Finite Group Databasebeta

← #11714040

Subgroups — #11714040

Order 768.

499 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 47 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#3428⋊C3C211yesmaximal, characteristic, complemented
2#85523C211yesmaximal, characteristic
3#73149C211yesmaximal, characteristic, complemented
4#73078C211yesmaximal, characteristic, complemented
5#73078C211yesmaximal, characteristic, complemented
6#73076C211yesmaximal, characteristic, complemented
7#73075C211yesmaximal, characteristic, complemented
8#32998—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C2×#986C2211yescharacteristic
11#520⋊C3C2211yescharacteristic
12#520⋊C3C2211yescharacteristic
13#519⋊C3—21no—
14C4×#41⋊C3—21no—
15#5427—21no—
16#5426C2211yescharacteristic
17C2×GU(2,𝔽3)C2211yescharacteristic
18C2×GU(2,𝔽3)C2211yescharacteristic
19#5421—21no—
20#5003—41nomaximal
21#3428D311yescharacteristic, complemented
22#3094—31no—
23#3006—31no—
24#2500—31no—
25#1883—31no—
26#1883—31no—
27#1883—31no—
28#1883—31no—
29#1860—31no—
30#1860—31no—
31#1859—31no—
32#1859—31no—
33#1826—31no—
34#1822—31no—
35#1789—31no—
36C2×#41⋊C3—21no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3D411yescharacteristic
39C2×#41⋊C3C2311yescharacteristic
40C22×SL(2,𝔽3)D411yescharacteristic
41#986—21no—
42#986—21no—
43C2×GL(2,𝔽3)—21no—
44C3×#117—41no—
45#930—41no—
46C4×SL(2,𝔽3)—21no—
47C4×SL(2,𝔽3)—21no—
48GU(2,𝔽3)—21no—
49GU(2,𝔽3)—21no—
50#859—21no—
51#830—41no—
52#809—41no—
53#809—41no—
54#808—41no—
55#807—41no—
56C22×#41C2×D311yescharacteristic
57C2×AGL(1,ℤ/8)—31no—
58C2×AGL(1,ℤ/8)—31no—
59C2×#121—31no—
60#520—31no—
61#520—31no—
62#520C2×D311yescharacteristic
63#520C2×D311yescharacteristic
64#519—21no—
65#518—61no—
66C4×#41—21no—
67C4×#41—61no—
68C2×#117—31no—
69C2×#117—31no—
70#482—61no—
71#482—61no—
72#481—61no—
73#480—61no—
74#449—61no—
75#449—61no—
76#448—61no—
77#447—61no—
78#428—31no—
79#428—31no—
80C2×#104—31no—
81C2×#104—31no—
82C2×#104—31no—
83C2×#104—31no—
84#418—31no—
85#418—31no—
86#417—31no—
87#417—31no—
88#407—31no—
89#407—31no—
90#394—62no—
91#390—31no—
92#390—31no—
93#388—62no—
94#384—31no—
95#382—31no—
96C4×C22⋊C4—31no—
97#343—31no—
98#343—31no—
99#343—31no—
100#343—31no—
101#339—31no—
102#339—31no—
103#339—31no—
104#339—31no—
105C22×C3×C4—41no—
106C2×#67—81no—
107C2×D12—41no—
108#41⋊C3—21no—
109#41⋊C3#4111yescharacteristic
110#41⋊C3#4111yescharacteristic
111C2×SL(2,𝔽3)—21no—
112C2×SL(2,𝔽3)C2×D411yescharacteristic
113GL(2,𝔽3)—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C22⋊C4—81no—
117C3×C42—81no—
118(C2×D3)⋊C4—81no—
119C4⋊Dic3—41no—
120C2×C3⋊C8—41no—
121C2×C3⋊C8—41no—
122C2×#41—21no—
123C2×#41—21no—
124C2×#41—31no—
125C2×#41—31no—
126C2×#41—61no—
127C2×#41—61no—
128C2×#41#6711yescharacteristic
129C2×#41C22×D311yescharacteristic
130C22×Q8#6711yescharacteristic
131C22×D4—31no—
132C22×D4—31no—
133C22×D4—61no—
134C23×C4—31no—
135AGL(1,ℤ/8)—61no—
136AGL(1,ℤ/8)—61no—
137AGL(1,ℤ/8)—62no—
138AGL(1,ℤ/8)—62no—
139AGL(1,ℤ/8)—61no—
140AGL(1,ℤ/8)—61no—
141C2×SD16—61no—
142C2×SD16—61no—
143C2×D8—61no—
144C2×D8—61no—
145C2×M16—31no—
146C2×M16—31no—
147C2×M16—31no—
148C2×M16—31no—
149#121—61no—
150#121—61no—
151#121—61no—
152#121—61no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—61no—
156C4×Q8—61no—
157C4×D4—62no—
158C4×D4—62no—
159C4×D4—61no—
160C4×D4—61no—
161C4×D4—61no—
162C4×D4—61no—
163#117—31no—
164#117—31no—
165#117—31no—
166#117—61no—
167#117—61no—
168#117—61no—
169#117—61no—
170#117—61no—
171#117S411yescharacteristic
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—62no—
178C2×C22⋊C4—32no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—62no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—62no—
184C2×C42—32no—
185C2×C42—31no—
186C2×C42—31no—
187C2×C42—61no—
188#107—61no—
189#107—61no—
190#106—61no—
191#106—61no—
