Finite Group Databasebeta

← #11714049

Subgroups — #11714049

Order 768.

528 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C2×C4×#41⋊C3C211yesmaximal, characteristic
2C2×#5425C211yesmaximal, characteristic
3#73151C211yesmaximal, characteristic
4#73062C211yesmaximal, characteristic, complemented
5#73061C211yesmaximal, characteristic, complemented
6#73061C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#33011—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C22×#226C2211yescharacteristic
11C4×#41⋊C3—22no—
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5425—22no—
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18#4933—41nomaximal
19C2×C4×#41D311yescharacteristic
20C2×#485—31no—
21C2×#452—31no—
22#2855—31no—
23#1878—31no—
24#1878—31no—
25#1877—31no—
26#1877—31no—
27#1865—31no—
28#1864—31no—
29#1863—31no—
30#1862—31no—
31#1825—31no—
32#1817—31no—
33#1788—31no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3—21no—
36C2×#41⋊C3D412yes—
37C22×SL(2,𝔽3)C2311yescharacteristic
38C2×#226—21no—
39C2×#226—21no—
40C2×#226—21no—
41C2×C3×C42—41no—
42C2×Dic3⋊C4—41no—
43C2×Dic3⋊C4—41no—
44C4×SL(2,𝔽3)—21no—
45C4×SL(2,𝔽3)—21no—
46#859—21no—
47#859—21no—
48#859—21no—
49C3⋊#95—41no—
50C3⋊#95—41no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C22×#41C2×D311yescharacteristic
54C22×Q16—31no—
55C22×SD16—31no—
56C2×C22⋊Q8—31no—
57C4×#41—22no—
58C4×#41—62no—
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×Q8C2×D311yescharacteristic
61C2×C4×D4—31no—
62C2×C4×D4—31no—
63C2×#117—31no—
64C22×C42—31no—
65#485—64no—
66#452—62no—
67#452—62no—
68C2×#107—31no—
69C2×#106—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#103—31no—
75C2×#102—31no—
76C2×#102—31no—
77C2×#102—31no—
78C2×C22⋊C8—31no—
79C2×C22⋊C8—31no—
80#391—31no—
81#391—31no—
82#389—31no—
83#389—31no—
84#387—61no—
85#387—61no—
86#385—61no—
87#381—61no—
88#380—31no—
89#340—31no—
90#340—31no—
91#340—31no—
92#340—31no—
93#336—31no—
94#336—31no—
95#336—31no—
96#336—31no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×Dic3—41no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—41no—
104#41⋊C3—22no—
105C2×SL(2,𝔽3)C2×D411yescharacteristic
106C2×SL(2,𝔽3)#4111yescharacteristic
107C2×SL(2,𝔽3)#4111yescharacteristic
108#226—41no—
109C3×C42—82no—
110Dic3⋊C4—44no—
111Dic3⋊C4—82no—
112C2×#41—21no—
113C2×#41—21no—
114C2×#41—32no—
115C2×#41—61no—
116C2×#41—61no—
117C2×#41#6712yes—
118C22×Q8—31no—
119C22×Q8C22×D311yescharacteristic
120C22×D4—31no—
121C23×C4—31no—
122C23×C4—31no—
123C23×C4—31no—
124C2×Q16—61no—
125C2×Q16—61no—
126C2×Q16—61no—
127C2×Q16—61no—
128C2×Q16—61no—
129C2×Q16—61no—
130C2×SD16—62no—
131C2×SD16—62no—
132C2×SD16—62no—
133C22×C8—31no—
134C22×C8—31no—
135C22×C8—31no—
136C22×C8—31no—
137C22⋊Q8—62no—
138C22⋊Q8—62no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—61no—
142C4×Q8—61no—
143C4×D4—62no—
144C4×D4—62no—
145C4×D4—62no—
146C4×D4—62no—
147#117—34no—
148#117—62no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C42—31no—
161C2×C42—31no—
162C2×C42—31no—
163C2×C42—62no—
164C2×C42—62no—
165C2×C42S411yescharacteristic
166#107—62no—
167#106—62no—
168#103—62no—
169#103—62no—
170#103—61no—
171#103—61no—
172#103—61no—
173#103—61no—
174#103—61no—
175#103—61no—
176#102—62no—
177#102—62no—
178#102—62no—
179C22⋊C8—62no—
180C22⋊C8—62no—
181#95—61no—
