Finite Group Databasebeta

← #11714046

Subgroups — #11714046

Order 768.

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

normal subgroups only

This group has 577 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, complemented
2C2×#5427C211yesmaximal, characteristic
3#73151C211yesmaximal, characteristic, complemented
4#73065C211yesmaximal, characteristic, complemented
5#73065C211yesmaximal, characteristic, complemented
6#73062C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#33006—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C22×GL(2,𝔽3)C2211yescharacteristic
11C4×#41⋊C3—22no—
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5427—22no—
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18#4940—41nomaximal
19C2×C4×#41D311yescharacteristic, complemented
20C2×#482—31no—
21C2×#449—31no—
22#2854—31no—
23#1880—31no—
24#1880—31no—
25#1879—31no—
26#1879—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×GL(2,𝔽3)—21no—
39C2×GL(2,𝔽3)—21no—
40C2×GL(2,𝔽3)—21no—
41C2×C3×C42—41no—
42C2×(C2×D3)⋊C4—41no—
43C2×(C2×D3)⋊C4—41no—
44C2×(C2×D3)⋊C4—41no—
45C2×Dic3⋊C4—41no—
46C4×SL(2,𝔽3)—21no—
47C4×SL(2,𝔽3)—21no—
48#859—21no—
49#859—21no—
50#859—21no—
51C3⋊#95—41no—
52C3⋊#95—41no—
53C22×#41C2×D311yescharacteristic
54C22×SD16—31no—
55C22×D8—31no—
56C2×#123—61no—
57C2×#121—31no—
58C4×#41—22no—
59C4×#41—62no—
60C2×C4×Q8C2×D311yescharacteristic
61C2×C4×Q8C2×D311yescharacteristic
62C2×C4×D4—31no—
63C2×C4×D4—31no—
64C2×#117—31no—
65C22×C42—31no—
66#482—64no—
67#449—62no—
68#449—62no—
69C2×#107—31no—
70C2×#106—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#103—31no—
74C2×#102—31no—
75C2×#102—31no—
76C2×#102—31no—
77C2×#102—31no—
78C2×#102—31no—
79C2×C22⋊C8—31no—
80C2×C22⋊C8—31no—
81#390—31no—
82#390—31no—
83#389—31no—
84#389—31no—
85#388—61no—
86#388—61no—
87#386—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—
97C23×D3—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—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
108GL(2,𝔽3)—41no—
109C3×C42—82no—
110(C2×D3)⋊C4—44no—
111(C2×D3)⋊C4—82no—
112(C2×D3)⋊C4—82no—
113Dic3⋊C4—82no—
114C2×#41—21no—
115C2×#41—21no—
116C2×#41—32no—
117C2×#41—61no—
118C2×#41—61no—
119C2×#41#6712yes—
120C22×Q8C22×D311yescharacteristic
121C22×D4—31no—
122C22×D4—31no—
123C22×D4—61no—
124C23×C4—31no—
125C23×C4—31no—
126C23×C4—31no—
127C2×SD16—61no—
128C2×SD16—61no—
129C2×SD16—61no—
130C2×SD16—61no—
131C2×SD16—61no—
132C2×SD16—61no—
133C2×D8—62no—
134C2×D8—62no—
135C2×D8—62no—
136C22×C8—31no—
137C22×C8—31no—
138C22×C8—31no—
139C22×C8—31no—
140#123—124no—
141#121—62no—
142#121—62no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—61no—
146C4×Q8—61no—
147C4×D4—62no—
148C4×D4—62no—
149C4×D4—62no—
150C4×D4—62no—
151#117—34no—
152#117—62no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—61no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C42—31no—
168C2×C42—31no—
169C2×C42—31no—
170C2×C42—62no—
171C2×C42—62no—
172C2×C42S411yescharacteristic
173#107—62no—
174#106—62no—
175#103—61no—
176#103—61no—
177#103—61no—
178#103—61no—
179#103—61no—
180#103—61no—
181#102—62no—
182#102—62no—
183#102—62no—
184#102—62no—
185#102—62no—
186C22⋊C8—62no—
187C22⋊C8—62no—
188#95—61no—
189#95—61no—
190#95—61no—
191C23×C3—41no—
192C22×D3—42no—
193C22×D3—81no—
194C22×D3—81no—
195C22×D3—81no—
196C22×D3—81no—
197C22×D3—81no—
198C22×D3—81no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×Dic3—42no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214C2×Dic3—81no—
