Finite Group Databasebeta

← #10824895

Subgroups — #10824895

Order 768.

808 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×AGL(1,ℤ/8)C212yesmaximal, complemented
2C2×#5152C211yesmaximal, characteristic, complemented
3#78649C211yesmaximal, characteristic, complemented
4C3×#2018C211yesmaximal, characteristic
5#72093C212yesmaximal, complemented
6#72042C211yesmaximal, characteristic, complemented
7#70075C212yesmaximal, complemented
8Dic3×#100C212yesmaximal, complemented
9#69995C211yesmaximal, characteristic, complemented
10#68924C211yesmaximal, characteristic, complemented
11#68917C211yesmaximal, characteristic, complemented
12#23719—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—
27#5152—22no—
28#5152—22no—
29Dic3×D8—22no—
30C2×#822C2212yes—
31#5121C2212yes—
32#5119C2211yescharacteristic
33Dic3×M16C2212yes—
34Dic3×M16C2211yescharacteristic
35C2×#817C2211yescharacteristic
36#5102C2211yescharacteristic
37C2×#815C2211yescharacteristic
38#4720—22no—
39#4719—22no—
40C3×#341C2211yescharacteristic
41#4554C2212yes—
42#4554C2212yes—
43#4552C2211yescharacteristic
44C4×AGL(1,ℤ/8)—32no—
45C2×#442—31no—
46#2923—31no—
47#2090—31no—
48#2079—31no—
49#2063—32no—
50#2051—32no—
51#2045—31no—
52#2018D311yescharacteristic
53#1778—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×Dic3D412yes—
69D4×Dic3—22no—
70D4×Dic3—22no—
71D4×Dic3—22no—
72D4×Dic3—22no—
73D4×Dic3—22no—
74D4×Dic3D412yes—
75#926—21no—
76#926C2311yescharacteristic
77C2×C4⋊Dic3C2312yes—
78C2×C4×Dic3—21no—
79C2×C4×Dic3C2311yescharacteristic
80C2×#208C2312yes—
81C2×#208C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C3×#100—24no—
84C3×C22⋊C8—22no—
85#835—22no—
86#833—24no—
87#832—22no—
88#832—22no—
89#832—22no—
90#832—24no—
91#830—22no—
92#822—24no—
93#818—22no—
94#817—22no—
95#817—22no—
96#815—22no—
97#815—21no—
98#815D412yes—
99#815D412yes—
100Dic3⋊C8—22no—
101Dic3⋊C8—22no—
102C8×Dic3—22no—
103C8×Dic3—21no—
104C2×AGL(1,ℤ/8)C2×D312yes—
105C22×D8C2×D311yescharacteristic
106C4×#41—31no—
107C2×C4×D4—32no—
108#442—62no—
109#442—62no—
110#442—62no—
111#440—62no—
112C4×SD16—62no—
113C4×D8—62no—
114#433—62no—
115#432—62no—
116C2×#108—32no—
117#428—32no—
118C2×#106—31no—
119#424—31no—
120#424—31no—
121#418—32no—
122#417—32no—
123C2×#102—32no—
124C2×#100—32no—
125C2×#100C2×D312yes—
126#408—31no—
127#408C2×D311yescharacteristic
128C4×M16—31no—
129C4×M16—32no—
130C2×#97—31no—
131#343—32no—
132#343—32no—
133#341—31no—
134#341C2×D311yescharacteristic
135C24×C3—22no—
136C3×#41—22no—
137C3×#41—22no—
138C2×C3×Q8C2×D411yescharacteristic
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4C2×D411yescharacteristic
146C2×C3×D4#4114yes—
147C22×C3×C4—21no—
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—42no—
152C22×Dic3—42no—
153C22×Dic3—84no—
154C22×Dic3#4112yes—
155C3×SD16—28no—
156C3×D8—28no—
157C3×D8—28no—
158C3×M16—22no—
159C3×M16—22no—
160C3×M16#4114yes—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—21no—
164C2×C3×C8C2×D412yes—
165#217—21no—
166#217—22no—
167#217—42no—
168#217—42no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3C2×D412yes—
173C4×Dic3—21no—
174C4×Dic3—22no—
175C4×Dic3C2×D412yes—
176C4×Dic3C2×D411yescharacteristic
177C4×Dic3C2×D411yescharacteristic
178#208—22no—
179#208—22no—
180#208—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8C2×D412yes—
185C2×#41C22×D311yescharacteristic
186C22×D4C22×D312yes—
187C23×C4—62no—
188AGL(1,ℤ/8)—24no—
189AGL(1,ℤ/8)—24no—
190C2×SD16—22no—
191C2×D8—22no—
192C2×D8—24no—
193C2×D8—22no—
194C2×M16—32no—
195C2×M16—31no—
196C2×M16C22×D311yescharacteristic
197C2×M16C22×D312yes—
198C22×C8—31no—
199C22×C8C22×D311yescharacteristic
200C4×Q8—32no—
201C4×D4—32no—
202C4×D4—62no—
203C4×D4—62no—
204C4×D4—62no—
205C4×D4—62no—
206C4×D4—62no—
207#117—31no—
208#117—61no—
209C2×C4⋊C4—32no—
