta

← #11790572

Subgroups — #11790572

Order 1152.

1005 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#638C211yesmaximal, characteristic
2C2×C3×#830C212yesmaximal, complemented
3C2×#63572C211yesmaximal, characteristic, complemented
4C2×#63557C212yesmaximal, complemented
5C2×#63553C211yesmaximal, characteristic, complemented
6#69916—32nomaximal
7C23×C32×C4C2211yescharacteristic
8C22×#638C411yescharacteristic
9C22×C4×C3⋊D3C2211yescharacteristic
10C2×#3757C411yescharacteristic
11C4×#638—22no—
12C4×#638—22no—
13C2×#3752C411yescharacteristic
14C2×#3750C411yescharacteristic
15C2×C4×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊C8C2212yes—
17C22×C3×C3⋊C8C2212yes—
18C3×#830—24no—
19#63572—22no—
20#63557—24no—
21#63553—22no—
22C2×C4×#67—32no—
23C2×C4×#67—61no—
24C2×C3×C22⋊C8—32no—
25C2×#830D312yescomplemented
26C2×(C2×D3)⋊C8—32no—
27C2×(C2×D3)⋊C8—32no—
28C2×Dic3⋊C8—32no—
29C2×C8×Dic3—32no—
30C24×C32C2×C411yescharacteristic
31C23×C3⋊D3C2×C411yescharacteristic
32C22×C32×C4—21no—
33C22×C32×C4—21no—
34C22×C32×C4—22no—
35C22×C32×C4—22no—
36C22×C32×C4C2×C411yescharacteristic
37C22×C32×C4C2311yescharacteristic
38C2×#638—22no—
39C2×#638—22no—
40C2×#638C814yescomplemented
41C22×C3⋊Dic3C2×C411yescharacteristic
42C22×C3⋊Dic3C2×C411yescharacteristic
43C22×C3⋊Dic3C2×C411yescharacteristic
44C2×C4×C3⋊D3—21no—
45C2×C4×C3⋊D3—21no—
46C2×C4×C3⋊D3—22no—
47C2×C4×C3⋊D3—22no—
48#3757—22no—
49#3752—22no—
50#3750—22no—
51C4×C3⋊Dic3D412yes—
52C4×C3⋊Dic3D412yes—
53C2×C3×C3⋊C8—22no—
54C2×C3×C3⋊C8—22no—
55C2×C3×C3⋊C8—22no—
56C2×C3×C3⋊C8—22no—
57C2×C3×C3⋊C8—22no—
58C2×C3×C3⋊C8D414yescomplemented
59#1954—91noSylow
60C23×C3×C4—21no—
61C23×C3×C4C2×D312yes—
62C22×#67—32no—
63C22×#67—61no—
64C22×C4×D3—32no—
65C22×C4×D3—61no—
66C22×C3×C8—32no—
67C22×C3×C8—32no—
68C2×#217—32no—
69C2×#217—61no—
70C4×#67—64no—
71C4×#67—64no—
72C4×#67—64no—
73C4×#67—64no—
74C2×(C2×D3)⋊C4—32no—
75C2×(C2×D3)⋊C4—61no—
76C2×Dic3⋊C4—32no—
77C2×Dic3⋊C4—61no—
78C2×C4×Dic3—32no—
79C2×C4×Dic3—61no—
80C22×C3⋊C8C2×D312yes—
81C22×C3⋊C8C2×D312yes—
82C3×C22⋊C8—64no—
83#830—24no—
84(C2×D3)⋊C8—64no—
85(C2×D3)⋊C8—64no—
86Dic3⋊C8—64no—
87C8×Dic3—64no—
88C23×C32—21no—
89C23×C32—21no—
90C23×C32—22no—
91C23×C32—22no—
92C23×C32C2×C812yes—
93C23×C32C22×C411yescharacteristic
94C22×C3⋊D3—21no—
95C22×C3⋊D3—21no—
96C22×C3⋊D3—22no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3C2×C812yes—
99C2×C32×C4—21no—
100C2×C32×C4—22no—
101C2×C32×C4—42no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4C2×D411yescharacteristic
105C2×C32×C4C2×D411yescharacteristic
106C2×C32×C4C2×D412yes—
107C2×C32×C4#4112yes—
108#638—28no—
109C2×C3⋊Dic3—21no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3C22⋊C412yes—
113C2×C3⋊Dic3C22⋊C412yes—
114C2×C3⋊Dic3C2×C812yes—
115C2×C3⋊Dic3M1612yes—
116C4×C3⋊D3—42no—
117C4×C3⋊D3—41no—
118C4×C3⋊D3—41no—
119C3×C3⋊C8—24no—
120C3×C3⋊C8—42no—
121C2×C4×D4—91no—
122C2×C4⋊C8—91no—
123C2×C22⋊C8—32no—
124C2×C22⋊C8—92no—
125C2×C4×C8—91no—
126C24×C3—21no—
127C24×C3C4×D312yes—
128C23×D3—32no—
129C23×D3—61no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4C4×D312yes—
