Finite Group Databasebeta

← #11713229

Subgroups — #11713229

Order 768.

541 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 57 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⋊C3)C211yesmaximal, characteristic
2C2×#5424C211yesmaximal, characteristic, complemented
3#73151C211yesmaximal, characteristic, complemented
4#73151C211yesmaximal, characteristic, complemented
5#73151C211yesmaximal, characteristic, complemented
6#73062C211yesmaximal, characteristic, complemented
7#73059C211yesmaximal, characteristic, complemented
8#24034—31nomaximal, Sylow
9C22×#41⋊C3C2211yescharacteristic
10C22×#41⋊C3C2211yescharacteristic
11C4⋊(#41⋊C3)—22no—
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13#5424—22no—
14C2×#859C2211yescharacteristic
15C2×#859C2211yescharacteristic
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18#4978—41nomaximal
19C2×#550D311yescharacteristic
20C2×#482—31no—
21C2×#416—31no—
22#2485—31no—
23#2041—31no—
24#2041—31no—
25#2034—31no—
26#2033—31no—
27#1893—31no—
28#1869—31no—
29#1865—31no—
30#1863—31no—
31#1788—31no—
32#1788—31no—
33#1788—31no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3—21no—
36C2×#41⋊C3—21no—
37C2×#41⋊C3—21no—
38C2×#41⋊C3—21no—
39C2×#41⋊C3D412yes—
40C22×SL(2,𝔽3)C2311yescharacteristic
41C2×C3×C4⋊C4—41no—
42C2×Dic3⋊C4—41no—
43C2×C4×Dic3—41no—
44C4×SL(2,𝔽3)—21no—
45#859—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×#41C2×D311yescharacteristic
55C23×C8—31no—
56#550—22no—
57#550—62no—
58C2×#127—31no—
59C2×#121—31no—
60C2×#121—31no—
61C2×C4×Q8C2×D311yescharacteristic
62C2×C4×D4—31no—
63C2×#117—31no—
64#482—64no—
65C2×#107—31no—
66C2×#106—31no—
67#416—62no—
68#416—62no—
69C2×#103—31no—
70C2×#103—31no—
71C2×#103—31no—
72C2×#103—31no—
73C2×#102—31no—
74C2×#102—31no—
75C2×#102—31no—
76C2×#102—31no—
77C2×#102—31no—
78C2×#102—31no—
79C2×#102—31no—
80C2×#102—31no—
81C2×C22⋊C8—31no—
82C2×C22⋊C8—31no—
83C2×C22⋊C8—31no—
84#397—61no—
85#394—61no—
86#389—31no—
87#389—31no—
88#388—61no—
89#387—61no—
90#380—31no—
91#380—31no—
92#380—31no—
93#340—31no—
94#340—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—
105#41⋊C3—41no—
106C2×SL(2,𝔽3)C2×D411yescharacteristic
107C2×SL(2,𝔽3)#4111yescharacteristic
108C2×SL(2,𝔽3)#4111yescharacteristic
109C3×C4⋊C4—82no—
110Dic3⋊C4—44no—
111C4×Dic3—82no—
112C2×#41—21no—
113C2×#41—21no—
114C2×#41—21no—
115C2×#41—21no—
116C2×#41—21no—
117C2×#41—32no—
118C2×#41—61no—
119C2×#41—61no—
120C2×#41—61no—
121C2×#41—61no—
122C2×#41—61no—
123C2×#41#6712yes—
124C22×Q8C22×D311yescharacteristic
125C22×D4—31no—
126C22×D4—31no—
127C22×D4—31no—
128C22×D4—61no—
129C23×C4—31no—
130C23×C4—31no—
131C22×C8—31no—
132C22×C8—31no—
133C22×C8—31no—
134C22×C8—31no—
135C22×C8—62no—
136C22×C8—121no—
137C22×C8—121no—
138#127—62no—
139#127—62no—
140#121—34no—
141#121—62no—
142#121—62no—
143#121—62no—
144C4×Q8—21no—
145C4×Q8—61no—
146C4×D4—62no—
147C4×D4—62no—
148#117—62no—
149#117—62no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4S411yescharacteristic
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C42—31no—
165C2×C42—61no—
166#107—62no—
167#106—62no—
168#103—61no—
169#103—61no—
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—
179#102—62no—
180#102—121no—
181#102—121no—
182#102—121no—
183#102—121no—
184C22⋊C8—62no—
185C22⋊C8—121no—
186C22⋊C8—121no—
187#95—61no—
188#95—61no—
189#95—61no—
190#95—61no—
191#95—61no—
