Finite Group Databasebeta

← #10695650

Subgroups — #10695650

Order 768.

692 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#82177C211yesmaximal, characteristic, complemented
2C2×#4967C211yesmaximal, characteristic, complemented
3#78611C211yesmaximal, characteristic
4C3×#1823C211yesmaximal, characteristic, complemented
5#69528C211yesmaximal, characteristic, complemented
6#69509C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9952—31nomaximal, Sylow
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#930C2211yescharacteristic
12C2×#929—22no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C4×#217C2211yescharacteristic
16C2×#830C2211yescharacteristic
17#5115—22no—
18#4993—22no—
19#4988—21no—
20#4975—21no—
21#4967—22no—
22#4967—22no—
23C2×#809—21no—
24C2×#807—21no—
25C2×#806—22no—
26#4736—21no—
27#4736—21no—
28#4732—21no—
29#4732—21no—
30C3×#340—22no—
31#4451—22no—
32#3186—31no—
33C2×#480—31no—
34#2501—31no—
35#1964—31no—
36#1823D311yescharacteristic, complemented
37#1519—31no—
38#1282—31no—
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×#67—21no—
43C22×D12D411yescharacteristic
44C22×D12D411yescharacteristic
45C22×C3×C8D412yes—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C22⋊C4—21no—
54C2×#217D412yes—
55#930—21no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62#930—21no—
63#929—44no—
64#929—44no—
65C2×(C2×D3)⋊C4—24no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—21no—
68C2×C4⋊Dic3D411yescharacteristic
69C2×C4⋊Dic3D411yescharacteristic
70C2×Dic3⋊C4—24no—
71C2×C4×Dic3—21no—
72C22×C3⋊C8—21no—
73C2×D24—22no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×#95—21no—
77C3⋊#95—21no—
78#830—21no—
79#830D412yes—
80#821—22no—
81#821—22no—
82#818—22no—
83#809—24no—
84#807—24no—
85#806—44no—
86C2×#123—62no—
87C2×#121—31no—
88C2×#121—31no—
89C22×C4⋊C4C2×D311yescharacteristic
90#480—61no—
91#480—61no—
92#480—62no—
93#480—62no—
94#466—62no—
95#447—61no—
96#447—61no—
97C2×#107—61no—
98C2×#102—61no—
99C2×#99—62no—
100C2×C22⋊C8—31no—
101C2×C22⋊C8C2×D311yescharacteristic
102#399—62no—
103#390—61no—
104#384—61no—
105#380C2×D311yescharacteristic
106C4×C22⋊C4—31no—
107#340—22no—
108#327—62no—
109C24×C3C2×D411yescharacteristic
110C23×D3—21no—
111C23×D3—21no—
112C22×C3×C4—22no—
113C22×C3×C4—42no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—42no—
117C22×C3×C4—42no—
118C22×C3×C4—41no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C2×#67—42no—
130C2×#67—42no—
131C2×#67—42no—
132C2×#67—42no—
133C22×Dic3—22no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—42no—
141C2×D12—42no—
142C2×C3×C8—22no—
143C2×C3×C8—42no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—42no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C22⋊C4—42no—
150#217—22no—
151#217D814yescomplemented
152(C2×D3)⋊C4—48no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161Dic3⋊C4—48no—
162C4×Dic3—42no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—42no—
165D24—82no—
166C22×D4—31no—
167C22×D4—31no—
168C22×D4—61no—
169C22×D4—61no—
170C23×C4—21no—
171C23×C4C22×D311yescharacteristic
172C2×D8—62no—
173C22×C8—61no—
174C22×C8#6712yes—
175#123—124no—
176#123—124no—
177#121—61no—
178#121—61no—
179#121—61no—
180#121—61no—
181#121—61no—
182#121—61no—
183#121—61no—
184#121—61no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—64no—
194C2×C4⋊C4D1212yes—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—64no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C42—61no—
201#107—64no—
202#107—62no—
203#102—64no—
204#102—62no—
205#102—62no—
206#99—124no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—32no—
210C22⋊C8—61no—
211#95—21no—
212#95—61no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3#12011yescharacteristic
216C23×C3#12111yescharacteristic
217C23×C3#12711yescharacteristic
218C23×C3C2×D812yes—