210C2×C22⋊C4—62no—
211C2×C42—31no—
212C2×C42—61no—
213#108—64no—
214#107—62no—
215#106—62no—
216#106—62no—
217C4⋊C8—62no—
218C4⋊C8—62no—
219#103—62no—
220#102—62no—
221#102—62no—
222#102—64no—
223#102—62no—
224#100—24no—
225#100—64no—
226C22⋊C8—22no—
227C22⋊C8—62no—
228#97—32no—
229#97—32no—
230#97—61no—
231#97—62no—
232C4×C8—61no—
233C4×C8—62no—
234C23×C3—21no—
235C23×C3—24no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—82no—
239C23×C3—82no—
240C23×C3C2×#4111yescharacteristic
241C3×Q8—22no—
242C3×D4—24no—
243C3×D4—22no—
244C3×D4—24no—
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—42no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4C22×D411yescharacteristic
252C2×C3×C4C22×D411yescharacteristic
253C2×C3×C4C2×#4112yes—
254C2×C3×C4C2×#4112yes—
255C2×Dic3—22no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—82no—
259C2×Dic3—84no—
260C2×Dic3—84no—
261C2×Dic3—82no—
262C2×Dic3#12112yes—
263C2×Dic3#12112yes—
264C3×C8—22no—
265C3×C8—24no—
266C3×C8—41no—
267C3⋊C8—22no—
268C3⋊C8—41no—
269C3⋊C8—42no—
270C24—22no—
271#41—22no—
272#41—22no—
273C2×Q8D3×D411yescharacteristic
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4D3×D411yescharacteristic
281C2×D4#23714yes—
282C22×C4—21no—
283C22×C4—32no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—122no—
287C22×C4—122no—
288C22×C4—244no—
289C22×C4C23×D311yescharacteristic
290SD16—28no—
291D8—28no—
292D8—28no—
293M16—22no—
294M16—22no—
295M16—62no—
296M16—62no—
297M16—62no—
298M16#23714yes—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—32no—
303C2×C8—62no—
304C2×C8—61no—
305C2×C8—62no—
306C2×C8D3×D412yes—
307C4⋊C4—32no—
308C4⋊C4—62no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—61no—
313C22⋊C4—122no—
314C22⋊C4—122no—
315C42—31no—
316C42—32no—
317C42—31no—
318C42—62no—
319C42—61no—
320C22×C3—22no—
321C22×C3—41no—
322C22×C3—44no—
323C22×C3—42no—
324C22×C3—84no—
325C22×C3—84no—
326C22×C3—82no—
327C22×C3—82no—
328C22×C3C2×#12111yescharacteristic
329C22×C3#52912yes—
330C3×C4—41no—
331C3×C4#52511yescharacteristic
332C3×C4#52512yes—
333C3×C4#53211yescharacteristic
334Dic3—22no—
335Dic3—22no—
336Dic3—42no—
337Dic3—82no—
338C23—21no—
339C23—24no—
340C23—42no—
341C23—42no—
342C23—82no—
343C23—82no—
344C23D3×#4111yescharacteristic
345Q8—22no—
346D4—24no—
347D4—22no—
348D4—24no—
349D4—42no—
350D4—42no—
351D4—42no—
352C2×C4—21no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—62no—
358C2×C4—122no—
359C2×C4—122no—
360C2×C4—242no—
361C2×C4—244no—
362C2×C4—244no—
363C2×C4—242no—
364C2×C4C2×D3×D411yescharacteristic
365C2×C4C2×D3×D411yescharacteristic
366C2×C4C2×#23712yes—
367C2×C4C2×#23712yes—
368C8—22no—
369C8—24no—
370C8—41no—
371C8—62no—
372C8—121no—
373C8—122no—
374C2×C3—22no—
375C2×C3—81no—
376C2×C3—84no—
377C2×C3#239211yescharacteristic
378C2×C3#306812yes—
379C22—22no—
380C22—42no—
381C22—41no—
382C22—44no—
383C22—84no—
384C22—84no—
385C22—82no—
386C22—82no—
387C22#560012yes—
388C22D3×#12111yescharacteristic
389C4—41no—
390C4—62no—
391C4—62no—
392C4—122no—
393C4—242no—
394C4#558611yescharacteristic
395C4#558612yes—
396C4#571611yescharacteristic
397C3#2371911yescharacteristic, complemented, Sylow
398C2—22no—
399C2—81no—
400C2—84no—
401C2#7666611yescharacteristic
402C2#8093812yes—
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#2018C2
socleC22×C3C2×#121
hypercenter#2018D3
hypocenterC3#23719
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×#5152C2semidirect
3. #78649C2semidirect
5. #72093C2semidirect
6. #72042C2semidirect
7. #70075C2semidirect
8. Dic3×#100C2semidirect
9. #69995C2semidirect
10. #68924C2semidirect
11. #68917C2semidirect
397. C3#23719semidirect

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×#51522
#786492
C3×#20182
#720932
#720422
#700752
Dic3×#1002
#699952
#689242
#689172
#237193