Finite Group Databasebeta

← D3×#1354 #10705486

Subgroups — D3×#1354 #10705486

Order 768.

652 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 91 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
1D3×#395C211yesmaximal, characteristic
2D3×#343C211yesmaximal, characteristic
3D3×#324C211yesmaximal, characteristic
4C3×#1354C211yesmaximal, characteristic, complemented
5#67648C211yesmaximal, characteristic, complemented
6#67494C211yesmaximal, characteristic, complemented
7#67457C211yesmaximal, characteristic, complemented
8C2×#1354—31nomaximal, Sylow
9C22×D3×Q8C411yescharacteristic
10C2×D3×M16—21no—
11C2×D3×C4⋊C4—21no—
12C2×D3×C4⋊C4C2211yescharacteristic
13C3×#395C2211yescharacteristic
14#5230C411yescharacteristic
15#4979—21no—
16#4979C2211yescharacteristic
17D3×C22⋊C8—21no—
18D3×#95—21no—
19D3×#95C411yescharacteristic
20#4656—21no—
21#4656C411yescharacteristic
22C3×#343C2211yescharacteristic
23C3×#324C2211yescharacteristic
24#4554—21no—
25#4530C2211yescharacteristic
26#4528C2211yescharacteristic
27#4469—21no—
28C2×#395—31no—
29C2×#343—31no—
30C2×#324—31no—
31#1354—31no—
32#1354—31no—
33#1354—31no—
34#1354D311yescharacteristic, direct product factor
35C22×C3×Q8C2×C411yescharacteristic
36C2×D3×Q8—44no—
37C2×D3×Q8—41no—
38C2×D3×Q8—41no—
39C22×C4×D3—21no—
40C22×C4×D3—21no—
41C22×C4×D3C2×C411yescharacteristic
42C22×C4×D3D411yescharacteristic
43C22×C4×D3D411yescharacteristic
44C22×Dic6—21no—
45C22×Dic6C2×C411yescharacteristic
46C2×C3×M16—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2311yescharacteristic
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4C2×C411yescharacteristic
54C2×Dic3⋊C4D411yescharacteristic
55C2×#208—21no—
56D3×M16—24no—
57D3×M16—42no—
58C2×ΓL(1,𝔽25)—41no—
59C2×D3×C8—41no—
60C2×D3×C8—41no—
61D3×C4⋊C4—41no—
62D3×C4⋊C4—41no—
63D3×C4⋊C4—44no—
64C3×C22⋊C8—21no—
65C3×#95—21no—
66C3×#95C2×C411yescharacteristic
67C3⋊#95—21no—
68C3⋊#95—21no—
69C3⋊#95—21no—
70C3⋊#95—21no—
71C3⋊#95C2×C411yescharacteristic
72#830—21no—
73(C2×D3)⋊C8—41no—
74C23×Q8—31no—
75C22×M16—61no—
76C22×C4⋊C4—31no—
77C22×C4⋊C4—61no—
78C2×C22⋊C8—61no—
79#395—31no—
80#395—31no—
81#395—31no—
82#395—61no—
83#395—61no—
84#395C2×D311yescharacteristic
85C2×#95—31no—
86C2×#95—61no—
87#343—31no—
88#343—61no—
89#343C2×D311yescharacteristic
90#324—31no—
91#324—61no—
92#324C2×D311yescharacteristic
93C23×D3C22⋊C411yescharacteristic
94C2×C3×Q8—42no—
95C2×C3×Q8—41no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—21no—
102C22×Dic3—21no—
103C22×Dic3C22⋊C411yescharacteristic
104C22×Dic3C22⋊C411yescharacteristic
105C22×Dic3C22⋊C411yescharacteristic
106D3×Q8—44no—
107D3×Q8—44no—
108D3×Q8—44no—
109D3×Q8—44no—
110C2×C4×D3—21no—
111C2×C4×D3—21no—
112C2×C4×D3—22no—
113C2×C4×D3—24no—
114C2×C4×D3—24no—
115C2×C4×D3—21no—
116C2×C4×D3—21no—
117C2×C4×D3—21no—
118C2×C4×D3—21no—
119C2×C4×D3—42no—
120C2×C4×D3—42no—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C4×D3—42no—
124C2×C4×D3—42no—
125C2×C4×D3—42no—
126C2×C4×D3SD1612yes—
127C2×C4×D3Q1612yes—
128C2×Dic6—42no—
129C2×Dic6—42no—
130C2×Dic6—42no—
131C2×Dic6—41no—
132C2×Dic6—42no—
133C3×M16—22no—
134C3×M16—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—42no—
139C4⋊Dic3—41no—
140C4⋊Dic3—42no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143Dic3⋊C4—42no—
144Dic3⋊C4—42no—
145#208—22no—
146#208—41no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149ΓL(1,𝔽25)—44no—
150D3×C8—44no—
151D3×C8—82no—
152C22×Q8—31no—
153C22×Q8—61no—
154C22×Q8—124no—
155C22×Q8—121no—
156C22×Q8—121no—
157C22×Q8C4×D311yescharacteristic
158C23×C4—31no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—61no—
162C23×C4—61no—
163C2×M16—21no—
164C2×M16—64no—
165C2×M16—61no—
166C2×M16—121no—
167C2×M16—122no—
168C22×C8—121no—
169C22×C8—121no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—121no—
178C2×C4⋊C4—121no—
179C2×C4⋊C4—124no—
180C2×C4⋊C4C22×D311yescharacteristic
181C22⋊C8—21no—
182C22⋊C8—61no—
183C22⋊C8—121no—
184#95—21no—
185#95—31no—
186#95—61no—
187#95—61no—
188#95—61no—
189#95—61no—
190#95C4×D311yescharacteristic
191C23×C3C2×C22⋊C411yescharacteristic
192C22×D3—22no—
193C22×D3—22no—
