Finite Group Databasebeta

← #11779219

Subgroups — #11779219

Order 1152.

559 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 100 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×#5727C211yesmaximal, characteristic, complemented
2#10592787C211yesmaximal, characteristic, complemented
3#10590367C211yesmaximal, characteristic, complemented
4#10590349C211yesmaximal, characteristic
5C3×#4547C211yesmaximal, characteristic, complemented
6#10589083C211yesmaximal, characteristic, complemented
7#10589080C211yesmaximal, characteristic, complemented
8#71914—31nomaximal
9#68457—31nomaximal
10C2×C3×D3×Q8—21no—
11C32×C4⋊Q8C2211yescharacteristic
12#64083C2211yescharacteristic
13C3×#946—21no—
14C3×#896—22no—
15C3×#893—21no—
16C3×#888—21no—
17C3×C4×D12C2211yescharacteristic
18C3×C4×Dic6C2211yescharacteristic
19C2×#3717—21no—
20C2×#3714—21no—
21#63636—22no—
22C4×C3⋊(C3⋊C8)C2211yescharacteristic
23C3×C4⋊(C3⋊C8)C2211yescharacteristic
24C3×C4⋊(C3⋊C8)C2211yescharacteristic
25#63564—21no—
26#63564—21no—
27C3×D4×Q8—31no—
28#5727D311yescharacteristic, complemented
29#5088—31no—
30#5088—31no—
31#5088—61no—
32#5027—31no—
33#5015—31no—
34#4848—31no—
35#4843—31no—
36C3×#332—31no—
37#4547D311yescharacteristic, complemented
38#4535—31no—
39D4⋊(C3⋊C8)—31no—
40#4484—31no—
41#4483—31no—
42C2×C32×Q8—21no—
43C2×C4⋊(C3⋊D3)—21no—
44C2×C4⋊(C3⋊D3)—21no—
45C2×C4⋊(C3⋊D3)—21no—
46C3×D3×Q8—42no—
47C3×D3×Q8—42no—
48C2×C3×D12D411yescharacteristic
49C2×C3×C4×D3—22no—
50C2×C3×C4×D3—21no—
51C2×C3×Dic6—22no—
52C2×C3×Dic6D411yescharacteristic
53C32×C4⋊C4—21no—
54C32×C4⋊C4D412yes—
55C32×C42C2311yescharacteristic
56C2×C3⋊(C3⋊C8)—21no—
57C3×(C2×D3)⋊C4—22no—
58C3×(C2×D3)⋊C4—21no—
59C3×C4⋊Dic3D411yescharacteristic
60C3×C4⋊Dic3D411yescharacteristic
61C3×Dic3⋊C4—22no—
62C3×Dic3⋊C4—21no—
63C3×C4×Dic3—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66#3717—42no—
67#3714—42no—
68#1627—91noSylow
69C22×C3×Q8—61no—
70C2×D3×Q8—21no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—61no—
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4⋊Q8C2×D311yescharacteristic
75C3×C22⋊Q8—61no—
76C3×C22⋊Q8—62no—
77C3×C4×Q8—31no—
78C3×C4×D4—31no—
79C3×C4×D4—61no—
80#946—21no—
81C2×#215—61no—
82C2×#213—61no—
83C2×D24—61no—
84C2×#204—61no—
85#896—22no—
86#893—21no—
87#888—21no—
88#874—31no—
89#874—31no—
90#874—61no—
91C4×D12C2×D311yescharacteristic
92C4×Dic6C2×D311yescharacteristic
93C3×C4⋊C8—31no—
94C3×C4⋊C8—31no—
95#821—61no—
96#821—61no—
97#809—61no—
98#809—61no—
99#809—64no—
100#809—62no—
101#809—62no—
102C4⋊(C3⋊C8)C2×D311yescharacteristic
103C4⋊(C3⋊C8)C2×D311yescharacteristic
104C4×C3⋊C8—31no—
105C4×C3⋊C8—31no—
106C4×C3⋊C8—61no—
107C22×C3⋊D3—41no—
108C22×C3×D3—21no—
109C32×Q8—41no—
110C32×Q8—41no—
111C2×C32×C4—22no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C4⋊(C3⋊D3)—42no—
116C4⋊(C3⋊D3)—41no—
117C4⋊(C3⋊D3)—41no—
118C4⋊(C3⋊D3)—41no—
119C4⋊(C3⋊D3)—41no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C3×D12—21no—
124C3×D12SD1612yescomplemented
125C3×C4×D3—44no—
126C3×C4×D3—41no—
127C3×C4×D3—41no—
128C3×Dic6—21no—
129C3×Dic6—44no—
130C3×Dic6D812yescomplemented
131C3⋊(C3⋊C8)—41no—
132C3×C3⋊C8—41no—
133C3×C3⋊C8—41no—
134D4×Q8—31no—
135#486—91no—
136#460—91no—
