Finite Group Databasebeta

← #10694726

Subgroups — #10694726

Order 768.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 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×#5048C211yesmaximal, characteristic
2C3×#1996C211yesmaximal, characteristic, complemented
3#71690C211yesmaximal, characteristic, complemented
4#69512C211yesmaximal, characteristic, complemented
5#69509C211yesmaximal, characteristic
6#69456C211yesmaximal, characteristic
7#69447C211yesmaximal, characteristic, complemented
8#9786—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×#215C411yescharacteristic, complemented
11C2×#930C2211yescharacteristic
12C23×C3⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C3×C4×C22⋊C4C2211yescharacteristic
15C2×#835C411yescharacteristic, complemented
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#5048—21no—
19#5048—21no—
20#5048—21no—
21#5048—21no—
22C2×#809C411yescharacteristic, complemented
23C2×#808C411yescharacteristic, complemented
24C2×#807—21no—
25C2×#806—22no—
26C2×#803—21no—
27C3×#342—22no—
28#4538—22no—
29#4529—21no—
30#4524—21no—
31#4451—22no—
32C2×#465—31no—
33#1996D311yescharacteristic, complemented
34#1964—31no—
35#1808—31no—
36#1752—31no—
37#1520—31no—
38#1519—31no—
39C22×C3×Q8C2×C411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C22×#67—21no—
42C22×D12C2×C411yescharacteristic
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4D412yes—
53C2×C3×C42—21no—
54C2×#217—21no—
55C2×#215—21no—
56C2×#215—21no—
57C2×#215—42no—
58#930—21no—
59#930—21no—
60#930—21no—
61#930—21no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3C2×C411yescharacteristic
65C22×C3⋊C8—21no—
66C22×C3⋊C8—42no—
67C22×C3⋊C8C2×C411yescharacteristic
68C22×C3⋊C8C2×C411yescharacteristic
69C22×C3⋊C8D411yescharacteristic
70C22×C3⋊C8D411yescharacteristic
71C22×C3⋊C8D411yescharacteristic
72C22×C3⋊C8D411yescharacteristic
73C3×#95—21no—
74C3×#95—21no—
75#835—21no—
76#835—21no—
77C3⋊#95—21no—
78#830—21no—
79#830D412yes—
80#809—21no—
81#809—21no—
82#808—21no—
83#808—21no—
84#807—42no—
85#806—44no—
86#803—42no—
87C22×SD16—31no—
88C23×C8—31no—
89C2×C22⋊Q8C2×D311yescharacteristic
90C2×#121—31no—
91#465—61no—
92#465—61no—
93#465—61no—
94#465—61no—
95C2×#107—61no—
96C2×#106—31no—
97C2×#103—31no—
98C2×#102—31no—
99C2×#99—62no—
100C2×C22⋊C8—31no—
101C2×#97—61no—
102#380—31no—
103#380C2×D311yescharacteristic
104C4×C22⋊C4C2×D311yescharacteristic
105#342—22no—
106#340—61no—
107#336—61no—
108#328—62no—
109#327—62no—
110C24×C3C2×D411yescharacteristic
111C23×D3—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—41no—
115C2×C3×Q8—41no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C22×C411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—41no—
129C2×#67—41no—
130C2×#67—42no—
131C2×#67—42no—
132C22×Dic3—21no—
133C22×Dic3—41no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—41no—
137C2×D12—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—22no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—42no—
149C3×C42—41no—
150C3×C42—41no—
151#217—42no—
152#215—42no—
153#215—42no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168C2×C3⋊C8—41no—
169C2×C3⋊C8C22⋊C411yescharacteristic
170C2×C3⋊C8C22⋊C411yescharacteristic
171C2×C3⋊C8C22⋊C411yescharacteristic
172C2×C3⋊C8C22⋊C411yescharacteristic
173C22×Q8C4×D311yescharacteristic
174C22×D4—31no—
175C22×D4—61no—
176C23×C4C22×D311yescharacteristic
177C2×SD16—61no—
178C2×SD16—61no—
179C2×SD16—122no—
180C22×C8—31no—
181C22×C8—31no—
182C22×C8—31no—
183C22×C8—31no—
184C22×C8—31no—
185C22×C8—31no—
186C22×C8—61no—
187C22×C8—122no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192#121—61no—
193#121—61no—
194#121—61no—
195#121—61no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4C4×D311yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4#6712yes—
206C2×C42—21no—
207#107—122no—
208#106—61no—
209#106—61no—
210#103—61no—
211#103—61no—
212#102—61no—
213#102—61no—
214#99—124no—
215C22⋊C8—32no—
216C22⋊C8—61no—
217#97—122no—
218#95—21no—
219#95—21no—
