Finite Group Databasebeta

← #11878900

Subgroups — #11878900

Order 1152.

817 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5728C211yesmaximal, characteristic
2D3×#869C211yesmaximal, characteristic
3#10592792C211yesmaximal, characteristic, complemented
4C3×#5093C211yesmaximal, characteristic, complemented
5#10590604C214yesmaximal
6#10590561C214yesmaximal, complemented
7#10590387C211yesmaximal, characteristic, complemented
8#10590386C211yesmaximal, characteristic, complemented
9#10590261C211yesmaximal, characteristic
10#83370—31nomaximal
11#81604—31nomaximal
12C2×D3×Dic6C2212yes—
13C32×C4⋊Q8C2211yescharacteristic
14#64078C2211yescharacteristic
15C3×#946—21no—
16C3×Q8×Dic3—21no—
17C3×#897C2212yes—
18C3×#892—21no—
19C3×#890—21no—
20C3×#887C2212yes—
21C3×C42×D3C2211yescharacteristic
22C3×#869C2211yescharacteristic
23C3×#869C2211yescharacteristic
24#63918—21no—
25#63918—21no—
26D3×C4⋊Dic3—21no—
27D3×C4⋊Dic3—21no—
28#63866C2212yes—
29C2×#3719C2214yes—
30C2×#3718C2214yes—
31#63637C2214yes—
32C4×C3⋊(C3⋊C8)C2211yescharacteristic
33C3×#810C2214yes—
34C3×C4×C3⋊C8C2211yescharacteristic
35Dic3⋊(C3⋊C8)C2212yes—
36(C2×D3)⋊(C3⋊C8)C2212yes—
37C3×#533—31no—
38#5728D311yescharacteristic
39D3×C4⋊Q8—31no—
40C2×#869—31no—
41C3×#487—31no—
42#5093—31no—
43#5093—31no—
44#5093—61no—
45#5093D311yescharacteristic, complemented
46#5092—31no—
47#5092—31no—
48#5092—31no—
49#5050—34no—
50#5031—34no—
51#4925—31no—
52#4809—34no—
53#4795—34no—
54#4699—31no—
55#4685—31no—
56C2×C32×Q8—21no—
57C2×#634C2312yes—
58C2×C3×C4×D3D411yescharacteristic
59C2×C3×C4×D3D411yescharacteristic
60C2×C3×C4×D3C2311yescharacteristic
61C2×C3×Dic6C2312yes—
62C2×C3×Dic6C2312yes—
63C2×#627—22no—
64C2×D3×Dic3—22no—
65D3×Dic6—44no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2312yes—
68C32×C42C2311yescharacteristic
69C4⋊(C3⋊Dic3)—21no—
70C4⋊(C3⋊Dic3)—21no—
71C2×C3⋊(C3⋊C8)C2312yes—
72C3×(C2×D3)⋊C4—22no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—22no—
78C3×Dic3⋊C4—22no—
79C3×C4×Dic3—22no—
80C3×C4×Dic3D411yescharacteristic
81C3×C4×Dic3D411yescharacteristic
82C3×C4×Dic3C2311yescharacteristic
83C2×C3×C3⋊C8C2312yes—
84Dic3⋊Dic3—22no—
85Dic3⋊Dic3—22no—
86#3719—44no—
87#3718—44no—
88#3209—91noSylow
89C2×D3×Q8—32no—
90C22×Dic6—32no—
91C3×C4⋊Q8—21no—
92C3×C4⋊Q8—31no—
93C3×C4⋊Q8—31no—
94C3×C4⋊Q8C2×D311yescharacteristic
95C3×C4⋊Q8C2×D311yescharacteristic
96C3×#125—61no—
97C3×C22⋊Q8—32no—
98C3×C22⋊Q8—61no—
99C3×C4×Q8—32no—
100C3×C4×Q8—61no—
101C3×#117—61no—
102C2×C3×C42—31no—
103#946—21no—
104Q8×Dic3—21no—
105#944—32no—
106C2×#216—34no—
107C2×#216—34no—
108C2×#216—34no—
109C2×C4⋊Dic3—61no—
110C2×C4⋊Dic3—61no—
111C2×Dic12—34no—
112#897C2×D312yes—
113#892—21no—
114D3×C4⋊C4—61no—
115D3×C4⋊C4—61no—
116#890—21no—
117#888—61no—
118#888—61no—
119#887C2×D312yes—
120C42×D3C2×D311yescharacteristic
121#869—32no—
122#869—31no—
123#869—31no—
124#869—61no—
125#869—61no—
126#869—61no—
127#869C2×D311yescharacteristic
128#869C2×D311yescharacteristic
129C3×#103—34no—
130C3×C4×C8—31no—
131#830—32no—
132(C2×D3)⋊C8—32no—
133Dic3⋊C8—32no—
134#810—34no—
135#810—34no—
136#810—64no—
137#810C2×D314yes—
138C4⋊(C3⋊C8)—32no—
139C4×C3⋊C8—31no—
140C4×C3⋊C8—31no—
141C4×C3⋊C8—61no—
142C4×C3⋊C8C2×D311yescharacteristic
143C22×C3×D3C2×D411yescharacteristic
144C32×Q8—42no—
145C2×C32×C4—22no—
