Finite Group Databasebeta

← #10691252

Subgroups — #10691252

Order 768.

622 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 113 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#82675C211yesmaximal, characteristic
2#73869C211yesmaximal, characteristic, complemented
3C3×#1519C211yesmaximal, characteristic
4#71675C211yesmaximal, characteristic
5C2×#4469C211yesmaximal, characteristic, complemented
6#67676C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9204—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×#935—21no—
11C22×Dic3⋊C4C2211yescharacteristic
12C2×#877—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15#5222C412yescomplemented
16#5220—21no—
17#5212C2211yescharacteristic
18#5212C2211yescharacteristic
19C2×#830C2211yescharacteristic
20#4983C412yescomplemented
21#4976—22no—
22#4948—22no—
23#4942—21no—
24#4647—22no—
25C3×#327—22no—
26#4537—22no—
27#4469—22no—
28#2618—31no—
29#2316—31no—
30#1519—31no—
31#1519D311yescharacteristic
32C2×#324—31no—
33#1289—31no—
34#1289—31no—
35C22×C3×D4—21no—
36C22×C3×D4C2×C411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3D411yescharacteristic
39C23×Dic3D411yescharacteristic
40C22×C3×C8—21no—
41C3×#121—21no—
42C3×#121—21no—
43C3×#121—21no—
44C3×#121—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—22no—
52C2×#217—22no—
53#935—42no—
54#935—42no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—22no—
62C2×Dic3⋊C4—22no—
63C2×Dic3⋊C4—22no—
64C2×Dic3⋊C4C2×C412yes—
65C2×Dic3⋊C4D412yes—
66C2×C4×Dic3—22no—
67C22×C3⋊C8—21no—
68#877—42no—
69#877—42no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95D412yes—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95C2×C412yes—
76C3⋊#95D412yes—
77#830—21no—
78#830—21no—
79C2×#123—61no—
80C2×#121C2×D311yescharacteristic
81C2×#117—61no—
82C22×C4⋊C4—31no—
83C2×C22⋊C8—31no—
84C2×C22⋊C8C2×D311yescharacteristic
85#398—32no—
86#396—62no—
87#394—32no—
88#389—62no—
89#389—62no—
90#386—61no—
91#380—31no—
92#380—31no—
93#380—61no—
94#380C2×D311yescharacteristic
95#327—22no—
96#327—62no—
97#324—62no—
98C24×C3—21no—
99C24×C3C2×D411yescharacteristic
100C2×C3×D4—21no—
101C2×C3×D4—21no—
102C2×C3×D4—41no—
103C2×C3×D4—41no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—81no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—81no—
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—21no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—22no—
125C22×Dic3—22no—
126C22×Dic3—81no—
127C22×Dic3C22⋊C412yes—
128C22×Dic3C22⋊C412yes—
129C22×Dic3D812yes—
130C22×Dic3SD1612yes—
131C2×C3×C8—41no—
132C2×C3×C8—81no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—81no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140#217—41no—
141#217—41no—
142#217—41no—
143#217—41no—
144#217—44no—
145#217—82no—
146C4⋊Dic3—81no—
147C4⋊Dic3—81no—
148Dic3⋊C4—22no—
149Dic3⋊C4—22no—
150Dic3⋊C4—22no—
151Dic3⋊C4—22no—
152Dic3⋊C4—44no—
153Dic3⋊C4—44no—
154Dic3⋊C4—82no—
155C4×Dic3—44no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—81no—
158C22×D4—21no—
159C22×D4C4×D311yescharacteristic
160C23×C4—31no—
161C23×C4—31no—
162C23×C4C22×D311yescharacteristic
163C22×C8—21no—
164C22×C8—61no—
165#123—122no—
166#123—122no—
167#121—21no—
168#121—21no—
169#121—21no—
170#121—21no—
171#117—122no—
172#117—122no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—32no—
175C2×C4⋊C4—32no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—62no—
181C2×C4⋊C4—62no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4C4×D311yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—62no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—62no—
192C2×C42—62no—
193C22⋊C8—21no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197#95—32no—
198#95—32no—
199#95—62no—
200#95—62no—
201#95#6712yes—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—42no—
205C23×C3—42no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3C2×C22⋊C411yescharacteristic
210C23×C3#12011yescharacteristic
211C23×C3#12311yescharacteristic
212C23×C3C2×D811yescharacteristic
213C23×C3C2×SD1611yescharacteristic
214C23×C3AGL(1,ℤ/8)11yescharacteristic
215C23×C3#13711yescharacteristic
216C3×D4—81no—
217C3×D4—81no—
218C3×D4—81no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4—81no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—22no—
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—81no—
254C2×Dic3#10214yes—
255C3×C8—81no—
256C3⋊C8—81no—
257C24—21no—
258C24D3×D411yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—81no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—32no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—32no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—81no—
283C22×C4—241no—
284C22×C4C2×C4×D311yescharacteristic
285C22×C4#23511yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288C22×C4#23711yescharacteristic
289C22×C4C2×#6711yescharacteristic
290C2×C8—41no—
291C2×C8—81no—
292C2×C8—121no—
293C2×C8—241no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—62no—
297C4⋊C4—62no—
298C4⋊C4—62no—
299C4⋊C4—62no—
300C4⋊C4—81no—
301C4⋊C4—124no—
302C4⋊C4—124no—
303C4⋊C4—242no—
304C4⋊C4—241no—
305C4⋊C4—241no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—124no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C22⋊C4—242no—
316C42—124no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—82no—
328C22×C3—82no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3#38611yescharacteristic
333C22×C3C2×#10211yescharacteristic
334C22×C3#41711yescharacteristic
335C22×C3#44711yescharacteristic
336C22×C3#44811yescharacteristic
337C22×C3#48011yescharacteristic
338C22×C3#48111yescharacteristic
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—82no—
344C3×C4—81no—
345Dic3—24no—
346Dic3—42no—
347Dic3—82no—
348Dic3—82no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23D3×C22⋊C411yescharacteristic
357C23#88311yescharacteristic
358C23D3×D811yescharacteristic
359C23#91111yescharacteristic
360C23D3×SD1611yescharacteristic
361C23#91511yescharacteristic
362C23#93711yescharacteristic
363D4—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369C2×C4—34no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—241no—
398C2×C4#67⋊C411yescharacteristic
399C2×C4#88411yescharacteristic
400C2×C4C4×#6711yescharacteristic
401C2×C4#93811yescharacteristic
402C8—81no—
403C8—241no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—161no—
411C2×C3#187411yescharacteristic
412C2×C3#191211yescharacteristic
413C2×C3#191311yescharacteristic
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22D3×#10211yescharacteristic
430C22#477111yescharacteristic
431C22#477411yescharacteristic
432C22#477711yescharacteristic
433C22#495611yescharacteristic
434C22#515611yescharacteristic
435C22#517011yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—64no—
441C4—82no—
442C4—81no—
443C4—122no—
444C4—242no—
445C4—242no—
446C3#920411yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—161no—
454C2#7003411yescharacteristic
455C2#7020011yescharacteristic
456C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1519D3
hypocenterC3#9204
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #73869C2semidirect
5. C2×#4469C2semidirect
6. #67676C2semidirect
7. #67625C2semidirect
15. #5222C4semidirect
20. #4983C4semidirect
446. C3#9204semidirect

Maximal subgroups

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

subgroupindex
#826752
#738692
C3×#15192
#716752
C2×#44692
#676762
#676252
#92043