219C23×C3AGL(1,ℤ/8)12yes—
220C22×D3—42no—
221C22×D3—42no—
222C22×D3—41no—
223C22×D3—41no—
224C22×D3—42no—
225C22×D3—42no—
226C22×D3—81no—
227C22×D3—81no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—44no—
236C2×C3×C4—82no—
237C2×C3×C4—81no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246#67—82no—
247#67—82no—
248#67—81no—
249#67—81no—
250#67—81no—
251#67—81no—
252C2×Dic3—24no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259D12—42no—
260D12—42no—
261D12—42no—
262D12—81no—
263C3×C8—42no—
264C3⋊C8—42no—
265C24—61no—
266C24—61no—
267C24D3×D411yescharacteristic
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—61no—
272C2×D4—122no—
273C2×D4—122no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—122no—
277C2×D4—122no—
278C2×D4—122no—
279C2×D4—122no—
280C2×D4—121no—
281C2×D4—121no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—62no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4C2×D1211yescharacteristic
293C22×C4#23511yescharacteristic
294C22×C4D3×D411yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4C2×#6711yescharacteristic
298D8—242no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—62no—
302C2×C8—122no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—128no—
313C22⋊C4—34no—
314C22⋊C4—42no—
315C22⋊C4—62no—
316C22⋊C4—128no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C42—122no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—42no—
329C22×C3—42no—
330C22×C3#39211yescharacteristic
331C22×C3#44712yes—
332C22×C3#45912yes—
333C22×C3#49311yescharacteristic
334C22×C3#49611yescharacteristic
335C2×D3—84no—
336C2×D3—81no—
337C2×D3—81no—
338C2×D3—81no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—81no—
342C2×D3—162no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—82no—
347C3×C4—161no—
348C3×C4#45812yes—
349Dic3—44no—
350Dic3—81no—
351Dic3—81no—
352C23—41no—
353C23—41no—
354C23—122no—
355C23—122no—
356C23—121no—
357C23—121no—
358C23—122no—
359C23—122no—
360C23—241no—
361C23—241no—
362C23#88211yescharacteristic
363C23#88411yescharacteristic
364C23D3×D812yes—
365C23#91412yes—
366C23#94011yescharacteristic
367D4—122no—
368D4—122no—
369D4—122no—
370D4—241no—
371D4—241no—
372D4—241no—
373D4—241no—
374D4—241no—
375D4—242no—
376D4—242no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—64no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4#88211yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4#93011yescharacteristic
401C2×C4#93711yescharacteristic
402C8—42no—
403C8—122no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—21no—
408C2×C3#200911yescharacteristic
409C2×C3#201111yescharacteristic
410C2×C3#202211yescharacteristic
411D3—161no—
412D3—161no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—244no—
422C22—241no—
423C22—241no—
424C22—241no—
425C22—241no—
426C22—241no—
427C22—241no—
428C22—482no—
429C22#477312yes—
430C22#478612yes—
431C22#496011yescharacteristic
432C22#515111yescharacteristic
433C22#517611yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—82no—
438C4—124no—
439C4—161no—
440C4—241no—
441C4—241no—
442C4#505312yes—
443C3#995211yescharacteristic, complemented, Sylow
444C2—22no—
445C2—21no—
446C2—22no—
447C2—21no—
448C2—481no—
449C2—481no—
450C2#6953911yescharacteristic
451C2#7020911yescharacteristic
452C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1823C2
socleC22×C3#392
prime radical#78611C2
hypercenter#1823D3
hypocenterC3#9952
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
1. #82177C2semidirect
2. C2×#4967C2semidirect
4. C3×#1823C2semidirect
5. #69528C2semidirect
6. #69509C2semidirect
7. #67677C2semidirect
36. #1823D3semidirect
151. #217D8semidirect
443. C3#9952semidirect

Maximal subgroups

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

subgroupindex
#821772
C2×#49672
#786112
C3×#18232
#695282
#695092
#676772
#99523