Finite Group Databasebeta

← #11788319

Subgroups — #11788319

Order 1152.

993 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 993 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64104C211yesmaximal, characteristic
2C2×#63818C211yesmaximal, characteristic, complemented
3C2×#63559C212yesmaximal, complemented
4#10591074C211yesmaximal, characteristic, complemented
5C3×#4530C212yesmaximal, complemented
6#68992—32nomaximal
7C22×C4⋊(C3⋊D3)C412yes—
8C22×C4⋊(C3⋊D3)C2211yescharacteristic
9C22×C4×C3⋊D3C2211yescharacteristic
10C2×C32×C4⋊C4C2211yescharacteristic
11C2×#3752C412yes—
12#64104—21no—
13#64104—21no—
14#64104—21no—
15#64104—21no—
16C2×C3×#208C2212yes—
17C2×C3×#208C2212yes—
18C2×#3712—21no—
19#63818—24no—
20C2×#3710—21no—
21#63559—24no—
22#63557—21no—
23#63557—21no—
24C2×D3×M16—32no—
25C2×#895—32no—
26C2×#895—61no—
27C2×#823—32no—
28C2×#823—32no—
29#5126—32no—
30C3×#343—32no—
31#4530D312yescomplemented
32C23×C3⋊D3C2×C411yescharacteristic
33C23×C3⋊D3C2×C412yes—
34C22×C32×C4C2×C412yes—
35C22×C32×C4C2311yescharacteristic
36C22×C3⋊Dic3C2×C411yescharacteristic
37C2×C4⋊(C3⋊D3)—22no—
38C2×C4⋊(C3⋊D3)—22no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×C4⋊(C3⋊D3)—21no—
41C2×C4⋊(C3⋊D3)—22no—
42C2×C4⋊(C3⋊D3)—22no—
43C2×C4⋊(C3⋊D3)—44no—
44C2×C4×C3⋊D3—22no—
45C2×C4×C3⋊D3—22no—
46C2×C4×C3⋊D3D411yescharacteristic
47C2×C4×C3⋊D3D411yescharacteristic
48C2×C4×C3⋊D3D411yescharacteristic
49C2×C4×C3⋊D3D411yescharacteristic
50C32×C4⋊C4—21no—
51C32×C4⋊C4—21no—
52#3752—22no—
53#3752—22no—
54C3×#208—22no—
55C3×#208—22no—
56C3×#208—22no—
57C3×#208D414yescomplemented
58C2×C3×C3⋊C8—22no—
59C2×C3×C3⋊C8—22no—
60#3712—24no—
61#3710—24no—
62#1900—91noSylow
63C22×D12—32no—
64C22×D12—34no—
65C22×D12—62no—
66C22×D12—61no—
67C22×C4×D3—32no—
68C22×C4×D3—61no—
69C2×C3×M16—32no—
70C2×C3×M16—32no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2×D312yes—
73C2×(C2×D3)⋊C4—34no—
74C2×(C2×D3)⋊C4—62no—
75C2×#208C2×D312yes—
76C2×#208C2×D312yes—
77D3×M16—64no—
78D3×M16—64no—
79C2×ΓL(1,𝔽25)—62no—
80C2×D3×C8—62no—
81#895—62no—
82#895—62no—
83#895—62no—
84#895—62no—
85#895—62no—
86#895—62no—
87#895—62no—
88#895—62no—
89#823—64no—
90#823—64no—
91(C2×D3)⋊C8—62no—
92(C2×D3)⋊C8—62no—
93C23×C32C22×C411yescharacteristic
94C22×C3⋊D3—22no—
95C22×C3⋊D3—22no—
96C22×C3⋊D3—22no—
97C22×C3⋊D3—24no—
98C22×C3⋊D3—41no—
99C22×C3⋊D3—42no—
100C22×C3⋊D3—42no—
101C22×C3⋊D3C22⋊C411yescharacteristic
102C22×C3⋊D3C22⋊C411yescharacteristic
103C2×C32×C4—22no—
104C2×C32×C4—42no—
105C2×C32×C4C2×D412yes—
106C2×C32×C4C2×D411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4#4112yes—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3C22⋊C411yescharacteristic
111C2×C3⋊Dic3C22⋊C411yescharacteristic
112C4⋊(C3⋊D3)—44no—
113C4⋊(C3⋊D3)—44no—
114C4⋊(C3⋊D3)—41no—
115C4⋊(C3⋊D3)—42no—
116C4⋊(C3⋊D3)—41no—
117C4×C3⋊D3—21no—
118C4×C3⋊D3—21no—
119C4×C3⋊D3—22no—
120C4×C3⋊D3—22no—
121C4×C3⋊D3—21no—
122C4×C3⋊D3—21no—
123C3×C3⋊C8—24no—
124C3×C3⋊C8—42no—
125C22×M16—91no—
126C2×#121—91no—
127C2×#100—92no—
128#407—91no—
129#343—32no—
130C23×D3—32no—
131C23×D3—34no—
132C23×D3—61no—