192#104—61no—
193#104—61no—
194#104—62no—
195#104—62no—
196#104—61no—
197#104—61no—
198#103—61no—
199#103—61no—
200#102—61no—
201#102—61no—
202#102—61no—
203#102—61no—
204#102—61no—
205#102—61no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210#95—61no—
211#95—62no—
212C23×C3—41no—
213C22×D3—81no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220#67—162no—
221C2×Dic3—81no—
222D12—81no—
223D12—81no—
224SL(2,𝔽3)#12311yescharacteristic
225C3⋊C8—81no—
226C3⋊C8—81no—
227C24—31no—
228C24—61no—
229#41—21no—
230#41—31no—
231#41—31no—
232#41—32no—
233#41—32no—
234#41—61no—
235#41—61no—
236#41—62no—
237#41—61no—
238#41—61no—
239#41—61no—
240#41#23511yescharacteristic
241#41#23511yescharacteristic
242C2×Q8—21no—
243C2×Q8—31no—
244C2×Q8—61no—
245C2×Q8C2×#6711yescharacteristic
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—62no—
257C2×D4—61no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—242no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—32no—
267C22×C4—32no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4C2×S411yescharacteristic
278SD16—121no—
279SD16—122no—
280SD16—121no—
281D8—121no—
282D8—121no—
283D8—122no—
284M16—61no—
285M16—61no—
286M16—61no—
287M16—61no—
288M16—61no—
289M16—61no—
290M16—61no—
291M16—61no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—61no—
296C4⋊C4—31no—
297C4⋊C4—31no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—242no—
305C4⋊C4C2×S411yescharacteristic
306C4⋊C4C2×S411yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—62no—
311C22⋊C4—61no—
312C22⋊C4—121no—
313C22⋊C4—124no—
314C22⋊C4—122no—
315C22⋊C4—121no—
316C22⋊C4—242no—
317C42—21no—
318C42—61no—
319C42—61no—
320C42—62no—
321C42—61no—
322C42—61no—
323C42—61no—
324C42—61no—
325C42—61no—
326C22×C3—41no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C2×D3—161no—
331C2×D3—161no—
332C2×D3—161no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—81no—
336C3×C4—161no—
337C3×C4—161no—
338Dic3—161no—
339C23—31no—
340C23—31no—
341C23—61no—
342C23—61no—
343C23—121no—
344C23—121no—
345C23—121no—
346C23—121no—
347C23—122no—
348C23—122no—
349C23—121no—
350C23GL(2,ℤ/4)11yescharacteristic
351Q8—31no—
352Q8—32no—
353Q8—61no—
354Q8#92911yescharacteristic
355D4—31no—
356D4—31no—
357D4—32no—
358D4—61no—
359D4—61no—
360D4—62no—
361D4—62no—
362D4—121no—
363D4—121no—
364D4—121no—
365D4—121no—
366D4—242no—
367D4—242no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—62no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4—121no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4GL(2,ℤ/4)11yescharacteristic
401C2×C4C22×S411yescharacteristic
402C8—121no—
403C8—121no—
404C8—121no—
405C8—121no—
406C2×C3—41no—
407C2×C3—41no—
408C2×C3—41no—
409C2×C3—161no—
410D3—321no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—31no—
415C22—31no—
416C22—61no—
417C22—61no—
418C22—61no—
419C22—61no—
420C22—61no—
421C22—241no—
422C22—241no—
423C22—242no—
424C22—242no—
425C22—241no—
426C22—241no—
427C22—241no—
428C22C2×GL(2,ℤ/4)11yescharacteristic
429C4—21no—
430C4—31no—
431C4—31no—
432C4—41no—
433C4—41no—
434C4—61no—
435C4—122no—
436C4—121no—
437C4—121no—
438C4—241no—
439C4—241no—
440C4#591211yescharacteristic
441C4#591211yescharacteristic
442C3—41noSylow
443C2—41no—
444C2—61no—
445C2—61no—
446C2—121no—
447C2—481no—
448C2#8548511yescharacteristic
449C2#8554311yescharacteristic
450C2#8554411yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3428D3
prime radical#85523C2
hypercenter#117S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #3428⋊C3C2semidirect
3. #73149C2semidirect
4. #73078C2semidirect
5. #73078C2semidirect
6. #73076C2semidirect
7. #73075C2semidirect
21. #3428D3semidirect

Maximal subgroups

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

subgroupindex
#3428⋊C32
#855232
#731492
#730782
#730762
#730752
#329983
#50034