137#459—91no—
138#332—31no—
139Q8⋊C8—91no—
140D4⋊C8—91no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—31no—
144C2×C3×Q8—41no—
145C2×C3×Q8—62no—
146C2×C3×Q8—122no—
147C2×C3×Q8—122no—
148C2×C3×D4—31no—
149C22×C3×C4—62no—
150C22×C3×C4—61no—
151D3×Q8—42no—
152D3×Q8—42no—
153C2×D12—62no—
154C2×D12—61no—
155C2×D12—61no—
156C2×D12—61no—
157C2×D12—61no—
158C2×D12—61no—
159C2×D12—61no—
160C2×D12—121no—
161C2×D12—121no—
162C2×D12#6711yescharacteristic
163C2×C4×D3—22no—
164C2×C4×D3—21no—
165C2×Dic6—22no—
166C2×Dic6#6711yescharacteristic
167C2×C3×C8—61no—
168C2×C3×C8—61no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—41no—
175C3×C4⋊C4—62no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4D1212yes—
178C3×C4⋊C4#6712yes—
179C3×C22⋊C4—62no—
180C3×C22⋊C4—61no—
181C3×C42—21no—
182C3×C42—61no—
183C3×C42C22×D311yescharacteristic
184C3×C42C22×D311yescharacteristic
185#215—122no—
186#213—122no—
187(C2×D3)⋊C4—22no—
188(C2×D3)⋊C4—21no—
189C4⋊Dic3#6711yescharacteristic
190C4⋊Dic3#6711yescharacteristic
191Dic3⋊C4—22no—
192Dic3⋊C4—21no—
193C4×Dic3—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—62no—
197C2×C3⋊C8—61no—
198C2×C3⋊C8—61no—
199D24—122no—
200#204—122no—
201C22×C32#12011yescharacteristic
202C2×C3⋊D3—81no—
203C2×C3⋊D3—81no—
204C2×C3⋊D3—81no—
205C2×C3×D3—22no—
206C2×C3×D3—41no—
207C2×C3×D3—41no—
208C32×C4—21no—
209C32×C4—42no—
210C32×C4C2×D811yescharacteristic
211C32×C4C2×SD1611yescharacteristic
212C32×C4AGL(1,ℤ/8)11yescharacteristic
213C32×C4#13711yescharacteristic
214C3×Dic3—22no—
215C3×Dic3—41no—
216C3×Dic3—41no—
217C22×Q8—61no—
218C2×SD16—181no—
219C2×D8—181no—
220C4⋊Q8—61no—
221C4⋊Q8D3211yescharacteristic
222#127—91no—
223C22⋊Q8—61no—
224C22⋊Q8—62no—
225C4×Q8—31no—
226C4×D4—31no—
227C4×D4—61no—
228C4⋊C8—31no—
229C4⋊C8—31no—
230#102—181no—
231#102—181no—
232#102—182no—
233C4×C8—91no—
234C23×C3—61no—
235C22×D3—21no—
236C22×D3—122no—
237C22×D3—121no—
238C22×D3—121no—
239C3×Q8—32no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×Q8—41no—
243C3×Q8—41no—
244C3×Q8—61no—
245C3×Q8—81no—
246C3×Q8—81no—
247C3×Q8—124no—
248C3×D4—32no—
249C3×D4—61no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—42no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—124no—
260C2×C3×C4—121no—
261C2×C3×C4—121no—
262C2×C3×C4C2×D1211yescharacteristic
263C2×C3×C4D3×D411yescharacteristic
264C2×C3×C4D3×D411yescharacteristic
265C2×C3×C4C2×#6711yescharacteristic
266C2×C3×C4C2×#6711yescharacteristic
267C2×C3×C4C2×#6711yescharacteristic
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271D12—21no—
272D12—124no—
273D12—122no—
274D12—122no—
275D12—121no—
276D12—121no—
277D12—121no—
278D12—121no—
279D12—121no—
280D12—121no—
281D12—121no—
282D12—121no—
283D12—241no—
284D12—241no—
285D12—241no—
286D12—241no—
287D12#21512yescomplemented
288C4×D3—44no—
289C4×D3—41no—
290C4×D3—41no—
291Dic6—21no—
292Dic6—44no—
293Dic6#21312yescomplemented
294C3×C8—121no—
295C3×C8—121no—
296C3⋊C8—41no—
297C3⋊C8—41no—
298C3⋊C8—122no—
299C3⋊C8—121no—
300C3⋊C8—121no—
301C2×C32#44711yescharacteristic
302C2×C32#44811yescharacteristic
303C2×C32#45411yescharacteristic
304C3⋊D3—161no—
305C3×D3—42no—
306C2×Q8—21no—
307C2×Q8—31no—
308C2×Q8—62no—
