Finite Group Databasebeta

← #11778080

Subgroups — #11778080

Order 1152.

589 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 78 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
1#10594980C211yesmaximal, characteristic, complemented
2C3×#4731C211yesmaximal, characteristic, complemented
3#10591300C211yesmaximal, characteristic, complemented
4#10591162C211yesmaximal, characteristic
5#10589336C211yesmaximal, characteristic, complemented
6C3×#4469C211yesmaximal, characteristic, complemented
7#10589152C211yesmaximal, characteristic, complemented
8#70978—31nomaximal
9#68130—31nomaximal
10C22×C3×D12C2211yescharacteristic
11#64139C2211yescharacteristic
12C3×#924C2211yescharacteristic
13C2×C3×Dic3⋊C4C2211yescharacteristic
14C2×C3×#204—21no—
15#63966—21no—
16C2×(C2×D3)⋊Dic3—21no—
17C2×(C2×D3)⋊Dic3—21no—
18#63952—21no—
19#63897—21no—
20#63888—21no—
21#63848—21no—
22#63845—21no—
23C2×#3714—21no—
24C32×C22⋊C8C2211yescharacteristic
25#63658C2211yescharacteristic
26C3×#830C2211yescharacteristic
27C3×#821—21no—
28#63563—21no—
29#5645—31no—
30#5515—31no—
31C3×#450—31no—
32#5222—31no—
33#5217—31no—
34#4991—31no—
35#4732—31no—
36#4731D311yescharacteristic, complemented
37C3×#324—31no—
38#4538—31no—
39#4470—31no—
40#4470—31no—
41#4470—61no—
42#4469D311yescharacteristic, complemented
43#4467—31no—
44C23×C3×D3—21no—
45C22×C32×C4C2311yescharacteristic
46C2×#638—21no—
47C22×C3⋊Dic3—21no—
48C2×C4⋊(C3⋊D3)D411yescharacteristic
49C22×C3×Dic3D411yescharacteristic
50C22×C3×Dic3D411yescharacteristic
51C2×C3×D12—21no—
52C2×C3×D12—21no—
53C2×C3×D12—42no—
54C2×C3×Dic6D411yescharacteristic
55C2×#624—21no—
56C2×C32×C8—21no—
57#3752—21no—
58C4⋊(C3⋊Dic3)D411yescharacteristic
59C3×#217—21no—
60C3×C4⋊Dic3D411yescharacteristic
61C3×Dic3⋊C4—21no—
62C3×Dic3⋊C4—21no—
63C3×Dic3⋊C4—21no—
64C2×C3×C3⋊C8—21no—
65C3×#204—42no—
66#3724—21no—
67Dic3⋊Dic3—21no—
68Dic3⋊Dic3—21no—
69#3722—21no—
70#3722—21no—
71#3722—21no—
72(C2×D3)⋊Dic3—42no—
73(C2×D3)⋊Dic3—42no—
74Dic32—21no—
75#3714—42no—
76#1597—91noSylow
77C22×C3×D4—31no—
78C22×D12C2×D311yescharacteristic
79C2×C3×SD16—61no—
80C3×C22⋊Q8—31no—
81C2×C3×C4⋊C4—31no—
82C2×#217—61no—
83C2×#217—61no—
84#946—61no—
85C2×#213—61no—
86#930—31no—
87#930—31no—
88#930—61no—
89#929—61no—
90C4×#67—61no—
91C2×(C2×D3)⋊C4—61no—
92C2×(C2×D3)⋊C4—61no—
93#924C2×D311yescharacteristic
94C2×Dic3⋊C4C2×D311yescharacteristic
95C2×D24—61no—
96C2×#204—21no—
97#898—61no—
98#898—61no—
99#896—61no—
100#894—61no—
101#894—61no—
102#890—61no—
103#889—61no—
104#886—61no—
105#67⋊C4—61no—
106C3×#102—61no—
107C3×C22⋊C8—21no—
108C3×C22⋊C8—31no—
109C3×C22⋊C8C2×D311yescharacteristic
110C3×C22⋊C8C2×D311yescharacteristic
111#832—61no—
112C3⋊#95—31no—
113C3⋊#95—31no—
114C3⋊#95—61no—
115#830C2×D311yescharacteristic
116#821—21no—
117#821—61no—
118C23×C32C2×D411yescharacteristic
119C22×C3⋊D3—21no—
120C22×C3×D3—22no—
121C22×C3×D3—41no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C22×C3×D3—41no—
125C22×C3×D3—41no—
126C22×C3×D3—41no—
127C2×C32×C4—41no—
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130#638—41no—
131#638—41no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—41no—
134C2×C3⋊Dic3—41no—
135C2×C3⋊Dic3—41no—
136C4⋊(C3⋊D3)—41no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—41no—
144C3×D12—22no—
145C3×D12—22no—
146C3×D12—41no—
147C3×D12—41no—
148C3×Dic6—41no—
149#624—81no—
150C32×C8—41no—
151C3×C3⋊C8—41no—
152#539—91no—
153#450—31no—
154#447—91no—
155#394—91no—
156#328—91no—
157#327—91no—
158#324—31no—