143C22×C3×C4C22×D312yes—
144C2×#67—38no—
145C2×#67—64no—
146C2×#67—64no—
147C2×#67—64no—
148C2×#67—64no—
149C2×#67—64no—
150C22×Dic3—32no—
151C22×Dic3—32no—
152C22×Dic3—32no—
153C22×Dic3—61no—
154C22×Dic3—61no—
155C22×Dic3—61no—
156C2×C4×D3—62no—
157C2×C4×D3—62no—
158C2×C4×D3—62no—
159C2×C4×D3—62no—
160C2×C4×D3—62no—
161C2×C4×D3—62no—
162C2×C4×D3—62no—
163C2×C4×D3—62no—
164C2×C4×D3—122no—
165C2×C4×D3—122no—
166C2×C3×C8—34no—
167C2×C3×C8—62no—
168C2×C3×C8—62no—
169C2×C3×C8—62no—
170C2×C3×C8—62no—
171C2×C3×C8—62no—
172#217—64no—
173#217—64no—
174(C2×D3)⋊C4—64no—
175(C2×D3)⋊C4—64no—
176Dic3⋊C4—64no—
177Dic3⋊C4—64no—
178C4×Dic3—34no—
179C4×Dic3—34no—
180C4×Dic3—62no—
181C4×Dic3—62no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8#6714yescomplemented
188C22×C32—22no—
189C22×C32—24no—
190C22×C32—41no—
191C22×C32—41no—
192C22×C32—42no—
193C22×C32C2×C22⋊C411yescharacteristic
194C22×C32C4×D412yes—
195C22×C32C22×C811yescharacteristic
196C22×C32C2×M1611yescharacteristic
197C22×C32#13112yes—
198C2×C3⋊D3—22no—
199C2×C3⋊D3—24no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—42no—
203C32×C4—41no—
204C32×C4#12112yes—
205C32×C4#12411yescharacteristic
206C32×C4#12711yescharacteristic
207C3⋊Dic3—22no—
208C3⋊Dic3—41no—
209C3⋊Dic3C22⋊C814yescomplemented
210C22×D4—91no—
211C23×C4—91no—
212C23×C4D3211yescharacteristic
213C22×C8—32no—
214C22×C8—32no—
215C4×D4—182no—
216C4×D4—182no—
217C2×C4⋊C4—91no—
218C2×C22⋊C4—91no—
219C2×C22⋊C4—91no—
220C2×C42—91no—
221C4⋊C8—182no—
222C22⋊C8—64no—
223C22⋊C8—184no—
224C4×C8—182no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—21no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3D3×C814yes—
238C23×C3C2×C4×D312yes—
239C22×D3—34no—
240C22×D3—62no—
241C22×D3—62no—
242C22×D3—62no—
243C22×D3—62no—
244C22×D3—62no—
245C22×D3—62no—
246C22×D3—62no—
247C22×D3—62no—
248C22×D3—62no—
249C22×D3—122no—
250C22×D3—122no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—44no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4#23512yes—
268C2×C3×C4D3×D412yes—
269C2×C3×C4D3×D412yes—
270C2×C3×C4D3×D412yes—
271C2×C3×C4#23712yes—
272C2×C3×C4C2×#6712yes—
273#67—616no—
274#67—616no—
275C2×Dic3—34no—
276C2×Dic3—34no—
277C2×Dic3—34no—
278C2×Dic3—34no—
279C2×Dic3—62no—
280C2×Dic3—62no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3—62no—
284C2×Dic3—62no—
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—62no—
288C2×Dic3—62no—
289C2×Dic3—122no—
290C2×Dic3—122no—
291C4×D3—122no—
292C4×D3—124no—
293C4×D3—122no—
294C4×D3—122no—
295C4×D3—122no—
296C4×D3—122no—
297C4×D3—122no—
298C3×C8—64no—
299C3×C8—122no—
300C3⋊C8—24no—
301C3⋊C8—42no—
302C2×C32—42no—
303C2×C32#39011yescharacteristic
304C2×C32C2×C22⋊C811yescharacteristic
305C2×C32#40811yescharacteristic
306C2×C32C8×D412yes—
307C2×C32#43612yes—
308C3⋊D3—42no—
309C24—91no—
310C24#62411yescharacteristic
311C2×D4—94no—
312C2×D4—182no—
313C2×D4—182no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—91no—
319C22×C4—91no—
320C22×C4—91no—
321C22×C4—182no—
322C22×C4—182no—
323C22×C4—181no—
324C22×C4—181no—
325C22×C4#62411yescharacteristic