194C22×D3#9911yescharacteristic
195C22×D3#9911yescharacteristic
196C22×D3#10312yes—
197C22×D3#10412yes—
198C3×Q8—42no—
199C3×Q8—42no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4C2×SD1611yescharacteristic
213C2×C3×C4C2×Q1611yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—22no—
217C2×Dic3—22no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3#10312yes—
227C4×D3—22no—
228C4×D3—22no—
229C4×D3—22no—
230C4×D3—22no—
231C4×D3—48no—
232C4×D3—42no—
233C4×D3—44no—
234C4×D3—42no—
235C4×D3—44no—
236C4×D3—42no—
237C4×D3—81no—
238Dic6—48no—
239Dic6—42no—
240Dic6—42no—
241C3×C8—42no—
242C3×C8—81no—
243C3⋊C8—42no—
244C3⋊C8—81no—
245C24—31no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—122no—
249C2×Q8—121no—
250C2×Q8—122no—
251C2×Q8—122no—
252C2×Q8—122no—
253C2×Q8—124no—
254C2×Q8—124no—
255C2×Q8—124no—
256C2×Q8—124no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—31no—
260C22×C4—32no—
261C22×C4—32no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—62no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—64no—
271C22×C4—64no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—121no—
276C22×C4—121no—
277C22×C4—122no—
278C22×C4—122no—
279C22×C4—122no—
280C22×C4—122no—
281C22×C4—122no—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4D3×D411yescharacteristic
284C22×C4D3×D411yescharacteristic
285M16—22no—
286M16—41no—
287M16—62no—
288M16—124no—
289M16—121no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—124no—
293C2×C8—121no—
294C2×C8—121no—
295C2×C8—242no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C4⋊C4—121no—
301C4⋊C4—122no—
302C4⋊C4—122no—
303C4⋊C4—122no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3C2×#9911yescharacteristic
307C22×C3C2×#10311yescharacteristic
308C22×C3C2×#10411yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—24no—
312C2×D3—24no—
313C2×D3#32812yes—
314C2×D3#32812yes—
315C2×D3AΣL(1,𝔽16)12yes—
316C2×D3#35612yes—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—42no—
320C3×C4—41no—
321C3×C4—42no—
322C3×C4—41no—
323Dic3—22no—
324Dic3—42no—
325Dic3—41no—
326Dic3—42no—
327Dic3—41no—
328C23—31no—
329C23—31no—
330C23—32no—
331C23—32no—
332C23—62no—
333C23—62no—
334C23D3×C22⋊C411yescharacteristic
335Q8—42no—
336Q8—42no—
337Q8—122no—
338Q8—128no—
339Q8—122no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—32no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—62no—
360C2×C4—62no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—122no—
364C2×C4—124no—
365C2×C4—122no—
366C2×C4—124no—
367C2×C4—122no—
368C2×C4—128no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—241no—
376C2×C4D3×SD1611yescharacteristic
377C2×C4D3×Q1611yescharacteristic
378C8—42no—
379C8—81no—
380C8—122no—
381C8—241no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3C2×#32811yescharacteristic
385C2×C3C2×AΣL(1,𝔽16)11yescharacteristic
386C2×C3C2×#35611yescharacteristic
387D3—22no—
388D3#135414yesdirect product factor
389C22—21no—
390C22—21no—
391C22—32no—
392C22—32no—
393C22—32no—
394C22—32no—
395C22—64no—
396C22—64no—
397C22—61no—
398C22—61no—
399C22D3×#9911yescharacteristic
400C22D3×#10311yescharacteristic
401C22D3×#10411yescharacteristic
402C4—21no—
403C4—21no—
404C4—42no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—62no—
409C4—122no—
410C4—122no—
411C4—121no—
412C4—121no—
413C3C2×#135411yescharacteristic, complemented, Sylow
414C2—21no—
415C2—21no—
416C2—34no—
417C2—62no—
418C2D3×#32811yescharacteristic
419C2D3×AΣL(1,𝔽16)11yescharacteristic
420C2D3×#35611yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1354C2
socleC22×C3C2×#99
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1354D3
hypocenterC3C2×#1354
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1354

Complements

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

NH ≅ G/Ntype
4. C3×#1354C2semidirect
5. #67648C2semidirect
6. #67494C2semidirect
7. #67457C2semidirect
34. #1354D3direct (N × H)
388. D3#1354direct (N × H)
413. C3C2×#1354semidirect

Maximal subgroups

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

subgroupindex
D3×#3952
D3×#3432
D3×#3242
C3×#13542
#676482
#674942
#674572
C2×#13543