Finite Group Databasebeta

← #11779218

Subgroups — #11779218

Order 1152.

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

normal subgroups only

This group has 570 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
1#10593903C211yesmaximal, characteristic, complemented
2#10592771C211yesmaximal, characteristic
3C3×#5075C211yesmaximal, characteristic, complemented
4C3×#5070C211yesmaximal, characteristic, complemented
5#10589093C211yesmaximal, characteristic, complemented
6#10589084C211yesmaximal, characteristic, complemented
7#10589082C211yesmaximal, characteristic, complemented
8#71205—31nomaximal
9#71035—31nomaximal
10C2×#3797—21no—
11C32×#127C2211yescharacteristic
12C2×#3753—21no—
13C4×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×#214—21no—
15C3×#869C2211yescharacteristic
16C3×C4×Dic6C2211yescharacteristic
17#63918—21no—
18#63895—22no—
19#63894—21no—
20#63871—21no—
21#63659—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C3×#832—22no—
24C3×#832—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5727—31no—
28Q8×D12—31no—
29C3×#486—31no—
30C3×#460—31no—
31#5075D311yescharacteristic, complemented
32#5072—31no—
33#5072—31no—
34#5072—61no—
35#5070D311yescharacteristic, complemented
36#4547—31no—
37#4535—31no—
38#4483—31no—
39#4482—31no—
40#4459—31no—
41#4456—31no—
42C2×C32×D4—21no—
43C2×C32×D4—21no—
44C2×C32×D4—21no—
45C2×C4⋊(C3⋊D3)D411yescharacteristic
46C2×C4×C3⋊D3—21no—
47C2×C3×Dic6—21no—
48C2×C3×Dic6D411yescharacteristic
49C2×#627—22no—
50C2×#624—22no—
51#3797—42no—
52#3797—42no—
53C32×C42C2311yescharacteristic
54#3753—42no—
55#3752—21no—
56C4⋊(C3⋊Dic3)D411yescharacteristic
57C2×C3⋊(C3⋊C8)—21no—
58C3×#214—42no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3D411yescharacteristic
61C3×C4⋊Dic3D412yes—
62C3×Dic3⋊C4—21no—
63C3×C4×Dic3—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66Dic3⋊Dic3—22no—
67#3722—22no—
68#1627—91noSylow
69C2×D3×Q8—61no—
70C2×D3×Q8—61no—
71C2×C3×SD16—61no—
72C3×C4⋊Q8—31no—
73C3×#127—21no—
74C3×#127C2×D311yescharacteristic
75C3×#127C2×D311yescharacteristic
76C3×C4×Q8—31no—
77#946—61no—
78C2×#214—21no—
79C2×#213—61no—
80C2×#213—61no—
81C2×#213—62no—
82#897—61no—
83#897—62no—
84#896—62no—
85#893—61no—
86#888—61no—
87C4×D12—31no—
88C4×D12—31no—
89C4×D12—61no—
90C4×D12—61no—
91#869—61no—
92#869C2×D311yescharacteristic
93C4×Dic6C2×D311yescharacteristic
94C3×C4⋊C8—31no—
95C3×#102—61no—
96C3×#102—62no—
97C3×C4×C8—31no—
98#832—22no—
99#832—21no—
100#832—61no—
101#832—61no—
102#832—62no—
103C4⋊(C3⋊C8)—31no—
104C4⋊(C3⋊C8)—31no—
105C4⋊(C3⋊C8)—61no—
106C4⋊(C3⋊C8)C2×D311yescharacteristic
107C4×C3⋊C8C2×D311yescharacteristic
108C23×C32—41no—
109C22×C3⋊D3—21no—
110C32×D4—41no—
111C32×D4—41no—
112C32×D4—42no—
113C32×D4—41no—
114C32×D4—41no—
115C2×C32×C4C2×D411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C3⋊Dic3—21no—
119C4⋊(C3⋊D3)—21no—
120C4⋊(C3⋊D3)SD1612yescomplemented
121C4×C3⋊D3—41no—
122C4×C3⋊D3—41no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—22no—
126C3×Dic6—21no—
127C3×Dic6—41no—
128C3×Dic6—41no—
129C3×Dic6D812yescomplemented
130#627—44no—
131#624—44no—
132C3⋊(C3⋊C8)—41no—
133C3×C3⋊C8—41no—
134C3×C3⋊C8—41no—
135D4×Q8—91no—
136#486—31no—