326C22×C4C2×D3211yescharacteristic
327C2×C8—34no—
328C2×C8—62no—
329C2×C8—62no—
330C2×C8—62no—
331C2×C8—62no—
332C2×C8—62no—
333C4⋊C4—182no—
334C22⋊C4—182no—
335C22⋊C4—182no—
336C42—92no—
337C42—92no—
338C22×C3—24no—
339C22×C3—24no—
340C22×C3—24no—
341C22×C3—24no—
342C22×C3—21no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—42no—
349C22×C3—44no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3D3×C22⋊C412yes—
356C22×C3#67⋊C412yes—
357C22×C3C2×D3×C812yes—
358C22×C3#90112yes—
359C22×C3D3×M1612yes—
360C22×C3#90712yes—
361C22×C3C4×#6712yes—
362C2×D3—64no—
363C2×D3—64no—
364C2×D3—64no—
365C2×D3—64no—
366C2×D3—124no—
367C2×D3—122no—
368C2×D3—124no—
369C2×D3—122no—
370C2×D3—122no—
371C2×D3—122no—
372C2×D3—122no—
373C2×D3—122no—
374C3×C4—21no—
375C3×C4—22no—
376C3×C4—21no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4#88412yes—
380C3×C4#88512yes—
381C3×C4#93912yes—
382C3×C4#94012yes—
383Dic3—38no—
384Dic3—64no—
385Dic3—64no—
386Dic3—122no—
387Dic3—122no—
388Dic3—122no—
389C32#195411yescharacteristic, complemented, Sylow
390C23—22no—
391C23—21no—
392C23—22no—
393C23—21no—
394C23—92no—
395C23—182no—
396C23—181no—
397C23—182no—
398C23—181no—
399C23#371012yes—
400C23C2×#62411yescharacteristic
401D4—188no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—92no—
408C2×C4—92no—
409C2×C4—92no—
410C2×C4—92no—
411C2×C4—182no—
412C2×C4—182no—
413C2×C4—181no—
414C2×C4—361no—
415C2×C4—362no—
416C2×C4—361no—
417C2×C4#380412yes—
418C2×C4D3×#6712yes—
419C2×C4#381111yescharacteristic
420C2×C4#381111yescharacteristic
421C8—64no—
422C8—122no—
423C2×C3—22no—
424C2×C3—22no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—44no—
429C2×C3—44no—
430C2×C3D3×C22⋊C812yes—
431C2×C3#67⋊C812yes—
432C2×C3#472712yes—
433C2×C3#472912yes—
434C2×C3#495712yes—
435C2×C3C8×#6712yes—
436C2×C3#512812yes—
437D3—124no—
438D3—124no—
439C22—24no—
440C22—22no—
441C22—42no—
442C22—41no—
443C22—41no—
444C22—184no—
445C22—182no—
446C22—362no—
447C22—361no—
448C22—361no—
449C22C2×#371011yescharacteristic
450C22#6381712yes—
451C22#6381811yescharacteristic
452C22#6395212yes—
453C22#6397411yescharacteristic
454C4—41no—
455C4—94no—
456C4—182no—
457C4—361no—
458C4#6395311yescharacteristic
459C4#6397112yes—
460C4#6397911yescharacteristic
461C3—21no—
462C3#6991612yescomplemented
463C2—42no—
464C2—362no—
465C2#1059105112yes—
466C2#1059114812yes—
467C2#1059115211yescharacteristic
468C2#1059115311yescharacteristic
469C2#1059170411yescharacteristic
subgroupG/H
center, FrattiniC22×C4C2×D32
commutatorC22×C32C22×C8
FittingC23×C32×C4C22
socleC23×C32C22×C4
prime radicalC22×#638C4
cosocleC22×C32×C4C23
hypercenterC23×C4D32
hypocenterC32#1954
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#830C2semidirect
3. C2×#63572C2semidirect
4. C2×#63557C2semidirect
5. C2×#63553C2semidirect
25. C2×#830D3semidirect
40. C2×#638C8semidirect
58. C2×C3×C3⋊C8D4semidirect
187. C2×C3⋊C8#67semidirect
209. C3⋊Dic3C22⋊C8semidirect
389. C32#1954semidirect
462. C3#69916semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6382
C2×C3×#8302
C2×#635722
C2×#635572
C2×#635532
#699163