Finite Group Databasebeta

← #10691250

Subgroups — #10691250

Order 768.

762 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C3×#2315C211yesmaximal, characteristic, complemented
2#82130C211yesmaximal, characteristic
3C2×#4525C211yesmaximal, characteristic, complemented
4#69509C212yesmaximal, complemented
5#67625C212yesmaximal, complemented
6#9203—31nomaximal, Sylow
7C22×C3×C22⋊C4—21no—
8C22×C3×C22⋊C4C2211yescharacteristic
9C2×#953—21no—
10C2×#930C2211yescharacteristic
11C3×#385—24no—
12C3×#380—21no—
13C3×#380C2212yes—
14#5213—21no—
15#5212C2211yescharacteristic
16C2×#830C2212yes—
17#4961—22no—
18#4960C412yescomplemented
19#4953C412yescomplemented
20#4525—22no—
21#4451—24no—
22#2617—31no—
23#2315D311yescharacteristic, complemented
24#1519—32no—
25C2×#323—31no—
26#1289—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C22×#67—21no—
32C22×#67—22no—
33C22×D12C2×C411yescharacteristic, complemented
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—42no—
41C2×C3×C22⋊C4—42no—
42C2×C3×C22⋊C4C2×C412yes—
43C2×C3×C22⋊C4D412yes—
44#953—42no—
45#953—42no—
46C2×#217—22no—
47C2×#217—21no—
48#930—22no—
49#930—22no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—22no—
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3C2×C411yescharacteristic, complemented
54C22×C3⋊C8—22no—
55C3×#95—22no—
56C3×#95D414yescomplemented
57C3⋊#95—22no—
58C3⋊#95C2×C412yescomplemented
59#830—22no—
60#830—22no—
61C2×#121—31no—
62C2×#120—61no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4C2×D311yescharacteristic
65C2×C22⋊C8—32no—
66#397—32no—
67#394—62no—
68#392—32no—
69#386—61no—
70#385—24no—
71#380—21no—
72#380—31no—
73#380C2×D312yes—
74#327—64no—
75#323—62no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—21no—
80C24×C3—81no—
81C24×C3—81no—
82C24×C3C22⋊C412yes—
83C24×C3D812yes—
84C24×C3SD1612yes—
85C24×C3C2×D411yescharacteristic
86C23×D3—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—24no—
90C22×C3×C4—44no—
91C22×C3×C4—81no—
92C22×C3×C4—81no—
93C22×C3×C4—82no—
94C22×C3×C4—82no—
95C22×C3×C4C22⋊C412yes—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D412yes—
99C22×C3×C4#4112yes—
100C2×#67—44no—
101C2×#67—41no—
102C2×#67—41no—
103C2×#67—41no—
104C2×#67—41no—
105C2×#67—41no—
106C2×#67—41no—
107C2×#67—82no—
108C2×#67—82no—
109C22×Dic3—21no—
110C22×Dic3—22no—
111C2×D12—22no—
112C2×D12—81no—
113C3×C4⋊C4—82no—
114C3×C4⋊C4—82no—
115C3×C22⋊C4—24no—
116C3×C22⋊C4—24no—
117C3×C22⋊C4—44no—
118C3×C22⋊C4—42no—
119C3×C22⋊C4—42no—
120#217—44no—
121#217—41no—
122#217—41no—
123(C2×D3)⋊C4—42no—
124(C2×D3)⋊C4—82no—
125C4⋊Dic3—22no—
126C4⋊Dic3—81no—
127C2×C3⋊C8—42no—
128C2×C3⋊C8—82no—
129C25#6711yescharacteristic
130C22×D4—31no—
131C22×D4—61no—
132C22×D4—62no—
133C23×C4—22no—
134C23×C4#6711yescharacteristic
135C23×C4C22×D311yescharacteristic
136C22×C8—62no—
137#121—62no—
138#121—62no—
139#120—122no—
140#120—122no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—61no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—62no—
152C2×C22⋊C4—61no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—62no—
155C2×C22⋊C4C4×D312yes—
156C2×C22⋊C4D1212yes—
157C22⋊C8—62no—
158C22⋊C8—62no—
159#95—22no—
160#95—32no—
161#95—62no—
162#95#6714yescomplemented
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3—22no—
168C23×C3—22no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—81no—
176C23×C3—81no—
177C23×C3—82no—
178C23×C3—82no—
179C23×C3—82no—
180C23×C3—82no—
181C23×C3#10214yes—
182C23×C3C2×C22⋊C411yescharacteristic
183C23×C3#12011yescharacteristic
184C23×C3#12311yescharacteristic
185C23×C3C2×D811yescharacteristic
186C23×C3C2×SD1611yescharacteristic
187C23×C3AGL(1,ℤ/8)11yescharacteristic
188C23×C3#13711yescharacteristic
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—41no—
192C22×D3—41no—
193C22×D3—41no—
194C2×C3×C4—22no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—44no—
202C2×C3×C4—44no—
203C2×C3×C4—44no—
204C2×C3×C4—81no—
205C2×C3×C4—88no—
206C2×C3×C4C4×D412yes—
