Finite Group Databasebeta

← #10690942

Subgroups — #10690942

Order 768.

946 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#380C211yesmaximal, characteristic
2#84031C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic
4C2×#4452C212yesmaximal, complemented
5#67612C212yesmaximal, complemented
6#9151—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C4⋊C4—21no—
9C22×C3×C22⋊C4—21no—
10C2×#924C412yescomplemented
11C2×#924C2211yescharacteristic
12C3×#380—24no—
13C2×C3×#95C2212yes—
14#5247—21no—
15#5212C412yescomplemented
16#5212C2211yescharacteristic
17#5210—21no—
18#5210—22no—
19C2×#830—22no—
20C2×#830C2212yes—
21#4452—24no—
22#2848—31no—
23C2×#380D311yescharacteristic
24#1779—31no—
25C2×#328—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—22no—
30C23×C3×C4—22no—
31C23×C3×C4—44no—
32C23×C3×C4C2×C412yes—
33C23×C3×C4D412yes—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C22×Dic6C2×C411yescharacteristic
37C2×C3×C4⋊C4—24no—
38C2×C3×C4⋊C4—44no—
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4—42no—
41C2×C3×C22⋊C4—42no—
42C2×#217—22no—
43C2×#217—22no—
44C2×#217—21no—
45C2×#217—21no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×C4⋊Dic3C2×C412yes—
48#924—22no—
49#924—22no—
50#924—44no—
51C2×Dic3⋊C4—22no—
52C2×Dic3⋊C4—21no—
53C22×C3⋊C8—22no—
54C3×#95—22no—
55C3×#95D414yes—
56C3⋊#95—22no—
57C3⋊#95—21no—
58#830—22no—
59#830—22no—
60#830—44no—
61C24×C4C2×D311yescharacteristic
62C2×C22⋊Q8—31no—
63C2×C22⋊Q8—32no—
64C22×C4⋊C4—21no—
65C22×C22⋊C4—21no—
66C2×C22⋊C8—32no—
67C2×C22⋊C8—62no—
68#385—62no—
69#385—61no—
70#380—24no—
71#380—32no—
72#380—31no—
73#379—61no—
74C2×#95C2×D312yes—
75#328—64no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—21no—
80C24×C3—42no—
81C24×C3—42no—
82C24×C3C22⋊C412yes—
83C24×C3SD1612yes—
84C24×C3Q1612yes—
85C24×C3C2×D411yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—24no—
93C22×C3×C4—24no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4—48no—
98C22×C3×C4—48no—
99C22×C3×C4—44no—
100C22×C3×C4—44no—
101C22×C3×C4—42no—
102C22×C3×C4—42no—
103C22×C3×C4C22⋊C412yes—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D412yes—
107C22×C3×C4#4112yes—
108C22×Dic3—21no—
109C22×Dic3—22no—
110C22×Dic3—21no—
111C2×Dic6—22no—
112C2×Dic6—42no—
113C3×C4⋊C4—48no—
114C3×C22⋊C4—48no—
115#217—44no—
116#217—44no—
117#217—41no—
118#217—41no—
119#217—41no—
120#217—41no—
121C4⋊Dic3—22no—
122C4⋊Dic3—42no—
123Dic3⋊C4—44no—
124Dic3⋊C4—42no—
125C2×C3⋊C8—44no—
126C2×C3⋊C8—42no—
127C25#6711yescharacteristic
128C22×Q8—31no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—44no—
133C23×C4C4×D312yes—
134C23×C4D1212yes—
135C23×C4#6711yescharacteristic
136C23×C4C22×D311yescharacteristic
137C22×C8—62no—
138C22⋊Q8—62no—
139C22⋊Q8—62no—
140C22⋊Q8—124no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—32no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—62no—
146C2×C4⋊C4—61no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—62no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C22⋊C8—62no—
155C22⋊C8—62no—
156C22⋊C8—124no—
157#95—22no—
158#95—62no—
159#95—61no—
160#95#6714yes—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—22no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3—44no—
174C23×C3—44no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—42no—
179C23×C3—44no—
180C23×C3—44no—
181C23×C3#10314yes—
182C23×C3C2×C22⋊C411yescharacteristic
183C23×C3#12011yescharacteristic
184C23×C3#12311yescharacteristic
185C23×C3C2×SD1611yescharacteristic
186C23×C3C2×Q1611yescharacteristic
187C23×C3AGL(1,ℤ/8)11yescharacteristic
188C23×C3#13711yescharacteristic
