Finite Group Databasebeta

← #11778028

Subgroups — #11778028

Order 1152.

793 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 793 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C3×#5483C211yesmaximal, characteristic, complemented
2#10592659C211yesmaximal, characteristic
3#10591183C212yesmaximal, complemented
4C3×#4525C211yesmaximal, characteristic, complemented
5#10589152C212yesmaximal, complemented
6#71892—31nomaximal
7#68114—31nomaximal
8C22×#638—21no—
9C22×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C32×C22⋊C4C2211yescharacteristic
11C2×#3752—21no—
12C2×C3×#217—21no—
13C3×#924—21no—
14C3×#924C2212yes—
15C3×#878—22no—
16C3×#878—22no—
17C2×#3717—22no—
18#63658C2211yescharacteristic
19C3×#830C2212yes—
20#63564—22no—
21C3×#543—31no—
22#5483D311yescharacteristic, complemented
23#5046—32no—
24#4960—31no—
25#4960—31no—
26#4960—61no—
27#4731—32no—
28C3×#323—31no—
29#4538—32no—
30#4525D311yescharacteristic, complemented
31#4467—32no—
32C24×C32D411yescharacteristic
33C23×C3⋊D3—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4C2311yescharacteristic
36C2×#638—42no—
37C2×#638—42no—
38C2×#638—42no—
39C22×C3⋊Dic3—21no—
40C2×C4⋊(C3⋊D3)—22no—
41C2×C4⋊(C3⋊D3)—42no—
42C22×C3×Dic3—21no—
43C22×C3×Dic3—21no—
44C2×C3×Dic6—21no—
45C2×C3×Dic6D412yescomplemented
46C32×C22⋊C4—22no—
47#3752—42no—
48C3×#217—22no—
49C3×#217—22no—
50C3×C4⋊Dic3—21no—
51C3×C4⋊Dic3D412yescomplemented
52C3×Dic3⋊C4—22no—
53C3×Dic3⋊C4—22no—
54C2×C3×C3⋊C8—22no—
55#3717—44no—
56#1596—91noSylow
57C22×#67—62no—
58C22×#67—61no—
59C22×#67—61no—
60C22×D12—31no—
61C22×D12—31no—
62C22×D12—61no—
63C3×C22⋊Q8—32no—
64C3×C22⋊Q8—62no—
65C3×C22⋊Q8—62no—
66C3×C22⋊Q8—61no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—61no—
69C2×C3×C22⋊C4C2×D311yescharacteristic
70C2×C3×C22⋊C4C2×D311yescharacteristic
71C2×#217—21no—
72C2×#215—62no—
73C2×(C2×D3)⋊C4—62no—
74C2×(C2×D3)⋊C4—61no—
75C2×(C2×D3)⋊C4—61no—
76#924—21no—
77#924C2×D312yes—
78C2×#204—62no—
79#878—22no—
80#878—22no—
81C3×C22⋊C8—32no—
82C3⋊#95—31no—
83C3⋊#95—31no—
84C3⋊#95—61no—
85#830C2×D312yes—
86#821—62no—
87#809—62no—
88C23×C32—21no—
89C23×C32—21no—
90C23×C32—41no—
91C23×C32—41no—
92C23×C32SD1612yescomplemented
93C23×C32C2×D411yescharacteristic
94C22×C3⋊D3—22no—
95C22×C3⋊D3—41no—
96C22×C3⋊D3—41no—
97C22×C3⋊D3—41no—
98C22×C3⋊D3—41no—
99C22×C3⋊D3—41no—
100C22×C3⋊D3—41no—
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4C2×D412yes—
105#638—48no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—41no—
108C2×C3⋊Dic3—41no—
109C4⋊(C3⋊D3)—24no—
110C4⋊(C3⋊D3)—42no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—81no—
114C2×C3×Dic3—81no—
115C3×Dic6—42no—
116C3×Dic6—81no—
117C3×C3⋊C8—42no—
118#543—31no—
119#450—92no—
120#392—91no—
121#328—92no—
122#323—31no—
123C24×C3—21no—
124C24×C3D1211yescharacteristic
125C24×C3#6711yescharacteristic
126C23×D3—62no—
127C23×D3—61no—
128C23×D3—61no—
129C2×C3×Q8—32no—
130C2×C3×Q8—61no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—61no—
134C22×C3×C4—61no—
135C22×C3×C4D1211yescharacteristic
136C22×C3×C4#6711yescharacteristic
137C22×C3×C4C22×D311yescharacteristic
138C22×C3×C4C22×D311yescharacteristic
139C2×#67—122no—
140C2×#67—122no—
141C2×#67—124no—
142C2×#67—122no—
