Finite Group Databasebeta

← #10824858

Subgroups — #10824858

Order 768.

816 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×AGL(1,ℤ/8)C212yesmaximal, complemented
2C2×Dic3×D8C211yesmaximal, characteristic, complemented
3#78648C211yesmaximal, characteristic, complemented
4C3×#2018C211yesmaximal, characteristic
5#72093C212yesmaximal, complemented
6#72042C211yesmaximal, characteristic, complemented
7#70075C212yesmaximal, complemented
8Dic3×#100C212yesmaximal, complemented
9#69994C211yesmaximal, characteristic, complemented
10#68923C211yesmaximal, characteristic, complemented
11#68916C211yesmaximal, characteristic, complemented
12#23716—31nomaximal, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C22×C3×D8C2211yescharacteristic
15Dic3×#41C2211yescharacteristic
16C2×D4×Dic3C2212yes—
17C2×C3×#100C2212yes—
18C3×#408C2211yescharacteristic
19#5234C2211yescharacteristic
20#5232C2212yes—
21C2×#833C2212yes—
22#5215C2212yes—
23C2×#832C2212yes—
24#5164—22no—
25Dic3×SD16—22no—
26#5152—22no—
27Dic3×D8—22no—
28Dic3×D8—22no—
29Dic3×D8—22no—
30C2×#822C2212yes—
31#5121C2212yes—
32#5119C2211yescharacteristic
33Dic3×M16C2212yes—
34Dic3×M16C2211yescharacteristic
35C2×#818C2211yescharacteristic
36#5102C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4720—22no—
39#4718—22no—
40C3×#341C2211yescharacteristic
41#4554C2212yes—
42#4554C2212yes—
43#4552C2211yescharacteristic
44C4×AGL(1,ℤ/8)—32no—
45C2×C4×D8—31no—
46#2922—31no—
47#2089—31no—
48#2078—31no—
49#2063—32no—
50#2051—32no—
51#2045—31no—
52#2018D311yescharacteristic
53C4×#108—31no—
54C4×#100—32no—
55C2×C3×#41C2311yescharacteristic
56C22×C3×D4C2312yes—
57C23×Dic3—22no—
58C3×AGL(1,ℤ/8)—24no—
59C3×AGL(1,ℤ/8)—24no—
60C2×C3×SD16—22no—
61C2×C3×D8—22no—
62C2×C3×D8—24no—
63C2×C3×D8—22no—
64C2×C3×M16C2311yescharacteristic
65C2×C3×M16C2312yes—
66C22×C3×C8C2311yescharacteristic
67C2×#217—22no—
68Q8×Dic3D411yescharacteristic
69Q8×Dic3D411yescharacteristic
70D4×Dic3—21no—
71D4×Dic3—21no—
72D4×Dic3—22no—
73D4×Dic3—22no—
74D4×Dic3—22no—
75D4×Dic3—22no—
76D4×Dic3D411yescharacteristic
77D4×Dic3D411yescharacteristic
78#926—21no—
79#926C2311yescharacteristic
80C2×C4⋊Dic3C2312yes—
81C2×C4×Dic3—21no—
82C2×C4×Dic3C2311yescharacteristic
83C2×#208C2312yes—
84C2×#208C2311yescharacteristic
85C22×C3⋊C8C2311yescharacteristic
86C3×#100—24no—
87C3×C22⋊C8—22no—
88#835—22no—
89#833—24no—
90#832—22no—
91#832—22no—
92#832—22no—
93#832—24no—
94#830—22no—
95#822—24no—
96#818—22no—
97#818—22no—
98#817—22no—
99#815—22no—
100#815—21no—
101Dic3⋊C8—22no—
102Dic3⋊C8—22no—
103C8×Dic3—22no—
104C8×Dic3—21no—
105C8×Dic3D412yes—
106C8×Dic3D412yes—
107C2×AGL(1,ℤ/8)C2×D312yes—
108C22×D8C2×D311yescharacteristic
109C4×#41—31no—
110C2×C4×D4—32no—
111#442—62no—
112#440—62no—
113C4×SD16—62no—
114C4×D8—62no—
115C4×D8—62no—
116C4×D8—62no—
117#433—62no—
118#431—62no—
119C2×#108—32no—
120#428—32no—
121C2×#107—31no—
122#424—31no—
123#424—31no—
124#418—32no—
125#417—32no—
126C2×#102—32no—
127C2×#100—32no—
128C2×#100C2×D312yes—
129#408—31no—
130#408C2×D311yescharacteristic
131C4×M16—31no—