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2411yescharacteristic
149C2×C3⋊Dic3—22no—
150#634—24no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3C2×D411yescharacteristic
154C2×C3×Dic3C2×D411yescharacteristic
155C2×C3×Dic3C2×D411yescharacteristic
156C3×C4×D3—22no—
157C3×C4×D3—22no—
158C3×C4×D3Q1614yescomplemented
159C3×Dic6—24no—
160C3×Dic6—24no—
161#627—44no—
162D3×Dic3—44no—
163C3⋊(C3⋊C8)—22no—
164C3×C3⋊C8—22no—
165#533—31no—
166C2×C4⋊Q8—91no—
167#494—91no—
168#494—91no—
169#487—31no—
170#487—91no—
171#462—94no—
172#451—94no—
173#431—91no—
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—32no—
177C2×C3×Q8—32no—
178C2×C3×Q8—41no—
179C22×C3×C4—31no—
180C22×C3×C4—31no—
181C22×C3×C4—31no—
182C22×Dic3—62no—
183D3×Q8—124no—
184C2×C4×D3—62no—
185C2×C4×D3#6711yescharacteristic
186C2×C4×D3#6711yescharacteristic
187C2×C4×D3C22×D311yescharacteristic
188C2×Dic6—32no—
189C2×Dic6—32no—
190C2×Dic6—62no—
191C2×Dic6—62no—
192C2×Dic6—62no—
193C2×Dic6—124no—
194C2×Dic6C22×D312yes—
195C2×Dic6C22×D312yes—
196C2×C3×C8—32no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—62no—
202C3×C4⋊C4—62no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4C22×D312yes—
208C3×C4⋊C4C22×D312yes—
209C3×C22⋊C4—62no—
210C3×C42—21no—
211C3×C42—31no—
212C3×C42—31no—
213C3×C42—31no—
214C3×C42—62no—
215C3×C42C22×D311yescharacteristic
216C3×C42C22×D311yescharacteristic
217#216—124no—
218#216—124no—
219#216—124no—
220(C2×D3)⋊C4—22no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—22no—
226C4⋊Dic3—62no—
227C4⋊Dic3—62no—
228C4⋊Dic3—61no—
229C4⋊Dic3—61no—
230C4⋊Dic3—61no—
231C4⋊Dic3—61no—
232C4⋊Dic3—121no—
233C4⋊Dic3—121no—
234Dic3⋊C4—22no—
235Dic3⋊C4—62no—
236Dic3⋊C4—62no—
237C4×Dic3—22no—
238C4×Dic3#6711yescharacteristic
239C4×Dic3#6711yescharacteristic
240C4×Dic3C22×D311yescharacteristic
241C2×C3⋊C8—32no—
242C2×C3⋊C8—32no—
243C2×C3⋊C8—62no—
244C2×C3⋊C8C22×D312yes—
245Dic12—124no—
246C22×C32C22×D411yescharacteristic
247C2×C3×D3—21no—
248C2×C3×D3—21no—
249C2×C3×D3C2×Q1612yes—
250C32×C4—42no—
251C32×C4C2×Q1612yes—
252C32×C4#13712yes—
253C32×C4H(5,𝔽2)12yes—
254C3⋊Dic3—42no—
255C3×Dic3—21no—
256C3×Dic3—21no—
257C3×Dic3—42no—
258C3×Dic3—42no—
259C3×Dic3C2×Q1612yes—
260C22×Q8—92no—
261C2×Q16—94no—
262C2×Q16—94no—
263C4⋊Q8—31no—
264C4⋊Q8—31no—
265C4⋊Q8—92no—
266C4⋊Q8—91no—
267C4⋊Q8—181no—
268C4⋊Q8—181no—
269C4⋊Q8D3211yescharacteristic
270#125—61no—
271C22⋊Q8—32no—
272C22⋊Q8—61no—
273C4×Q8—32no—
274C4×Q8—61no—
275#117—61no—
276C2×C4⋊C4—181no—
277C2×C4⋊C4—181no—
278C2×C42—31no—
279C4⋊C8—92no—
280#103—34no—
281#103—94no—
282C22⋊C8—92no—
283C4×C8—31no—
284C4×C8—91no—
285C23×C3—31no—
286C22×D3C2×#6711yescharacteristic
287C3×Q8—42no—
288C3×Q8—42no—
289C3×Q8—64no—
290C3×Q8—64no—
291C3×Q8—82no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—31no—
298C2×C3×C4—34no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—42no—
302C2×C3×C4—62no—
303C2×C3×C4—62no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4D3×D411yescharacteristic
307C2×C3×C4D3×D411yescharacteristic
308C2×C3×C4C2×#6711yescharacteristic
309C2×C3×C4C2×#6711yescharacteristic
310C2×C3×C4C23×D311yescharacteristic
311C2×C3×C4C23×D311yescharacteristic
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—62no—
315C2×Dic3—62no—
316C2×Dic3—122no—
317C2×Dic3—124no—
318C2×Dic3C2×#6711yescharacteristic