143C2×#67—122no—
144C2×#67—124no—
145C2×#67—124no—
146C2×#67—122no—
147C2×#67—122no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—62no—
151C22×Dic3—61no—
152C22×Dic3—61no—
153C2×D12—62no—
154C2×D12—62no—
155C2×D12—122no—
156C2×D12—122no—
157C2×D12—122no—
158C2×D12—124no—
159C2×Dic6—21no—
160C2×Dic6#6712yescomplemented
161C2×C3×C8—62no—
162C3×C4⋊C4—32no—
163C3×C4⋊C4—62no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—62no—
166C3×C22⋊C4—22no—
167C3×C22⋊C4—22no—
168C3×C22⋊C4—42no—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—62no—
171#217—22no—
172#217—22no—
173#215—124no—
174(C2×D3)⋊C4—122no—
175(C2×D3)⋊C4—122no—
176(C2×D3)⋊C4—124no—
177C4⋊Dic3—21no—
178C4⋊Dic3#6712yescomplemented
179Dic3⋊C4—22no—
180Dic3⋊C4—22no—
181C2×C3⋊C8—22no—
182#204—124no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—22no—
186C22×C32—42no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32—81no—
190C22×C32#12011yescharacteristic
191C22×C32C2×SD1611yescharacteristic
192C22×C32#13711yescharacteristic
193C2×C3⋊D3—42no—
194C2×C3⋊D3—42no—
195C2×C3⋊D3—42no—
196C2×C3⋊D3—81no—
197C2×C3⋊D3—81no—
198C2×C3⋊D3—81no—
199C2×C3⋊D3—81no—
200C32×C4—22no—
201C32×C4—41no—
202C3⋊Dic3—42no—
203C3×Dic3—42no—
204C3×Dic3—42no—
205C22×D4—91no—
206C22×D4—181no—
207C2×SD16—182no—
208C22⋊Q8—32no—
209C22⋊Q8—62no—
210C22⋊Q8—62no—
211C22⋊Q8—61no—
212C2×C22⋊C4—61no—
213C2×C22⋊C4—181no—
214C2×C22⋊C4D3211yescharacteristic
215#102—182no—
216C22⋊C8—32no—
217#95—91no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—22no—
223C23×C3—21no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3#20412yescomplemented
233C23×C3#21512yescomplemented
234C23×C3C2×D1211yescharacteristic
235C23×C3C2×#6711yescharacteristic
236C22×D3—64no—
237C22×D3—62no—
238C22×D3—62no—
239C22×D3—122no—
240C22×D3—121no—
241C22×D3—121no—
242C22×D3—122no—
243C22×D3—121no—
244C22×D3—121no—
245C22×D3—122no—
246C22×D3—122no—
247C22×D3—121no—
248C22×D3—121no—
249C22×D3—122no—
250C22×D3—122no—
251C22×D3—121no—
252C22×D3—121no—
253C22×D3—121no—
254C22×D3—121no—
255C22×D3—121no—
256C22×D3—121no—
257C3×Q8—122no—
258C3×Q8—241no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—241no—
272C2×C3×C4—241no—
273C2×C3×C4D3×D412yes—
274C2×C3×C4C2×#6712yes—
275#67—128no—
276#67—128no—
277#67—1216no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—64no—
281C2×Dic3—62no—
282C2×Dic3—62no—
283C2×Dic3—81no—
284C2×Dic3—81no—
285C2×Dic3—122no—
286C2×Dic3—121no—
287C2×Dic3—121no—
288C2×Dic3—122no—
289C2×Dic3—121no—
290C2×Dic3—121no—
291D12—64no—
292D12—64no—
293D12—124no—
294D12—122no—
295D12—122no—
296D12—124no—
297Dic6—42no—
298Dic6—81no—
299C3×C8—122no—
300C3⋊C8—42no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—42no—
304C2×C32#44811yescharacteristic
305C2×C32#45311yescharacteristic
306C2×C32B(4,𝔽2)11yescharacteristic
307C3⋊D3—82no—
308C24—181no—
309C24#62611yescharacteristic
310C2×Q8—32no—
311C2×Q8—61no—
312C2×D4—182no—
313C2×D4—362no—
314C2×D4—362no—
315C2×D4—362no—
316C2×D4—362no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—181no—
320C22×C4#62611yescharacteristic
321C22×C4C2×D3211yescharacteristic
322SD16—364no—
323C2×C8—62no—
324C4⋊C4—32no—
325C4⋊C4—62no—
326C4⋊C4—61no—
327C4⋊C4—62no—
328C22⋊C4—22no—