137#460—31no—
138#459—91no—
139#332—91no—
140Q8⋊C8—91no—
141D4⋊C8—91no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—41no—
152C2×C3×D4—41no—
153D3×Q8—122no—
154D3×Q8—122no—
155D3×Q8—122no—
156D3×Q8—122no—
157C2×D12—31no—
158C2×D12—31no—
159C2×D12—61no—
160C2×C4×D3—62no—
161C2×C4×D3—62no—
162C2×C4×D3—61no—
163C2×C4×D3—61no—
164C2×C4×D3—121no—
165C2×Dic6—21no—
166C2×Dic6—62no—
167C2×Dic6—62no—
168C2×Dic6#6711yescharacteristic
169C3×SD16—122no—
170C2×C3×C8—61no—
171C2×C3×C8—61no—
172C3×C4⋊C4—32no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—61no—
175C3×C4⋊C4—61no—
176C3×C42—21no—
177C3×C42—61no—
178C3×C42C22×D311yescharacteristic
179C3×C42C22×D311yescharacteristic
180#214—42no—
181#213—122no—
182#213—122no—
183#213—124no—
184(C2×D3)⋊C4—62no—
185(C2×D3)⋊C4—62no—
186(C2×D3)⋊C4—61no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—121no—
189C4⋊Dic3—21no—
190C4⋊Dic3—31no—
191C4⋊Dic3—31no—
192C4⋊Dic3—61no—
193C4⋊Dic3—62no—
194C4⋊Dic3#6711yescharacteristic
195C4⋊Dic3#6712yes—
196Dic3⋊C4—21no—
197Dic3⋊C4—62no—
198C4×Dic3—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—62no—
202C2×C3⋊C8—61no—
203C2×C3⋊C8—61no—
204C22×C32—81no—
205C22×C32—81no—
206C22×C32—81no—
207C22×C32#12011yescharacteristic
208C2×C3⋊D3—22no—
209C2×C3⋊D3—41no—
210C2×C3⋊D3—41no—
211C32×C4—21no—
212C32×C4C2×D811yescharacteristic
213C32×C4C2×SD1611yescharacteristic
214C32×C4AGL(1,ℤ/8)11yescharacteristic
215C32×C4#13711yescharacteristic
216C3⋊Dic3—41no—
217C3×Dic3—22no—
218C3×Dic3—42no—
219C3×Dic3—41no—
220C22×Q8—181no—
221C2×SD16—61no—
222C2×D8—181no—
223C4⋊Q8—31no—
224C4⋊Q8—181no—
225#127D3211yescharacteristic
226C22⋊Q8—182no—
227C22⋊Q8—181no—
228C4×Q8—31no—
229C4×D4—91no—
230C4×D4—181no—
231C4⋊C8—31no—
232C4⋊C8—91no—
233#102—61no—
234#102—62no—
235#102—181no—
236C4×C8—31no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—41no—
240C22×D3—61no—
241C22×D3—61no—
242C22×D3—121no—
243C3×Q8—32no—
244C3×Q8—61no—
245C3×Q8—121no—
246C3×Q8—121no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—42no—
251C3×D4—41no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—81no—
260C3×D4—81no—
261C3×D4—82no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—61no—
266C2×C3×C4—62no—
267C2×C3×C4—61no—
268C2×C3×C4D3×D411yescharacteristic
269C2×C3×C4D3×D411yescharacteristic
270C2×C3×C4D3×D411yescharacteristic
271C2×C3×C4D3×D411yescharacteristic
272C2×C3×C4C2×#6711yescharacteristic
273C2×C3×C4C2×#6711yescharacteristic
274C2×Dic3—21no—
275C2×Dic3—21no—
276C2×Dic3—22no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279C2×Dic3—121no—
280D12—32no—
281D12—32no—
282D12—62no—
283D12—61no—
284D12—61no—
285D12—121no—
286C4×D3—124no—
287C4×D3—124no—
288C4×D3—121no—
289C4×D3—121no—
290C4×D3—121no—
291C4×D3—121no—
292C4×D3—241no—
293C4×D3—241no—
294Dic6—21no—
295Dic6—41no—
296Dic6—41no—
297Dic6—124no—
298Dic6—124no—
299Dic6#21312yescomplemented
300C3×C8—121no—
301C3×C8—121no—
302C3⋊C8—41no—
303C3⋊C8—41no—
304C3⋊C8—122no—
305C3⋊C8—121no—
306C3⋊C8—121no—
307C2×C32—161no—
308C2×C32#44711yescharacteristic
309C2×C32#44811yescharacteristic
310C2×C32#45411yescharacteristic
311C3⋊D3—42no—
312C2×Q8—31no—
313C2×Q8—61no—
314C2×Q8—182no—
315C2×Q8—362no—