319C2×Dic3C2×#6711yescharacteristic
320C2×Dic3C2×#6711yescharacteristic
321C4×D3—22no—
322C4×D3—22no—
323C4×D3—124no—
324C4×D3#21614yescomplemented
325Dic6—24no—
326Dic6—24no—
327Dic6—64no—
328Dic6—64no—
329Dic6—124no—
330Dic6—124no—
331Dic6—124no—
332C3×C8—62no—
333C3⋊C8—22no—
334C3⋊C8—62no—
335C3⋊C8—62no—
336C3⋊C8—122no—
337C2×C32#53511yescharacteristic
338C2×C32C22×Q1611yescharacteristic
339C2×C32C2×#13711yescharacteristic
340C3×D3—22no—
341C2×Q8—21no—
342C2×Q8—32no—
343C2×Q8—32no—
344C2×Q8—92no—
345C2×Q8—182no—
346C2×Q8—364no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—182no—
351Q16—364no—
352Q16—364no—
353C2×C8—32no—
354C2×C8—92no—
355C4⋊C4—21no—
356C4⋊C4—62no—
357C4⋊C4—62no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—61no—
362C4⋊C4—182no—
363C4⋊C4—182no—
364C4⋊C4—181no—
365C4⋊C4—181no—
366C4⋊C4C2×D3212yes—
367C22⋊C4—62no—
368C42—31no—
369C42—31no—
370C42—31no—
371C42—62no—
372C42C2×D3211yescharacteristic
373C22×C3—21no—
374C22×C3—32no—
375C22×C3—61no—
376C22×C3—61no—
377C22×C3C2×D3×D411yescharacteristic
378C22×C3C22×#6711yescharacteristic
379C2×D3—21no—
380C2×D3—21no—
381C2×D3C2×#21612yes—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—32no—
386C3×C4—42no—
387C3×C4—42no—
388C3×C4—61no—
389C3×C4—61no—
390C3×C4—82no—
391C3×C4—122no—
392C3×C4—122no—
393C3×C4#91512yes—
394C3×C4D3×Q1612yes—
395C3×C4C2×#21612yes—
396C3×C4#95112yes—
397C3×C4#100912yes—
398C3×C4#100912yes—
399Dic3—21no—
400Dic3—21no—
401Dic3—42no—
402Dic3—42no—
403Dic3—122no—
404Dic3—122no—
405Dic3—242no—
406Dic3C2×#21612yes—
407C32#320911yescharacteristic, complemented, Sylow
408C23—31no—
409Q8—42no—
410Q8—64no—
411Q8—64no—
412Q8—184no—
413Q8—364no—
414C2×C4—22no—
415C2×C4—31no—
416C2×C4—34no—
417C2×C4—31no—
418C2×C4—31no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—182no—
424C2×C4—364no—
425C2×C4D3×#6711yescharacteristic
426C2×C4D3×#6711yescharacteristic
427C2×C4C22×D3211yescharacteristic
428C8—62no—
429C8—182no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—62no—
434C2×C3#567411yescharacteristic
435C2×C3C2×#91511yescharacteristic
436C2×C3C2×D3×Q1611yescharacteristic
437C2×C3C22×#21611yescharacteristic
438C2×C3C2×#95111yescharacteristic
439C2×C3#583011yescharacteristic
440D3—22no—
441C22—32no—
442C22—61no—
443C22—61no—
444C22C2×D3×#6711yescharacteristic
445C4—32no—
446C4—42no—
447C4—61no—
448C4—61no—
449C4—122no—
450C4—122no—
451C4—362no—
452C4#6392912yes—
453C4D3×#21612yes—
454C4#6432012yes—
455C3—21no—
456C3#8160411yescharacteristic, complemented
457C3#8337011yescharacteristic, complemented
458C2—62no—
459C2C2×#6392911yescharacteristic
460C2C2×D3×#21611yescharacteristic
461C2#1059542611yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radicalC22×C3×D3C2×D4
hypercenterC4⋊Q8D32
hypocenterC32#3209
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #10592792C2semidirect
4. C3×#5093C2semidirect
6. #10590561C2semidirect
7. #10590387C2semidirect
8. #10590386C2semidirect
45. #5093D3semidirect
158. C3×C4×D3Q16semidirect
324. C4×D3#216semidirect
407. C32#3209semidirect
456. C3#81604semidirect
457. C3#83370semidirect

Maximal subgroups

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

subgroupindex
C3×#57282
D3×#8692
#105927922
C3×#50932
#105906042
#105905612
#105903872
#105903862
#105902612
#833703
#816043