159C24×C3—61no—
160C23×D3—21no—
161C2×C3×Q8—31no—
162C2×C3×D4—61no—
163C2×C3×D4—61no—
164C2×C3×D4—122no—
165C22×C3×C4—21no—
166C22×C3×C4—31no—
167C22×C3×C4—31no—
168C22×C3×C4C22×D311yescharacteristic
169C22×C3×C4C22×D311yescharacteristic
170C2×#67—61no—
171C2×#67—61no—
172C2×#67—121no—
173C22×Dic3—61no—
174C22×Dic3—61no—
175C22×Dic3—121no—
176C22×Dic3D1211yescharacteristic
177C22×Dic3D1211yescharacteristic
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—31no—
181C2×D12—31no—
182C2×D12—42no—
183C2×D12—61no—
184C2×C4×D3—61no—
185C2×C4×D3—61no—
186C2×Dic6#6711yescharacteristic
187C3×SD16—122no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—41no—
191C2×C3×C8—61no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C22⋊C4—61no—
197#217—21no—
198#217—122no—
199#217—122no—
200#213—122no—
201(C2×D3)⋊C4—61no—
202(C2×D3)⋊C4—61no—
203(C2×D3)⋊C4—61no—
204(C2×D3)⋊C4—61no—
205(C2×D3)⋊C4—61no—
206(C2×D3)⋊C4—61no—
207(C2×D3)⋊C4—61no—
208(C2×D3)⋊C4—61no—
209(C2×D3)⋊C4—121no—
210(C2×D3)⋊C4—122no—
211(C2×D3)⋊C4—122no—
212C4⋊Dic3—31no—
213C4⋊Dic3—31no—
214C4⋊Dic3—61no—
215C4⋊Dic3—61no—
216C4⋊Dic3—61no—
217C4⋊Dic3#6711yescharacteristic
218Dic3⋊C4—21no—
219Dic3⋊C4—21no—
220Dic3⋊C4—21no—
221Dic3⋊C4—61no—
222Dic3⋊C4—61no—
223Dic3⋊C4—61no—
224Dic3⋊C4—61no—
225C4×Dic3—61no—
226C4×Dic3—61no—
227C2×C3⋊C8—21no—
228D24—122no—
229#204—42no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32#12011yescharacteristic
233C22×C32#13511yescharacteristic
234C22×C32AGL(1,ℤ/8)11yescharacteristic
235C2×C3⋊D3—41no—
236C2×C3⋊D3—41no—
237C2×C3⋊D3—41no—
238C2×C3×D3—42no—
239C2×C3×D3—42no—
240C2×C3×D3—42no—
241C2×C3×D3—81no—
242C2×C3×D3—81no—
243C2×C3×D3—81no—
244C2×C3×D3—81no—
245C32×C4—21no—
246C32×C4—21no—
247C3⋊Dic3—41no—
248C3⋊Dic3—81no—
249C3×Dic3—41no—
250C3×Dic3—41no—
251C3×Dic3—41no—
252C3×Dic3—41no—
253C22×D4—31no—
254C2×SD16—61no—
255C2×D8—181no—
256#126—181no—
257#125—181no—
258#123—181no—
259#123—181no—
260C22⋊Q8—31no—
261C22⋊Q8—181no—
262#121—91no—
263C4×D4—181no—
264C2×C4⋊C4—31no—
265C2×C22⋊C4—181no—
266C2×C22⋊C4—181no—
267#102—61no—
268#102—181no—
269C22⋊C8—31no—
270C22⋊C8D3211yescharacteristic
271#95—91no—
272C23×C3—21no—
273C23×C3—62no—
274C23×C3—121no—
275C23×C3—121no—
276C23×C3—121no—
277C23×C3—121no—
278C23×C3—121no—
279C23×C3—121no—
280C23×C3C2×D1211yescharacteristic
281C23×C3D3×D411yescharacteristic
282C22×D3—22no—
283C22×D3—41no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—61no—
290C22×D3—61no—
291C22×D3—121no—
292C3×Q8—121no—
293C3×D4—62no—
294C3×D4—62no—
295C3×D4—121no—
296C3×D4—121no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—81no—
308C2×C3×C4—121no—
309C2×C3×C4D3×D411yescharacteristic
310C2×C3×C4D3×D411yescharacteristic
311C2×C3×C4D3×D411yescharacteristic
312C2×C3×C4C2×#6711yescharacteristic
313#67—121no—
314#67—121no—
315#67—121no—
316#67—121no—
317#67—241no—
318#67—241no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—41no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328C2×Dic3—121no—
329C2×Dic3—121no—
330C2×Dic3—121no—
331C2×Dic3—121no—
332C2×Dic3—121no—
333C2×Dic3—122no—
334C2×Dic3—121no—
335C2×Dic3—121no—
336C2×Dic3—241no—
337C2×Dic3—241no—
338D12—22no—
339D12—22no—
340D12—41no—
341D12—41no—
342D12—121no—
343D12—121no—
344D12—241no—
345C4×D3—241no—
346C4×D3—241no—
347Dic6—41no—
348C3×C8—41no—
349C3×C8—41no—
350C3×C8—81no—
351C3×C8—121no—
352C3⋊C8—41no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32#44911yescharacteristic
356C2×C32#45411yescharacteristic
357C2×C32B(4,𝔽2)11yescharacteristic