133C23×D3—62no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4C4×D314yes—
137C22×C3×C4C22×D312yes—
138C22×Dic3—32no—
139C22×Dic3—61no—
140C2×D12—64no—
141C2×D12—64no—
142C2×D12—64no—
143C2×D12—64no—
144C2×D12—62no—
145C2×D12—62no—
146C2×D12—62no—
147C2×D12—62no—
148C2×D12—62no—
149C2×D12—62no—
150C2×D12—62no—
151C2×D12—62no—
152C2×D12—122no—
153C2×D12—122no—
154C2×D12—128no—
155C2×D12—128no—
156C2×C4×D3—32no—
157C2×C4×D3—32no—
158C2×C4×D3—32no—
159C2×C4×D3—32no—
160C2×C4×D3—61no—
161C2×C4×D3—61no—
162C2×C4×D3—61no—
163C2×C4×D3—61no—
164C2×C4×D3—62no—
165C2×C4×D3—62no—
166C2×C4×D3—62no—
167C2×C4×D3—62no—
168C2×C4×D3—122no—
169C2×C4×D3—122no—
170C3×M16—34no—
171C3×M16—62no—
172C3×M16—62no—
173C3×M16—62no—
174C2×C3×C8—62no—
175C2×C3×C8—62no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180(C2×D3)⋊C4—64no—
181(C2×D3)⋊C4—64no—
182(C2×D3)⋊C4—64no—
183(C2×D3)⋊C4—64no—
184#208—22no—
185#208—22no—
186#208—22no—
187#208D1214yescomplemented
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190ΓL(1,𝔽25)—68no—
191D3×C8—68no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32C2×C22⋊C411yescharacteristic
195C22×C32C4×D412yes—
196C2×C3⋊D3—21no—
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3⋊D3—21no—
200C2×C3⋊D3—21no—
201C2×C3⋊D3—44no—
202C2×C3⋊D3—42no—
203C2×C3⋊D3—42no—
204C2×C3⋊D3—42no—
205C2×C3⋊D3—84no—
206C2×C3⋊D3#10012yescomplemented
207C32×C4—42no—
208C32×C4#12011yescharacteristic
209C32×C4#12112yes—
210C32×C4#12311yescharacteristic
211C3⋊Dic3—21no—
212C3⋊Dic3—21no—
213C22×D4—92no—
214C22×D4—91no—
215C23×C4—91no—
216C2×M16—32no—
217C2×M16—32no—
218C2×M16—181no—
219C2×M16—184no—
220C22×C8—181no—
221#121—181no—
222#121—181no—
223#121—181no—
224#121—181no—
225C2×C4⋊C4D3211yescharacteristic
226C2×C22⋊C4—92no—
227#100—184no—
228C22⋊C8—181no—
229C22⋊C8—181no—
230C23×C3—21no—
231C23×C3C2×C4×D312yes—
232C22×D3—32no—
233C22×D3—32no—
234C22×D3—61no—
235C22×D3—61no—
236C22×D3—64no—
237C22×D3—64no—
238C22×D3—62no—
239C22×D3—62no—
240C22×D3—64no—
241C22×D3—64no—
242C22×D3—64no—
243C22×D3—64no—
244C22×D3—122no—
245C22×D3—124no—
246C22×D3—122no—
247C22×D3—122no—
248C22×D3—124no—
249C22×D3—124no—
250C22×D3—124no—
251C22×D3—124no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4C2×D1212yes—
261C2×C3×C4#23512yes—
262C2×C3×C4D3×D412yes—
263C2×C3×C4D3×D412yes—
264C2×C3×C4D3×D412yes—
265C2×C3×C4#23912yes—
266C2×Dic3—32no—
267C2×Dic3—32no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—122no—
273D12—122no—
274D12—124no—
275D12—122no—
276D12—128no—
277D12—128no—
278D12—128no—
279D12—128no—
280D12—122no—
281D12—122no—
282D12—122no—
283D12—122no—
284C4×D3—62no—
285C4×D3—62no—
286C4×D3—62no—
287C4×D3—62no—
288C4×D3—62no—
289C4×D3—62no—
290C4×D3—62no—
291C4×D3—62no—
292C4×D3—121no—
293C4×D3—121no—
294C4×D3—122no—
295C4×D3—122no—
296C4×D3—121no—
297C4×D3—121no—
298C3×C8—64no—
299C3×C8—122no—
300C3⋊C8—24no—
301C3⋊C8—42no—
302C2×C32—22no—
303C2×C32#38611yescharacteristic
304C2×C32C2×#10011yescharacteristic
305C2×C32#41311yescharacteristic
306C3⋊D3—22no—
307C3⋊D3—41no—
308C3⋊D3—82no—
309C24—91no—
310C24—92no—
311C2×D4—182no—
312C2×D4—182no—
313C2×D4—182no—
314C2×D4—182no—
315C2×D4—181no—
