Finite Group Databasebeta

← #10950939

Subgroups — #10950939

Order 768.

590 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#467C211yesmaximal, characteristic
2C2×#4887C211yesmaximal, characteristic
3C3×#2236C211yesmaximal, characteristic, complemented
4#71087C211yesmaximal, characteristic, complemented
5#70603C211yesmaximal, characteristic, complemented
6#69331C212yesmaximal, complemented
7#69328C212yesmaximal, complemented
8#69227C212yesmaximal, complemented
9#69225C212yesmaximal, complemented
10#69113C211yesmaximal, characteristic, complemented
11D3×C22⋊C16C211yesmaximal, characteristic
12#33755—31nomaximal, Sylow
13C2×D3×Q16C2211yescharacteristic
14C22×Dic12C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16D3×C22⋊Q8—21no—
17C2×#897—21no—
18C3×#467C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20#5188C2211yescharacteristic
21C2×#829C2212yescomplemented
22#5116C2211yescharacteristic
23#5051—21no—
24C2×#810—21no—
25C2×Dic24C2212yescomplemented
26C2×D3×C16C2212yes—
27#4887—21no—
28#4887—21no—
29#4887C2212yes—
30#4887C2212yes—
31#4887C2211yescharacteristic
32D3×#107C2211yescharacteristic
33#4813—21no—
34D3×#103—21no—
35C3×#366C2212yescomplemented
36C3×#358C2212yescomplemented
37C3×C22⋊C16C2211yescharacteristic
38#4549C2211yescharacteristic
39(C2×D3)⋊C16C2212yes—
40#4503C2212yescomplemented
41#4492C2212yescomplemented
42#4492C2212yescomplemented
43#4489C2212yescomplemented
44C2×#467—31no—
45C2×#467—31no—
46#2236—31no—
47#2236—32no—
48#2236D311yescharacteristic, complemented
49#2216—32no—
50#2216—32no—
51#2188—31no—
52#2188—32no—
53#2188—31no—
54C2×C22⋊C16—31no—
55C2×D3×Q8—21no—
56C22×C4×D3D411yescharacteristic
57C22×Dic6—21no—
58C2×C3×Q16C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C3×C22⋊Q8—21no—
61C2×C3×C4⋊C4—21no—
62#946—21no—
63C2×#216—21no—
64C2×(C2×D3)⋊C4—41no—
65C2×C4⋊Dic3—41no—
66#924—21no—
67C22×C3⋊C8D411yescharacteristic
68D3×Q16—42no—
69C2×Dic12—21no—
70C2×Dic12—42no—
71C2×Dic12C2312yes—
72C2×Dic12C2311yescharacteristic
73C2×D3×C8—42no—
74C2×D3×C8D411yescharacteristic
75C2×D3×C8D411yescharacteristic
76C2×D3×C8C2311yescharacteristic
77C2×D3×C8C2311yescharacteristic
78#897—22no—
79#897—22no—
80#897—21no—
81#897—41no—
82#897—41no—
83#896—41no—
84D3×C4⋊C4—21no—
85D3×C4⋊C4—41no—
86D3×C22⋊C4—41no—
87#878—41no—
88C2×C3×C16C2312yes—
89C3×#107—21no—
90C3×#107C2311yescharacteristic
91C3×#107C2312yes—
92C3×#103—21no—
93#835—21no—
94#829—42no—
95#818C2311yescharacteristic
96#816—21no—
97C2×C3⋊C16C2312yes—
98#810—22no—
99#810—22no—
100#810—21no—
101#807—21no—
102Dic24—42no—
103D3×C16—42no—
104C22×Q16—31no—
105C22×Q16—31no—
106C23×C8—31no—
107C2×C22⋊Q8—61no—
108C2×C22⋊Q8—61no—
109C2×Q32—32no—
110C2×Q32—32no—
111C22×C16—32no—
112#467—31no—
113#467—31no—
114#467—32no—
115#467—31no—
116#467—32no—
117#467—61no—
118#467—61no—
119#467—61no—
120#467—61no—
121#467C2×D311yescharacteristic
122C2×#107—31no—
123C2×#107C2×D311yescharacteristic
124C2×#103—61no—
125C2×#103—61no—
126#366—32no—
127#366C2×D312yescomplemented
128#358—32no—
129#358—32no—
130#358—32no—
131#358C2×D312yescomplemented
132C22⋊C16—31no—
133C22⋊C16—32no—
134C22⋊C16C2×D311yescharacteristic