220#95—61no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3C4×D411yescharacteristic
224C23×C3#12111yescharacteristic
225C23×C3#12711yescharacteristic
226C23×C3AGL(1,ℤ/8)12yes—
227C23×C3#13712yes—
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—81no—
231C22×D3—81no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×Q8—81no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4C2×C22⋊C411yescharacteristic
255C2×C3×C4C4×D411yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12411yescharacteristic
258#67—82no—
259#67—81no—
260#67—81no—
261C2×Dic3—41no—
262C2×Dic3—42no—
263C2×Dic3—81no—
264C2×Dic3—81no—
265C2×Dic3—81no—
266D12—41no—
267D12—41no—
268D12—81no—
269C3⋊C8—21no—
270C3⋊C8—21no—
271C3⋊C8—21no—
272C3⋊C8—42no—
273C3⋊C8#9912yescomplemented
274C24—61no—
275C24C2×#6711yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—122no—
287C2×D4—122no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—61no—
295C22×C4—121no—
296C22×C4C2×C4×D311yescharacteristic
297C22×C4#23511yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4#23711yescharacteristic
302SD16—122no—
303SD16—122no—
304C2×C8—31no—
305C2×C8—31no—
306C2×C8—31no—
307C2×C8—31no—
308C2×C8—62no—
309C2×C8—62no—
310C2×C8—62no—
311C2×C8—62no—
312C2×C8—121no—
313C2×C8—121no—
314C2×C8—121no—
315C2×C8—121no—
316C2×C8—121no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—121no—
326C4⋊C4—121no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—42no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4—122no—
335C42—41no—
336C42—41no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3#39011yescharacteristic
346C22×C3#44012yes—
347C22×C3#46812yes—
348C22×C3#49611yescharacteristic
349C22×C3#49711yescharacteristic
350C2×D3—81no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—162no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—82no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4—161no—
361C3×C4C2×#9911yescharacteristic
362C3×C4#41011yescharacteristic
363Dic3—81no—
364Dic3—161no—
365C23—41no—
366C23—41no—
367C23—121no—
368C23—122no—
369C23—241no—
370C23—241no—
371C23C4×#6711yescharacteristic
372C23#93911yescharacteristic
373C23#94011yescharacteristic
374C23#94212yes—
375C23#94912yes—
376Q8—41no—
377Q8—41no—
378Q8—81no—
379D4—121no—
380D4—121no—
381D4—241no—
382D4—242no—
383D4—241no—
384D4—241no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—121no—
405C2×C4—122no—
406C2×C4—241no—
407C2×C4—241no—
408C2×C4—241no—
409C2×C4D3×C22⋊C411yescharacteristic
410C2×C4#67⋊C411yescharacteristic
411C2×C4#88411yescharacteristic
412C2×C4#88511yescharacteristic
413C8—32no—
414C8—61no—
415C8—61no—
416C8—61no—
417C8—122no—
418C2×C3—21no—
419C2×C3—22no—
420C2×C3—22no—
421C2×C3—21no—
422C2×C3#196411yescharacteristic
423C2×C3#197311yescharacteristic
424C2×C3#197611yescharacteristic
425D3—161no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—42no—
433C22—42no—
434C22—241no—
435C22—241no—
436C22—241no—
437C22—482no—
438C22#495711yescharacteristic
439C22#502612yes—
440C22#504912yes—
441C22#505811yescharacteristic
442C22#508611yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—82no—
447C4—81no—
448C4—81no—
449C4—161no—
450C4—241no—
451C4—481no—
452C4#474011yescharacteristic
453C4D3×#9911yescharacteristic
454C3#978611yescharacteristic, complemented, Sylow
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—481no—
460C2#6967911yescharacteristic
461C2#7003011yescharacteristic
462C2#7004211yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1996C2
socleC22×C3#390
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1996D3
hypocenterC3#9786
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
2. C3×#1996C2semidirect
3. #71690C2semidirect
4. #69512C2semidirect
7. #69447C2semidirect
10. C22×#215C4semidirect
15. C2×#835C4semidirect
22. C2×#809C4semidirect
23. C2×#808C4semidirect
33. #1996D3semidirect
273. C3⋊C8#99semidirect
454. C3#9786semidirect

Maximal subgroups

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

subgroupindex
C2×#50482
C3×#19962
#716902
#695122
#695092
#694562
#694472
#97863