316C2×D4—181no—
317C2×D4—364no—
318C22×C4—91no—
319C22×C4—91no—
320C22×C4—91no—
321C22×C4—91no—
322C22×C4—91no—
323C22×C4—182no—
324C22×C4—182no—
325C22×C4#62412yes—
326C22×C4C2×D3211yescharacteristic
327M16—34no—
328M16—62no—
329M16—62no—
330M16—62no—
331M16—184no—
332C2×C8—62no—
333C2×C8—62no—
334C2×C8—184no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C22⋊C4—182no—
338C22⋊C4—182no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—21no—
345C22×C3C4×D1212yes—
346C22×C3D3×C22⋊C412yes—
347C22×C3#89312yes—
348C2×D3—34no—
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—62no—
352C2×D3—62no—
353C2×D3—62no—
354C2×D3—62no—
355C2×D3—62no—
356C2×D3—121no—
357C2×D3—121no—
358C2×D3—121no—
359C2×D3—121no—
360C2×D3—124no—
361C2×D3—124no—
362C2×D3—128no—
363C2×D3—124no—
364C2×D3—124no—
365C2×D3—124no—
366C2×D3—124no—
367C2×D3—124no—
368C2×D3—124no—
369C2×D3—248no—
370C2×D3—244no—
371C2×D3—244no—
372C3×C4—21no—
373C3×C4—22no—
374C3×C4—21no—
375C3×C4—44no—
376C3×C4—44no—
377C3×C4#88212yes—
378C3×C4#88412yes—
379C3×C4#89412yes—
380C3×C4#89512yes—
381Dic3—62no—
382Dic3—62no—
383Dic3—121no—
384Dic3—121no—
385C32#190011yescharacteristic, complemented, Sylow
386C23—91no—
387C23—91no—
388C23—182no—
389C23—182no—
390C23—182no—
391C23—184no—
392C23—361no—
393C23—362no—
394C23—362no—
395C23C2×#62411yescharacteristic
396D4—364no—
397D4—364no—
398D4—361no—
399D4—362no—
400D4—361no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—91no—
404C2×C4—91no—
405C2×C4—182no—
406C2×C4—181no—
407C2×C4—181no—
408C2×C4—182no—
409C2×C4—182no—
410C2×C4—181no—
411C2×C4—181no—
412C2×C4#379912yes—
413C2×C4D3×D1212yes—
414C2×C4#381111yescharacteristic
415C2×C4#381111yescharacteristic
416C8—64no—
417C8—122no—
418C2×C3—24no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3#466512yes—
425C2×C3D3×#10012yes—
426C2×C3#475112yes—
427D3—64no—
428D3—122no—
429D3—122no—
430D3—122no—
431D3—244no—
432D3—244no—
433C22—21no—
434C22—21no—
435C22—92no—
436C22—181no—
437C22—181no—
438C22—181no—
439C22—181no—
440C22—181no—
441C22—364no—
442C22—362no—
443C22—362no—
444C22—362no—
445C22—724no—
446C22#6389412yes—
447C22#6397411yescharacteristic
448C4—42no—
449C4—181no—
450C4—181no—
451C4#6389711yescharacteristic
452C4#6391012yes—
453C4#6392311yescharacteristic
454C3—21no—
455C3#6899212yescomplemented
456C2—22no—
457C2—182no—
458C2—361no—
459C2—722no—
460C2#1059138011yescharacteristic
461C2#1059150311yescharacteristic
462C2#1059164011yescharacteristic
subgroupG/H
centerC22#63974
commutatorC23×C32C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32C2×C22⋊C4
prime radicalC2×#64104C2
cosocleC22×C32×C4C23
hypercenterC2×C4⋊C4D32
hypocenterC32#1900
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#63818C2semidirect
3. C2×#63559C2semidirect
4. #10591074C2semidirect
5. C3×#4530C2semidirect
31. #4530D3semidirect
57. C3×#208D4semidirect
187. #208D12semidirect
206. C2×C3⋊D3#100semidirect
385. C32#1900semidirect
455. C3#68992semidirect

Maximal subgroups

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

subgroupindex
C2×#641042
C2×#638182
C2×#635592
#105910742
C3×#45302
#689923