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—22no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—48no—
198C2×C3×C4—48no—
199C2×C3×C4—44no—
200C2×C3×C4—44no—
201C2×C3×C4—44no—
202C2×C3×C4—84no—
203C2×C3×C4—84no—
204C2×C3×C4#9914yes—
205C2×C3×C4C4×D412yes—
206C2×C3×C4#12112yes—
207C2×Dic3—41no—
208C2×Dic3—41no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214Dic6—82no—
215C3⋊C8—82no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—42no—
221C24—42no—
222C24(C2×D3)⋊C412yes—
223C24#21412yes—
224C24#21612yes—
225C24C2×#6711yescharacteristic
226C2×Q8—62no—
227C2×Q8—122no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—48no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—62no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4(C2×D3)⋊C412yes—
249C22×C4C2×C4×D311yescharacteristic
250C22×C4C2×D1211yescharacteristic
251C22×C4#23512yes—
252C22×C4C2×#6712yes—
253C2×C8—124no—
254C2×C8—122no—
255C4⋊C4—48no—
256C4⋊C4—62no—
257C4⋊C4—122no—
258C4⋊C4—124no—
259C4⋊C4—122no—
260C22⋊C4—48no—
261C22⋊C4—124no—
262C22⋊C4—124no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—24no—
276C22×C3—24no—
277C22×C3—24no—
278C22×C3—44no—
279C22×C3—44no—
280C22×C3—44no—
281C22×C3—44no—
282C22×C3—42no—
283C22×C3—42no—
284C22×C3—42no—
285C22×C3—42no—
286C22×C3#38611yescharacteristic
287C22×C3C2×#9912yes—
288C22×C3C2×#10311yescharacteristic
289C22×C3#41711yescharacteristic
290C22×C3#42112yes—
291C22×C3#45011yescharacteristic
292C22×C3#45111yescharacteristic
293C22×C3#48311yescharacteristic
294C22×C3#48411yescharacteristic
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—24no—
298C3×C4—84no—
299Dic3—81no—
300Dic3—81no—
301Dic3—82no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—44no—
315C23—44no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—44no—
320C23—42no—
321C23—42no—
322C23#81014yes—
323C23C2×(C2×D3)⋊C411yescharacteristic
324C23#92911yescharacteristic
325C23#93211yescharacteristic
326C23C2×#21411yescharacteristic
327C23#94211yescharacteristic
328C23C2×#21611yescharacteristic
329C23#95311yescharacteristic
330Q8—242no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—122no—
347C2×C4—122no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4#80614yes—
354C2×C4C4×#6712yes—
355C2×C4#93012yes—
356C8—242no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—44no—
362C2×C3#151312yes—
363C2×C3#152012yes—
364C2×C3#187211yescharacteristic
365C2×C3#189711yescharacteristic
366C2×C3#189811yescharacteristic
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—44no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22C2×#80612yes—
387C22#496611yescharacteristic
388C22C2×#81011yescharacteristic
389C22#497011yescharacteristic
390C22#497111yescharacteristic
391C22#500512yes—
392C22#521311yescharacteristic
393C22#521811yescharacteristic
394C22#522811yescharacteristic
395C4—22no—
396C4—24no—
397C4—22no—
398C4—84no—
399C4—241no—
400C4—242no—
401C4—241no—
402C3#915111yescharacteristic, complemented, Sylow
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—44no—
408C2#6950812yes—
409C2#6951012yes—
410C2#7154311yescharacteristic
411C2#7155911yescharacteristic
412C2#7159111yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#380C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenterC2×#380D3
hypocenterC3#9151
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. C2×#4452C2semidirect
5. #67612C2semidirect
10. C2×#924C4semidirect
15. #5212C4semidirect
402. C3#9151semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3802
#840312
#721262
C2×#44522
#676122
#91513