215C2×Dic3—81no—
216C2×Dic3—81no—
217SL(2,𝔽3)#12311yescharacteristic, complemented
218C24—31no—
219C24—61no—
220#41—22no—
221#41—64no—
222#41—62no—
223#41—62no—
224#41—62no—
225C2×Q8—31no—
226C2×Q8—31no—
227C2×Q8—31no—
228C2×Q8#23511yescharacteristic
229C2×Q8#23511yescharacteristic
230C2×Q8C2×#6711yescharacteristic
231C2×D4—32no—
232C2×D4—32no—
233C2×D4—32no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—122no—
244C2×D4—122no—
245C2×D4—122no—
246C22×C4—31no—
247C22×C4—32no—
248C22×C4—32no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4—62no—
256C22×C4—62no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—121no—
263C22×C4—121no—
264C22×C4C2×S411yescharacteristic
265C22×C4C2×S411yescharacteristic
266C22×C4C2×S411yescharacteristic
267SD16—121no—
268SD16—121no—
269SD16—121no—
270SD16—121no—
271D8—124no—
272C2×C8—61no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—61no—
284C4⋊C4—32no—
285C4⋊C4—32no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—122no—
295C4⋊C4—122no—
296C22⋊C4—64no—
297C22⋊C4—122no—
298C22⋊C4—122no—
299C22⋊C4—122no—
300C22⋊C4—122no—
301C22⋊C4—122no—
302C22⋊C4—122no—
303C42—22no—
304C42—61no—
305C42—61no—
306C42—61no—
307C42—61no—
308C42—62no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C2×D3—82no—
317C2×D3—82no—
318C2×D3—82no—
319C2×D3—161no—
320C2×D3—161no—
321C2×D3—161no—
322C2×D3—161no—
323C3×C4—82no—
324C3×C4—82no—
325C3×C4—82no—
326Dic3—82no—
327Dic3—161no—
328Dic3—161no—
329C23—32no—
330C23—61no—
331C23—61no—
332C23—61no—
333C23—61no—
334C23—61no—
335C23—61no—
336C23—121no—
337C23—121no—
338C23—121no—
339C23—121no—
340C23—121no—
341C23—121no—
342C23—121no—
343C23C22×S411yescharacteristic
344Q8—31no—
345Q8—31no—
346Q8—31no—
347Q8—61no—
348Q8#92911yescharacteristic
349D4—34no—
350D4—62no—
351D4—62no—
352D4—62no—
353D4—121no—
354D4—121no—
355D4—121no—
356D4—121no—
357D4—244no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—124no—
381C2×C4—124no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4#98512yes—
392C2×C4#98612yes—
393C2×C4GL(2,ℤ/4)12yes—
394C8—121no—
395C8—121no—
396C8—121no—
397C8—121no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3—41no—
405D3—162no—
406C22—62no—
407C22—62no—
408C22—62no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—241no—
414C22—241no—
415C22—241no—
416C22—241no—
417C22—241no—
418C22—241no—
419C22—241no—
420C22#591211yescharacteristic
421C22#591211yescharacteristic
422C22C2×#98511yescharacteristic
423C22C2×#98611yescharacteristic
424C22#592311yescharacteristic
425C22#592311yescharacteristic
426C22C2×GL(2,ℤ/4)11yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—31no—
431C4—31no—
432C4—31no—
433C4—31no—
434C4—121no—
435C4—121no—
436C4—241no—
437C4—241no—
438C4—241no—
439C3—41noSylow
440C2—122no—
441C2—481no—
442C2#8534311yescharacteristic
443C2#8534511yescharacteristic
444C2#8534811yescharacteristic
445C2#8534911yescharacteristic
446C2#8548511yescharacteristic
447C2#8552111yescharacteristic
448C2#8552311yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×C4×#41D3
prime radicalC2×#5427C2
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
1. C2×C4×#41⋊C3C2semidirect
3. #73151C2semidirect
4. #73065C2semidirect
5. #73065C2semidirect
6. #73062C2semidirect
7. #73059C2semidirect
19. C2×C4×#41D3semidirect
217. SL(2,𝔽3)#123semidirect

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×#54272
#731512
#730652
#730622
#730592
#330063
#49404