329C22⋊C4—62no—
330C22⋊C4—62no—
331C22⋊C4—362no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—44no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—81no—
352C22×C3—81no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3—82no—
356C22×C3#88211yescharacteristic
357C22×C3C2×#20411yescharacteristic
358C22×C3#90911yescharacteristic
359C22×C3C2×#21511yescharacteristic
360C22×C3#94211yescharacteristic
361C22×C3#95311yescharacteristic
362C2×D3—124no—
363C2×D3—124no—
364C2×D3—122no—
365C2×D3—122no—
366C2×D3—124no—
367C2×D3—122no—
368C2×D3—122no—
369C2×D3—122no—
370C2×D3—122no—
371C2×D3—242no—
372C2×D3—242no—
373C2×D3—241no—
374C2×D3—241no—
375C2×D3—241no—
376C2×D3—241no—
377C2×D3—242no—
378C2×D3—242no—
379C2×D3—241no—
380C2×D3—241no—
381C2×D3—241no—
382C2×D3—241no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—42no—
386C3×C4—41no—
387C3×C4—41no—
388C3×C4—81no—
389C3×C4—122no—
390C3×C4—122no—
391Dic3—42no—
392Dic3—42no—
393Dic3—124no—
394Dic3—122no—
395Dic3—122no—
396C32#159611yescharacteristic, complemented, Sylow
397C23—21no—
398C23—21no—
399C23—41no—
400C23—41no—
401C23—182no—
402C23—361no—
403C23—361no—
404C23—361no—
405C23—361no—
406C23—361no—
407C23—361no—
408C23#371712yescomplemented
409C23C2×#62611yescharacteristic
410Q8—122no—
411Q8—241no—
412D4—184no—
413D4—362no—
414D4—368no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—182no—
421C2×C4—241no—
422C2×C4—241no—
423C2×C4—361no—
424C2×C4—361no—
425C2×C4D3×#6712yes—
426C8—122no—
427C2×C3—22no—
428C2×C3—22no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—42no—
437C2×C3—42no—
438C2×C3—82no—
439C2×C3#473811yescharacteristic
440C2×C3#474111yescharacteristic
441C2×C3#474711yescharacteristic
442C2×C3#522711yescharacteristic
443C2×C3#524111yescharacteristic
444C2×C3#524411yescharacteristic
445D3—244no—
446D3—242no—
447D3—242no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—42no—
452C22—41no—
453C22—41no—
454C22—81no—
455C22—362no—
456C22—362no—
457C22—362no—
458C22—721no—
459C22—721no—
460C22—721no—
461C22—721no—
462C22C2×#371711yescharacteristic
463C22#6383311yescharacteristic
464C22#6397711yescharacteristic
465C4—22no—
466C4—41no—
467C4—122no—
468C4—122no—
469C4—362no—
470C3—21no—
471C3#6811411yescharacteristic, complemented
472C3#7189211yescharacteristic, complemented
473C2—21no—
474C2—21no—
475C2—42no—
476C2—722no—
477C2#1059117411yescharacteristic
478C2#1059151011yescharacteristic
479C2#1059153111yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C22⋊C4C22
socleC22×C32#120
prime radical#10592659C2
hypercenterC2×C22⋊C4D32
hypocenterC32#1596
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#5483C2semidirect
3. #10591183C2semidirect
4. C3×#4525C2semidirect
5. #10589152C2semidirect
22. #5483D3semidirect
30. #4525D3semidirect
45. C2×C3×Dic6D4semidirect
51. C3×C4⋊Dic3D4semidirect
92. C23×C32SD16semidirect
160. C2×Dic6#67semidirect
178. C4⋊Dic3#67semidirect
232. C23×C3#204semidirect
233. C23×C3#215semidirect
396. C32#1596semidirect
408. C23#3717semidirect
471. C3#68114semidirect
472. C3#71892semidirect

Maximal subgroups

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

subgroupindex
C3×#54832
#105926592
#105911832
C3×#45252
#105891522
#718923
#681143