309C2×Q8—122no—
310C2×Q8—122no—
311C2×D4—31no—
312C2×D4—181no—
313C2×D4—181no—
314C2×D4—181no—
315C22×C4—62no—
316C22×C4—61no—
317SD16—362no—
318D8—362no—
319C2×C8—61no—
320C2×C8—61no—
321C2×C8—181no—
322C4⋊C4—21no—
323C4⋊C4—31no—
324C4⋊C4—31no—
325C4⋊C4—62no—
326C4⋊C4—61no—
327C4⋊C4#62612yes—
328C22⋊C4—62no—
329C22⋊C4—61no—
330C42—61no—
331C42C2×D3211yescharacteristic
332C22×C3—21no—
333C22×C3—62no—
334C22×C3—121no—
335C22×C3—121no—
336C22×C3#88211yescharacteristic
337C22×C3#95311yescharacteristic
338C2×D3—22no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—242no—
342C2×D3—242no—
343C2×D3—242no—
344C2×D3—241no—
345C2×D3—241no—
346C2×D3—241no—
347C2×D3—241no—
348C2×D3—241no—
349C2×D3—241no—
350C3×C4—22no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—42no—
358C3×C4—42no—
359C3×C4—62no—
360C3×C4—82no—
361C3×C4—121no—
362C3×C4—121no—
363C3×C4D3×D811yescharacteristic
364C3×C4D3×SD1611yescharacteristic
365C3×C4#91411yescharacteristic
366C3×C4#91811yescharacteristic
367C3×C4C2×#21311yescharacteristic
368C3×C4#93211yescharacteristic
369C3×C4C2×#21511yescharacteristic
370C3×C4#94211yescharacteristic
371Dic3—22no—
372Dic3—41no—
373Dic3—41no—
374C32#162711yescharacteristic, complemented, Sylow
375C23—61no—
376C23—361no—
377Q8—32no—
378Q8—41no—
379Q8—41no—
380Q8—61no—
381Q8—124no—
382D4—32no—
383D4—61no—
384D4—362no—
385D4—361no—
386D4—361no—
387D4—361no—
388D4—361no—
389C2×C4—22no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—124no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4C2×#62611yescharacteristic
397C2×C4D3×#6711yescharacteristic
398C2×C4D3×#6711yescharacteristic
399C8—121no—
400C8—121no—
401C8—361no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—122no—
406C2×C3#475311yescharacteristic
407C2×C3#477311yescharacteristic
408C2×C3#480611yescharacteristic
409C2×C3#521711yescharacteristic
410C2×C3#522711yescharacteristic
411C2×C3#524511yescharacteristic
412D3—42no—
413D3—482no—
414D3—481no—
415D3—481no—
416C22—62no—
417C22—121no—
418C22—121no—
419C22—721no—
420C22—721no—
421C22—721no—
422C22#6397711yescharacteristic
423C4—21no—
424C4—42no—
425C4—62no—
426C4—121no—
427C4—121no—
428C4D3×#21311yescharacteristic
429C4#6393411yescharacteristic
430C4D3×#21511yescharacteristic
431C4#6394811yescharacteristic
432C3—21no—
433C3#6845711yescharacteristic, complemented
434C3#7191411yescharacteristic, complemented
435C2—122no—
436C2—1441no—
437C2#1059144811yescharacteristic
438C2#1059158411yescharacteristic
439C2#1059164611yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radical#10590349C2
hypercenterC4⋊Q8D32
hypocenterC32#1627
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5727C2semidirect
2. #10592787C2semidirect
3. #10590367C2semidirect
5. C3×#4547C2semidirect
6. #10589083C2semidirect
7. #10589080C2semidirect
28. #5727D3semidirect
37. #4547D3semidirect
124. C3×D12SD16semidirect
130. C3×Dic6D8semidirect
287. D12#215semidirect
293. Dic6#213semidirect
374. C32#1627semidirect
433. C3#68457semidirect
434. C3#71914semidirect

Maximal subgroups

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

subgroupindex
C3×#57272
#105927872
#105903672
#105903492
C3×#45472
#105890832
#105890802
#719143
#684573