316C2×Q8—362no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—91no—
321C22×C4—182no—
322C22×C4—181no—
323SD16—122no—
324D8—362no—
325C2×C8—61no—
326C2×C8—61no—
327C2×C8—181no—
328C4⋊C4—31no—
329C4⋊C4—32no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—91no—
333C4⋊C4—182no—
334C22⋊C4—182no—
335C22⋊C4—181no—
336C42—61no—
337C42C2×D3211yescharacteristic
338C22×C3—21no—
339C22×C3—82no—
340C22×C3—82no—
341C22×C3—82no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3#93711yescharacteristic
349C22×C3#93711yescharacteristic
350C2×D3—62no—
351C2×D3—62no—
352C2×D3—122no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—241no—
358C2×D3—241no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—41no—
366C3×C4—62no—
367C3×C4—122no—
368C3×C4—121no—
369C3×C4D3×D811yescharacteristic
370C3×C4#91111yescharacteristic
371C3×C4D3×SD1611yescharacteristic
372C3×C4D3×SD1611yescharacteristic
373C3×C4#91511yescharacteristic
374C3×C4#91511yescharacteristic
375C3×C4C2×#21311yescharacteristic
376C3×C4#94911yescharacteristic
377Dic3—22no—
378Dic3—42no—
379Dic3—41no—
380Dic3—121no—
381Dic3—121no—
382Dic3—241no—
383C32#162711yescharacteristic, complemented, Sylow
384C23—41no—
385C23—181no—
386Q8—32no—
387Q8—61no—
388Q8—121no—
389Q8—121no—
390Q8—364no—
391D4—42no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—92no—
397D4—181no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—62no—
401C2×C4—181no—
402C2×C4—364no—
403C2×C4—361no—
404C2×C4—361no—
405C2×C4D3×#6711yescharacteristic
406C2×C4D3×#6711yescharacteristic
407C2×C4#381111yescharacteristic
408C8—121no—
409C8—121no—
410C8—361no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—162no—
415C2×C3—161no—
416C2×C3—161no—
417C2×C3#503611yescharacteristic
418C2×C3#506211yescharacteristic
419C2×C3#515611yescharacteristic
420C2×C3#517011yescharacteristic
421C2×C3#517011yescharacteristic
422C2×C3#520111yescharacteristic
423D3—122no—
424D3—122no—
425D3—242no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—182no—
430C22—361no—
431C22—361no—
432C22#6398311yescharacteristic
433C4—21no—
434C4—62no—
435C4—122no—
436C4—121no—
437C4—361no—
438C4D3×#21311yescharacteristic
439C4#6392511yescharacteristic
440C4#6393011yescharacteristic
441C4#6393111yescharacteristic
442C3—21no—
443C3#7103511yescharacteristic, complemented
444C3#7120511yescharacteristic, complemented
445C2—161no—
446C2—362no—
447C2#1059151511yescharacteristic
448C2#1059151811yescharacteristic
449C2#1059152511yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×#127C22
socleC22×C32#120
prime radical#10592771C2
hypercenter#127D32
hypocenterC32#1627
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #10593903C2semidirect
3. C3×#5075C2semidirect
4. C3×#5070C2semidirect
5. #10589093C2semidirect
6. #10589084C2semidirect
7. #10589082C2semidirect
31. #5075D3semidirect
35. #5070D3semidirect
120. C4⋊(C3⋊D3)SD16semidirect
129. C3×Dic6D8semidirect
299. Dic6#213semidirect
383. C32#1627semidirect
443. C3#71035semidirect
444. C3#71205semidirect

Maximal subgroups

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

subgroupindex
#105939032
#105927712
C3×#50752
C3×#50702
#105890932
#105890842
#105890822
#712053
#710353