358C3⋊D3—81no—
359C3×D3—82no—
360C24—61no—
361C2×Q8—31no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—91no—
365C2×D4—122no—
366C2×D4—181no—
367C22×C4—31no—
368C22×C4—31no—
369C22×C4—181no—
370C22×C4—181no—
371C22×C4C2×D3211yescharacteristic
372SD16—122no—
373D8—362no—
374C2×C8—21no—
375C2×C8—61no—
376C4⋊C4—31no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—91no—
381C4⋊C4—181no—
382C4⋊C4—181no—
383C4⋊C4—181no—
384C22⋊C4—61no—
385C22⋊C4—181no—
386C22⋊C4—181no—
387C22⋊C4—181no—
388C22⋊C4—181no—
389C22⋊C4—362no—
390C22⋊C4—362no—
391C42—181no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—122no—
402C22×C3—122no—
403C22×C3—122no—
404C22×C3—241no—
405C22×C3—241no—
406C22×C3—241no—
407C22×C3—241no—
408C22×C3#88211yescharacteristic
409C22×C3#90411yescharacteristic
410C22×C3#90811yescharacteristic
411C22×C3#91411yescharacteristic
412C22×C3#91611yescharacteristic
413C22×C3#93711yescharacteristic
414C2×D3—42no—
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—81no—
418C2×D3—81no—
419C2×D3—81no—
420C2×D3—81no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—121no—
426C2×D3—121no—
427C2×D3—241no—
428C2×D3—241no—
429C2×D3—241no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—121no—
439C3×C4—121no—
440Dic3—41no—
441Dic3—41no—
442Dic3—41no—
443Dic3—41no—
444Dic3—121no—
445Dic3—121no—
446Dic3—241no—
447Dic3—241no—
448Dic3—241no—
449Dic3—242no—
450C32#159711yescharacteristic, complemented, Sylow
451C23—62no—
452C23—121no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—181no—
459C23D3×D1211yescharacteristic
460Q8—121no—
461D4—62no—
462D4—62no—
463D4—121no—
464D4—121no—
465D4—361no—
466D4—361no—
467D4—361no—
468C2×C4—41no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—121no—
476C2×C4—181no—
477C2×C4—361no—
478C2×C4—361no—
479C2×C4—361no—
480C2×C4—721no—
481C2×C4#380211yescharacteristic
482C2×C4D3×#6711yescharacteristic
483C8—41no—
484C8—121no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—242no—
495C2×C3#473411yescharacteristic
496C2×C3#474111yescharacteristic
497C2×C3#475311yescharacteristic
498C2×C3#503211yescharacteristic
499C2×C3#517211yescharacteristic
500C2×C3#520111yescharacteristic
501D3—82no—
502D3—241no—
503D3—241no—
504D3—481no—
505C22—21no—
506C22—21no—
507C22—122no—
508C22—122no—
509C22—122no—
510C22—241no—
511C22—241no—
512C22—241no—
513C22—241no—
514C22—361no—
515C22—361no—
516C22—361no—
517C22#6379411yescharacteristic
518C22#6380611yescharacteristic
519C22#6392211yescharacteristic
520C4—21no—
521C4—21no—
522C4—121no—
523C4—121no—
524C4—121no—
525C4—121no—
526C4—361no—
527C4—721no—
528C3—21no—
529C3#6813011yescharacteristic, complemented
530C3#7097811yescharacteristic, complemented
531C2—21no—
532C2—21no—
533C2—242no—
534C2—721no—
535C2#1059040711yescharacteristic
536C2#1059082311yescharacteristic
537C2#1059102011yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊C8C22
socleC22×C32#120
prime radical#10591162C2
hypercenterC22⋊C8D32
hypocenterC32#1597
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
1. #10594980C2semidirect
2. C3×#4731C2semidirect
3. #10591300C2semidirect
5. #10589336C2semidirect
6. C3×#4469C2semidirect
7. #10589152C2semidirect
36. #4731D3semidirect
42. #4469D3semidirect
450. C32#1597semidirect
529. C3#68130semidirect
530. C3#70978semidirect

Maximal subgroups

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

subgroupindex
#105949802
C3×#47312
#105913002
#105911622
#105893362
C3×#44692
#105891522
#709783
#681303