135C23×D3D811yescharacteristic
136C2×C3×Q8—21no—
137C22×C3×C4—41no—
138C22×C3×C4C2×D411yescharacteristic
139C22×Dic3—41no—
140C22×Dic3D811yescharacteristic
141D3×Q8—42no—
142D3×Q8—42no—
143C2×C4×D3—41no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×C4×D3D811yescharacteristic
148C2×C4×D3D811yescharacteristic
149C2×C4×D3C2×D411yescharacteristic
150C2×C4×D3C2×D411yescharacteristic
151C2×Dic6—22no—
152C2×Dic6—21no—
153C2×Dic6—42no—
154C2×Dic6—42no—
155C2×Dic6—41no—
156C2×Dic6—41no—
157C3×Q16—22no—
158C2×C3×C8—22no—
159C2×C3×C8C2×D411yescharacteristic
160C2×C3×C8C2411yescharacteristic
161C3×C4⋊C4—21no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—41no—
165C3×C22⋊C4—41no—
166#217—41no—
167#216—42no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—42no—
171(C2×D3)⋊C4—42no—
172C4⋊Dic3—21no—
173C4⋊Dic3—41no—
174C4⋊Dic3—42no—
175C4⋊Dic3—42no—
176Dic3⋊C4—41no—
177Dic3⋊C4—41no—
178Dic3⋊C4—41no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8C2×D411yescharacteristic
181C2×C3⋊C8C2×D411yescharacteristic
182Dic12—22no—
183Dic12—22no—
184Dic12—22no—
185Dic12—42no—
186D3×C8—22no—
187D3×C8—22no—
188D3×C8—41no—
189D3×C8—41no—
190C3×C16—22no—
191C3⋊C16—22no—
192C22×Q8—61no—
193C22×Q8—61no—
194C23×C4—31no—
195C2×Q16—31no—
196C2×Q16—32no—
197C2×Q16—61no—
198C2×Q16—61no—
199C2×Q16—122no—
200C2×Q16—122no—
201C2×Q16C22×D311yescharacteristic
202C22×C8—31no—
203C22×C8—31no—
204C22×C8—31no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8—122no—
208C22×C8C22×D311yescharacteristic
209C22⋊Q8—21no—
210C22⋊Q8—61no—
211C22⋊Q8—62no—
212C22⋊Q8—62no—
213C22⋊Q8—61no—
214C22⋊Q8—61no—
215C22⋊Q8—121no—
216C22⋊Q8—121no—
217C22⋊Q8—121no—
218C22⋊Q8—121no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—121no—
222C2×C4⋊C4—121no—
223C2×C22⋊C4—121no—
224C2×C22⋊C4—121no—
225Q32—122no—
226Q32—122no—
227C2×C16—32no—
228C2×C16—122no—
229C2×C16C22×D312yes—
230#107—21no—
231#107—31no—
232#107—61no—
233#107C22×D311yescharacteristic
234#107C22×D312yes—
235#103—21no—
236#103—61no—
237#103—62no—
238#103—62no—
239#103—61no—
240#103—61no—
241C23×C3C2×D811yescharacteristic
242C22×D3—41no—
243C22×D3—41no—
244C22×D3Q3214yescomplemented
245C22×D3C2×D811yescharacteristic
246C22×D3C2×D811yescharacteristic
247C3×Q8—41no—
248C3×Q8—41no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—82no—
255C2×C3×C4C2×D811yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×Dic3—41no—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—82no—
262C2×Dic3C2×D811yescharacteristic
263C2×Dic3C2×D811yescharacteristic
264C4×D3—21no—
265C4×D3—21no—
266C4×D3—21no—
267C4×D3—21no—
268C4×D3—41no—
269C4×D3—41no—
270C4×D3—82no—
271C4×D3—82no—
272Dic6—42no—
273Dic6—41no—
274Dic6—42no—
275Dic6—41no—
276Dic6—82no—
277Dic6—82no—
278C3×C8—21no—
279C3×C8H(5,𝔽2)12yes—
280C3⋊C8—21no—
281C3⋊C8—21no—
282C24—31no—
283C2×Q8—21no—
284C2×Q8—62no—
285C2×Q8—61no—
286C2×Q8—122no—
287C2×Q8—122no—
288C2×Q8—121no—
289C2×Q8—122no—
290C2×Q8—122no—
291C2×Q8—121no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—41no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4—121no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4D3×D411yescharacteristic