192C23×C3—41no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—81no—
196C2×C3×C4—81no—
197C2×C3×C4—81no—
198C2×C3×C4—81no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201C2×C3×C4—81no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204C2×Dic3—42no—
205C2×Dic3—81no—
206C2×Dic3—81no—
207C2×Dic3—81no—
208C2×Dic3—81no—
209C2×Dic3—81no—
210C2×Dic3—81no—
211C2×Dic3—81no—
212C2×Dic3—81no—
213C2×Dic3—81no—
214SL(2,𝔽3)#12311yescharacteristic
215C24—31no—
216C24—31no—
217#41—22no—
218#41—41no—
219#41—62no—
220#41—62no—
221#41—62no—
222#41—122no—
223#41—122no—
224#41—121no—
225#41—121no—
226#41—121no—
227C2×Q8—31no—
228C2×Q8—31no—
229C2×Q8—31no—
230C2×Q8#23711yescharacteristic
231C2×Q8#23711yescharacteristic
232C2×Q8C2×#6711yescharacteristic
233C2×D4—32no—
234C2×D4—32no—
235C2×D4—62no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—64no—
247C2×D4—62no—
248C2×D4—121no—
249C2×D4—241no—
250C2×D4—241no—
251C22×C4—31no—
252C22×C4—32no—
253C22×C4—32no—
254C22×C4—31no—
255C22×C4—61no—
256C22×C4—61no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4C2×S411yescharacteristic
270C22×C4C2×S411yescharacteristic
271C22×C4C2×S411yescharacteristic
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—
284C2×C8—124no—
285C4⋊C4—22no—
286C4⋊C4—32no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—122no—
297C4⋊C4—241no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—122no—
301C22⋊C4—122no—
302C22⋊C4—241no—
303C22⋊C4—241no—
304C42—61no—
305C42—61no—
306C42—122no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C3×C4—82no—
315C3×C4—82no—
316C3×C4—161no—
317Dic3—82no—
318Dic3—82no—
319Dic3—82no—
320Dic3—161no—
321C23—32no—
322C23—61no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23—61no—
327C23—61no—
328C23—61no—
329C23—61no—
330C23—61no—
331C23—61no—
332C23—61no—
333C23—61no—
334C23—61no—
335C23C22×S411yescharacteristic
336Q8—31no—
337Q8—31no—
338Q8—31no—
339Q8—61no—
340Q8#93511yescharacteristic
341D4—62no—
342D4—62no—
343D4—62no—
344D4—62no—
345D4—124no—
346D4—121no—
347D4—121no—
348D4—121no—
349D4—121no—
350D4—244no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—31no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4#98512yes—
388C2×C4GL(2,ℤ/4)12yes—
389C8—121no—
390C8—121no—
391C8—121no—
392C8—121no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C22—62no—
401C22—62no—
402C22—62no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22#591411yescharacteristic
415C22#591411yescharacteristic
416C22C2×#98511yescharacteristic
417C22#592311yescharacteristic
418C22#592411yescharacteristic
419C22#592511yescharacteristic
420C22C2×GL(2,ℤ/4)11yescharacteristic
421C4—22no—
422C4—22no—
423C4—31no—
424C4—31no—
425C4—31no—
426C4—31no—
427C4—41no—
428C4—121no—
429C4—241no—
430C4—241no—
431C4—241no—
432C4—241no—
433C3—41noSylow
434C2—122no—
435C2—241no—
436C2#8540311yescharacteristic
437C2#8540511yescharacteristic
438C2#8540811yescharacteristic
439C2#8541311yescharacteristic
440C2#8549111yescharacteristic
441C2#8552111yescharacteristic
442C2#8552911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#550D3
prime radicalC2×C4⋊(#41⋊C3)C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#5424C2semidirect
3. #73151C2semidirect
4. #73151C2semidirect
5. #73151C2semidirect
6. #73062C2semidirect
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⋊C3)2
C2×#54242
#731512
#730622
#730592
#240343
#49784