207C2×C3×C4#12112yes—
208#67—88no—
209#67—81no—
210#67—81no—
211#67—81no—
212#67—81no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—41no—
216C2×Dic3—42no—
217C2×Dic3—41no—
218C2×Dic3—41no—
219D12—82no—
220C3⋊C8—82no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—61no—
226C24—81no—
227C24—81no—
228C24(C2×D3)⋊C412yes—
229C24#21312yes—
230C24#21512yes—
231C24C2×#6711yescharacteristic
232C2×D4—62no—
233C2×D4—124no—
234C2×D4—121no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—241no—
241C2×D4—242no—
242C2×D4—242no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—44no—
247C22×C4—62no—
248C22×C4—61no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4(C2×D3)⋊C412yes—
254C22×C4C2×C4×D311yescharacteristic
255C22×C4C2×D1211yescharacteristic
256C22×C4#23512yes—
257C22×C4C2×#6712yes—
258C2×C8—122no—
259C2×C8—242no—
260C4⋊C4—62no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—241no—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—44no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—124no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—122no—
273C22⋊C4—242no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—24no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—44no—
286C22×C3—44no—
287C22×C3—44no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3—82no—
291C22×C3—82no—
292C22×C3—82no—
293C22×C3#38611yescharacteristic
294C22×C3C2×#10211yescharacteristic
295C22×C3#41711yescharacteristic
296C22×C3#44711yescharacteristic
297C22×C3#44811yescharacteristic
298C22×C3#48011yescharacteristic
299C22×C3#48111yescharacteristic
300C2×D3—82no—
301C2×D3—82no—
302C2×D3—81no—
303C2×D3—81no—
304C2×D3—81no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—42no—
308C3×C4—84no—
309Dic3—82no—
310Dic3—81no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—82no—
326C23—82no—
327C23—82no—
328C23—82no—
329C23—122no—
330C23—122no—
331C23—121no—
332C23—121no—
333C23—121no—
334C23#80914yes—
335C23C2×(C2×D3)⋊C411yescharacteristic
336C23#92911yescharacteristic
337C23C2×#21311yescharacteristic
338C23#93211yescharacteristic
339C23C2×#21511yescharacteristic
340C23#94211yescharacteristic
341C23#95311yescharacteristic
342D4—242no—
343D4—248no—
344D4—241no—
345D4—241no—
346D4—241no—
347D4—241no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—81no—
359C2×C4—88no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4C4×#6712yes—
367C2×C4#93012yes—
368C8—242no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—44no—
376C2×C3#187411yescharacteristic
377C2×C3#191111yescharacteristic
378C2×C3#191311yescharacteristic
379D3—161no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—24no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—242no—
400C22—242no—
401C22—241no—
402C22—241no—
403C22—241no—
404C22C2×#80911yescharacteristic
405C22#496611yescharacteristic
406C22#496711yescharacteristic
407C22#496811yescharacteristic
408C22#521311yescharacteristic
409C22#521711yescharacteristic
410C22#522711yescharacteristic
411C4—22no—
412C4—22no—
413C4—42no—
414C4—84no—
415C4—242no—
416C4—241no—
417C3#920311yescharacteristic, complemented, Sylow
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—21no—
423C2—21no—
424C2—44no—
425C2—481no—
426C2#7156011yescharacteristic
427C2#7165211yescharacteristic
428C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2315C2
socleC22×C3#386
prime radical#82130C2
cosocleC23×C3×C4C23
hypercenter#2315D3
hypocenterC3#9203
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
1. C3×#2315C2semidirect
3. C2×#4525C2semidirect
4. #69509C2semidirect
5. #67625C2semidirect
18. #4960C4semidirect
19. #4953C4semidirect
23. #2315D3semidirect
33. C22×D12C2×C4semidirect
53. C2×C4⋊Dic3C2×C4semidirect
56. C3×#95D4semidirect
58. C3⋊#95C2×C4semidirect
162. #95#67semidirect
417. C3#9203semidirect

Maximal subgroups

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

subgroupindex
C3×#23152
#821302
C2×#45252
#695092
#676252
#92033