182#95—61no—
183#95—61no—
184#95—61no—
185#95—61no—
186#95—61no—
187#95—61no—
188C23×C3—41no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—81no—
193C2×C3×C4—81no—
194C2×C3×C4—81no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210SL(2,𝔽3)#12311yescharacteristic
211C24—31no—
212#41—22no—
213#41—64no—
214#41—62no—
215#41—62no—
216#41—62no—
217C2×Q8—31no—
218C2×Q8—31no—
219C2×Q8—31no—
220C2×Q8—61no—
221C2×Q8—61no—
222C2×Q8—61no—
223C2×Q8—61no—
224C2×Q8—61no—
225C2×Q8—61no—
226C2×Q8#23511yescharacteristic
227C2×Q8#23511yescharacteristic
228C2×Q8C2×#6711yescharacteristic
229C2×D4—32no—
230C2×D4—32no—
231C2×D4—32no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C22×C4—31no—
236C22×C4—32no—
237C22×C4—32no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—62no—
245C22×C4—62no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—121no—
253C22×C4—121no—
254C22×C4C2×S411yescharacteristic
255C22×C4C2×S411yescharacteristic
256C22×C4C2×S411yescharacteristic
257Q16—121no—
258Q16—121no—
259Q16—121no—
260Q16—121no—
261SD16—124no—
262C2×C8—61no—
263C2×C8—61no—
264C2×C8—61no—
265C2×C8—61no—
266C2×C8—61no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—61no—
273C2×C8—61no—
274C4⋊C4—32no—
275C4⋊C4—32no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—122no—
285C4⋊C4—122no—
286C4⋊C4—122no—
287C22⋊C4—64no—
288C22⋊C4—122no—
289C22⋊C4—122no—
290C42—22no—
291C42—61no—
292C42—61no—
293C42—61no—
294C42—61no—
295C42—62no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C3×C4—82no—
304C3×C4—82no—
305C3×C4—82no—
306Dic3—82no—
307Dic3—82no—
308Dic3—161no—
309Dic3—161no—
310C23—32no—
311C23—61no—
312C23—61no—
313C23—61no—
314C23—61no—
315C23—61no—
316C23—61no—
317C23C22×S411yescharacteristic
318Q8—31no—
319Q8—31no—
320Q8—31no—
321Q8—61no—
322Q8—121no—
323Q8—121no—
324Q8—121no—
325Q8—121no—
326Q8#92911yescharacteristic
327D4—34no—
328D4—62no—
329D4—62no—
330D4—62no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—62no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—124no—
354C2×C4—124no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4#98412yes—
368C2×C4#98512yes—
369C2×C4GL(2,ℤ/4)12yes—
370C8—121no—
371C8—121no—
372C8—121no—
373C8—121no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C22—62no—
382C22—62no—
383C22—62no—
384C22—121no—
385C22—121no—
386C22—121no—
387C22—121no—
388C22#591211yescharacteristic
389C22#591211yescharacteristic
390C22C2×#98411yescharacteristic
391C22C2×#98511yescharacteristic
392C22#592311yescharacteristic
393C22#592311yescharacteristic
394C22C2×GL(2,ℤ/4)11yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—31no—
399C4—31no—
400C4—31no—
401C4—31no—
402C4—121no—
403C4—121no—
404C4—241no—
405C4—241no—
406C4—241no—
407C4—241no—
408C3—41noSylow
409C2—122no—
410C2#8533511yescharacteristic
411C2#8533811yescharacteristic
412C2#8534311yescharacteristic
413C2#8534911yescharacteristic
414C2#8548511yescharacteristic
415C2#8551711yescharacteristic
416C2#8552111yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×C4×#41D3
prime radicalC22×#41⋊C3C22
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
4. #73062C2semidirect
5. #73061C2semidirect
6. #73061C2semidirect
7. #73059C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#41⋊C32
C2×#54252
#731512
#730622
#730612
#730592
#330113
#49334