132C4×M16—32no—
133C2×C4×C8—31no—
134#343—32no—
135#343—32no—
136#341—31no—
137#341C2×D311yescharacteristic
138C24×C3—22no—
139C3×#41—22no—
140C3×#41—22no—
141C2×C3×Q8C2×D411yescharacteristic
142C2×C3×D4—21no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4C2×D411yescharacteristic
149C2×C3×D4#4114yes—
150C22×C3×C4—21no—
151C22×C3×C4C2411yescharacteristic
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—22no—
155C22×Dic3—42no—
156C22×Dic3—42no—
157C22×Dic3—84no—
158C22×Dic3#4111yescharacteristic
159C22×Dic3#4111yescharacteristic
160C3×SD16—28no—
161C3×D8—28no—
162C3×D8—28no—
163C3×M16—22no—
164C3×M16—22no—
165C3×M16#4114yes—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—21no—
169C2×C3×C8C2×D412yes—
170#217—21no—
171#217—21no—
172#217—21no—
173#217—42no—
174#217—42no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4⋊Dic3—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3C2×D411yescharacteristic
180C4⋊Dic3C2×D411yescharacteristic
181C4×Dic3—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3C2×D411yescharacteristic
185C4×Dic3C2×D411yescharacteristic
186C4×Dic3C2×D411yescharacteristic
187C4×Dic3C2×D411yescharacteristic
188#208—22no—
189#208—22no—
190#208—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8C2×D412yes—
195C2×#41C22×D311yescharacteristic
196C22×D4C22×D312yes—
197C23×C4—62no—
198AGL(1,ℤ/8)—24no—
199AGL(1,ℤ/8)—24no—
200C2×SD16—22no—
201C2×D8—22no—
202C2×D8—24no—
203C2×D8—22no—
204C2×M16—32no—
205C2×M16—31no—
206C2×M16C22×D311yescharacteristic
207C2×M16C22×D312yes—
208C22×C8—31no—
209C22×C8C22×D311yescharacteristic
210C4×Q8—31no—
211C4×Q8—31no—
212C4×D4—31no—
213C4×D4—31no—
214C4×D4—62no—
215C4×D4—62no—
216C4×D4—62no—
217C4×D4—62no—
218C4×D4—61no—
219C4×D4—61no—
220#117—31no—
221#117—61no—
222C2×C4⋊C4—32no—
223C2×C22⋊C4—62no—
224C2×C42—31no—
225C2×C42—61no—
226#108—64no—
227#107—62no—
228#107—62no—
229#106—62no—
230C4⋊C8—62no—
231C4⋊C8—62no—
232#103—62no—
233#102—62no—
234#102—62no—
235#102—64no—
236#102—62no—
237#100—24no—
238#100—64no—
239C22⋊C8—22no—
240C22⋊C8—62no—
241#97—61no—
242#97—62no—
243C4×C8—32no—
244C4×C8—32no—
245C4×C8—61no—
246C4×C8—62no—
247C23×C3—21no—
248C23×C3—24no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—82no—
252C23×C3—82no—
253C23×C3C2×#4111yescharacteristic
254C3×Q8—22no—
255C3×D4—24no—
256C3×D4—22no—
257C3×D4—24no—
258C3×D4—42no—
259C3×D4—42no—
260C3×D4—42no—
261C2×C3×C4—21no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C22×D411yescharacteristic
266C2×C3×C4C2×#4112yes—
267C2×C3×C4C2×#4112yes—
268C2×Dic3—22no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—41no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—84no—
276C2×Dic3—84no—
277C2×Dic3—82no—
278C2×Dic3#12111yescharacteristic
279C2×Dic3#12111yescharacteristic
280C2×Dic3#12111yescharacteristic
281C2×Dic3#12111yescharacteristic
282C3×C8—22no—
283C3×C8—24no—
284C3×C8—41no—
285C3⋊C8—22no—
286C3⋊C8—41no—
287C3⋊C8—42no—
288C24—22no—
289#41—22no—
290#41—22no—
291C2×Q8D3×D411yescharacteristic
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4D3×D411yescharacteristic