304Q16—22no—
305Q16—62no—
306Q16—62no—
307Q16—62no—
308Q16—122no—
309Q16—122no—
310C2×C8—22no—
311C2×C8—31no—
312C2×C8—31no—
313C2×C8—62no—
314C2×C8—62no—
315C2×C8—121no—
316C2×C8—121no—
317C2×C8—121no—
318C2×C8D3×D411yescharacteristic
319C2×C8C23×D311yescharacteristic
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—61no—
325C4⋊C4—121no—
326C4⋊C4—121no—
327C4⋊C4—121no—
328C4⋊C4—122no—
329C4⋊C4—122no—
330C4⋊C4—121no—
331C22⋊C4—41no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4—122no—
335C22⋊C4—122no—
336C22⋊C4—121no—
337C16—22no—
338C16—62no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3C2×Q3212yes—
342C22×C3C22×D811yescharacteristic
343C2×D3—21no—
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—21no—
347C2×D3—44no—
348C2×D3C2×Q3212yes—
349C2×D3C2×Q3212yes—
350C3×C4—21no—
351C3×C4—81no—
352C3×C4—82no—
353C3×C4—81no—
354C3×C4#53411yescharacteristic
355C3×C4#57511yescharacteristic
356Dic3—21no—
357Dic3—21no—
358Dic3—81no—
359Dic3—81no—
360Dic3—82no—
361C23—31no—
362C23—31no—
363C23—34no—
364C23—121no—
365C23—121no—
366C23D3×D811yescharacteristic
367Q8—41no—
368Q8—41no—
369Q8—122no—
370Q8—121no—
371Q8—122no—
372Q8—121no—
373Q8—242no—
374Q8—242no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—82no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—122no—
392C2×C4—121no—
393C2×C4—242no—
394C2×C4—242no—
395C2×C4—242no—
396C2×C4D3×D811yescharacteristic
397C2×C4C2×D3×D411yescharacteristic
398C8—21no—
399C8—61no—
400C8—61no—
401C8#100912yes—
402C2×C3—22no—
403C2×C3#318611yescharacteristic
404C2×C3C22×Q3211yescharacteristic
405C2×C3#341411yescharacteristic
406D3—22no—
407D3—22no—
408C22—21no—
409C22—21no—
410C22—32no—
411C22—32no—
412C22—61no—
413C22—61no—
414C22—61no—
415C22—61no—
416C22—124no—
417C22D3×Q3212yes—
418C22C2×D3×D811yescharacteristic
419C4—21no—
420C4—61no—
421C4—61no—
422C4—82no—
423C4—81no—
424C4—81no—
425C4—241no—
426C4—241no—
427C4—242no—
428C4#565311yescharacteristic
429C4#577711yescharacteristic
430C3#3375511yescharacteristic, complemented, Sylow
431C2—22no—
432C2—62no—
433C2—62no—
434C2#8129911yescharacteristic
435C2C2×D3×Q3211yescharacteristic
436C2#8203511yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2236C2
socleC22×C3C22×D8
prime radicalC23×D3D8
hypercenter#2236D3
hypocenterC3#33755
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2236

Complements

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

NH ≅ G/Ntype
3. C3×#2236C2semidirect
4. #71087C2semidirect
5. #70603C2semidirect
6. #69331C2semidirect
7. #69328C2semidirect
8. #69227C2semidirect
9. #69225C2semidirect
10. #69113C2semidirect
21. C2×#829C22semidirect
25. C2×Dic24C22semidirect
35. C3×#366C22semidirect
36. C3×#358C22semidirect
40. #4503C22semidirect
41. #4492C22semidirect
42. #4492C22semidirect
43. #4489C22semidirect
48. #2236D3semidirect
127. #366C2×D3semidirect
131. #358C2×D3semidirect
244. C22×D3Q32semidirect
430. C3#33755semidirect

Maximal subgroups

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

subgroupindex
D3×#4672
C2×#48872
C3×#22362
#710872
#706032
#693312
#693282
#692272
#692252
#691132
D3×C22⋊C162
#337553