Finite Group Databasebeta

← #11714043

Subgroups — #11714043

Order 768.

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

normal subgroups only

This group has 473 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
2#85517C211yesmaximal, characteristic
3#73149C211yesmaximal, characteristic
4#73077C211yesmaximal, characteristic, complemented
5#73077C211yesmaximal, characteristic, complemented
6#73076C211yesmaximal, characteristic, complemented
7#73075C211yesmaximal, characteristic, complemented
8#33001—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C2×#984C2211yescharacteristic
11#520⋊C3C2211yescharacteristic
12#520⋊C3C2211yescharacteristic
13#519⋊C3—21no—
14C4×#41⋊C3—21no—
15#5426C2211yescharacteristic
16#5425—21no—
17C2×GU(2,𝔽3)C2211yescharacteristic
18C2×GU(2,𝔽3)C2211yescharacteristic
19#5419—21no—
20#4996—41nomaximal
21#3428D311yescharacteristic
22#3095—31no—
23#3007—31no—
24#2499—31no—
25#1882—31no—
26#1882—31no—
27#1882—31no—
28#1882—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#984—21no—
42#984—21no—
43C2×#226—21no—
44C3×#117—41no—
45#924—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#810—41no—
53#810—41no—
54#808—41no—
55#807—41no—
56C22×#41C2×D311yescharacteristic
57C2×#137—31no—
58C2×#137—31no—
59C2×C22⋊Q8—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#485—61no—
71#485—61no—
72#484—61no—
73#483—61no—
74#452—61no—
75#452—61no—
76#451—61no—
77#450—61no—
78#428—31no—
79#428—31no—
80C2×#104—31no—
81C2×#104—31no—
82C2×#104—31no—
83C2×#104—31no—
84#419—31no—
85#419—31no—
86#417—31no—
87#417—31no—
88#407—31no—
89#407—31no—
90#397—62no—
91#391—31no—
92#391—31no—
93#387—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×Dic6—41no—
107#41⋊C3—21no—
108#41⋊C3#4111yescharacteristic
109#41⋊C3#4111yescharacteristic
110C2×SL(2,𝔽3)—21no—
111C2×SL(2,𝔽3)C2×D411yescharacteristic
112#226—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C22⋊C4—81no—
116C3×C42—81no—
117#217—81no—
118C4⋊Dic3—41no—
119Dic3⋊C4—81no—
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—31no—
131C22×Q8#6711yescharacteristic
132C22×D4—31no—
133C23×C4—31no—
134#137—61no—
135#137—61no—
136#137—62no—
137#137—62no—
138#137—61no—
139#137—61no—
140C2×Q16—61no—
141C2×Q16—61no—
142C2×SD16—61no—
143C2×SD16—61no—
144C2×M16—31no—
145C2×M16—31no—
146C2×M16—31no—
147C2×M16—31no—
148C22⋊Q8—61no—
149C22⋊Q8—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152C4×Q8—21no—
153C4×Q8—21no—
154C4×Q8—61no—
155C4×Q8—61no—
156C4×D4—62no—
157C4×D4—62no—
158C4×D4—61no—
159C4×D4—61no—
160C4×D4—61no—
161C4×D4—61no—
162#117—31no—
163#117—31no—
164#117—31no—
165#117—61no—
166#117—61no—
167#117—61no—
168#117—61no—
169#117—61no—
170#117S411yescharacteristic
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—62no—
178C2×C22⋊C4—32no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C42—32no—
183C2×C42—31no—
184C2×C42—31no—
185C2×C42—61no—
186#107—61no—
187#107—61no—
188#106—61no—
189#106—61no—
190#104—61no—
191#104—61no—
192#104—62no—
193#104—62no—
194#104—61no—
195#104—61no—
196#103—61no—
197#103—61no—
198#103—61no—
199#103—61no—
200#103—61no—
201#103—61no—
202#102—61no—
203#102—61no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208#95—61no—
209#95—62no—
210#95—62no—
211#95—62no—
212C23×C3—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—81no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×Dic3—81no—
220C2×Dic3—81no—
221Dic6—81no—
222Dic6—81no—
223SL(2,𝔽3)#12311yescharacteristic
224C3⋊C8—81no—
225C3⋊C8—81no—
226C24—31no—
227#41—21no—
228#41—31no—
229#41—31no—
230#41—32no—
231#41—32no—
232#41—61no—
233#41—61no—
234#41—62no—
235#41—61no—
236#41—61no—
237#41—61no—
238#41#23511yescharacteristic
239#41#23511yescharacteristic
240C2×Q8—21no—
241C2×Q8—31no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8C2×#6711yescharacteristic
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—62no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—32no—
262C22×C4—32no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4C2×S411yescharacteristic
274Q16—121no—
275Q16—122no—
276Q16—121no—
277SD16—121no—
278SD16—121no—
279SD16—122no—
280M16—61no—
281M16—61no—
282M16—61no—
283M16—61no—
284M16—61no—
285M16—61no—
286M16—61no—
287M16—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—61no—
291C2×C8—61no—
292C4⋊C4—31no—
293C4⋊C4—31no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—121no—
301C4⋊C4—124no—
302C4⋊C4—121no—
303C4⋊C4C2×S411yescharacteristic
304C4⋊C4C2×S411yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—61no—
307C22⋊C4—61no—
308C22⋊C4—62no—
309C22⋊C4—61no—
310C22⋊C4—122no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C42—21no—
314C42—61no—
315C42—61no—
316C42—62no—
317C42—61no—
318C42—61no—
319C42—61no—
320C42—61no—
321C42—61no—
322C22×C3—41no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3—81no—
326C3×C4—41no—
327C3×C4—41no—
328C3×C4—81no—
329C3×C4—161no—
330C3×C4—161no—
331Dic3—161no—
332Dic3—161no—
333C23—31no—
334C23—31no—
335C23—61no—
336C23—61no—
337C23—121no—
338C23—121no—
339C23GL(2,ℤ/4)11yescharacteristic
340Q8—31no—
341Q8—32no—
342Q8—61no—
343Q8—121no—
344Q8—121no—
345Q8—121no—
346Q8—121no—
347Q8#92911yescharacteristic
348D4—31no—
349D4—31no—
350D4—32no—
351D4—61no—
352D4—61no—
353D4—62no—
354D4—62no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—62no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—122no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4GL(2,ℤ/4)11yescharacteristic
391C2×C4C22×S411yescharacteristic
392C8—121no—
393C8—121no—
394C8—121no—
395C8—121no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—161no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—31no—
404C22—31no—
405C22—61no—
406C22—61no—
407C22—61no—
408C22—61no—
409C22—61no—
410C22—241no—
411C22—241no—
412C22C2×GL(2,ℤ/4)11yescharacteristic
413C4—21no—
414C4—31no—
415C4—31no—
416C4—41no—
417C4—41no—
418C4—61no—
419C4—122no—
420C4—121no—
421C4—121no—
422C4—241no—
423C4—241no—
424C4—241no—
425C4#591211yescharacteristic
426C4#591211yescharacteristic
427C3—41noSylow
428C2—41no—
429C2—61no—
430C2—61no—
431C2—121no—
432C2#8548511yescharacteristic
433C2#8554211yescharacteristic
434C2#8554411yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3428D3
prime radicalC22×#41⋊C3C22
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
4. #73077C2semidirect
5. #73077C2semidirect
6. #73076C2semidirect
7. #73075C2semidirect

Maximal subgroups

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

subgroupindex
#3428⋊C32
#855172
#731492
#730772
#730762
#730752
#330013
#49964