299C2×D4#23714yes—
300C22×C4—21no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—62no—
306C22×C4—122no—
307C22×C4—122no—
308C22×C4—244no—
309C22×C4C23×D311yescharacteristic
310SD16—28no—
311D8—28no—
312D8—28no—
313M16—22no—
314M16—22no—
315M16—62no—
316M16—62no—
317M16—62no—
318M16#23714yes—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—32no—
323C2×C8—62no—
324C2×C8—61no—
325C2×C8—62no—
326C2×C8D3×D412yes—
327C4⋊C4—31no—
328C4⋊C4—31no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—122no—
337C22⋊C4—122no—
338C42—31no—
339C42—31no—
340C42—31no—
341C42—31no—
342C42—61no—
343C42—61no—
344C42—61no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—41no—
348C22×C3—44no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—84no—
352C22×C3—84no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3C2×#12111yescharacteristic
356C22×C3#52912yes—
357C3×C4—41no—
358C3×C4#52511yescharacteristic
359C3×C4#52512yes—
360C3×C4#53211yescharacteristic
361Dic3—21no—
362Dic3—21no—
363Dic3—21no—
364Dic3—41no—
365Dic3—41no—
366Dic3—82no—
367Dic3#47212yes—
368C23—21no—
369C23—24no—
370C23—42no—
371C23—42no—
372C23—82no—
373C23—82no—
374C23D3×#4111yescharacteristic
375Q8—22no—
376D4—24no—
377D4—22no—
378D4—24no—
379D4—42no—
380D4—42no—
381D4—42no—
382C2×C4—21no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—62no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—244no—
397C2×C4—244no—
398C2×C4—242no—
399C2×C4C2×D3×D411yescharacteristic
400C2×C4C2×D3×D411yescharacteristic
401C2×C4C2×#23712yes—
402C2×C4C2×#23712yes—
403C8—22no—
404C8—24no—
405C8—41no—
406C8—62no—
407C8—121no—
408C8—122no—
409C2×C3—22no—
410C2×C3—81no—
411C2×C3—84no—
412C2×C3#239211yescharacteristic
413C2×C3C2×#47211yescharacteristic
414C2×C3#306711yescharacteristic
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—44no—
421C22—84no—
422C22—84no—
423C22—82no—
424C22—82no—
425C22#560012yes—
426C22D3×#12111yescharacteristic
427C4—32no—
428C4—41no—
429C4—61no—
430C4—61no—
431C4—61no—
432C4—121no—
433C4—121no—
434C4—242no—
435C4#558611yescharacteristic
436C4#558612yes—
437C4#571611yescharacteristic
438C3#2371611yescharacteristic, complemented, Sylow
439C2—22no—
440C2—81no—
441C2—84no—
442C2#7666611yescharacteristic
443C2D3×#47211yescharacteristic
444C2#8093711yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2018C2
socleC22×C3C2×#121
hypercenter#2018D3
hypocenterC3#23716
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2018

Complements

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

NH ≅ G/Ntype
1. Dic3×AGL(1,ℤ/8)C2semidirect
2. C2×Dic3×D8C2semidirect
3. #78648C2semidirect
5. #72093C2semidirect
6. #72042C2semidirect
7. #70075C2semidirect
8. Dic3×#100C2semidirect
9. #69994C2semidirect
10. #68923C2semidirect
11. #68916C2semidirect
438. C3#23716semidirect

Maximal subgroups

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

subgroupindex
Dic3×AGL(1,ℤ/8)2
C2×Dic3×D82
#786482
C3×#20182
#720932
#720422
#700752
Dic3×#1002
#699942
#689232
#689162
#237163