Finite Group Databasebeta

← #10691753

Subgroups — #10691753

Order 768.

522 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2368C211yesmaximal, characteristic
2#82741C211yesmaximal, characteristic
3#71690C211yesmaximal, characteristic
4#69457C211yesmaximal, characteristic
5C2×#4452C211yesmaximal, characteristic
6#67617C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#9276—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C3×#391—21no—
12C3×#387—21no—
13C3×#387—21no—
14C3×#385—21no—
15C3×C4×C4⋊C4—21no—
16C3×C4×C22⋊C4—21no—
17C3×C4×C22⋊C4C2211yescharacteristic
18C2×C3×#95C2211yescharacteristic
19#5228—21no—
20#5212C2211yescharacteristic
21C2×#830C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#5051—22no—
24#5050C412yescomplemented
25#5044C412yescomplemented
26#4970—21no—
27#4538—22no—
28#4527—21no—
29#4527—21no—
30#4452—21no—
31#4452—21no—
32#3189—31no—
33#2368D311yescharacteristic
34#1996—31no—
35#1520—31no—
36C2×#328—31no—
37#1294—31no—
38#1282—31no—
39C22×C3×Q8D411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2311yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8C2×C412yes—
46C3×C22⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C22⋊C4D411yescharacteristic
54C2×C3×C42—21no—
55C2×C3×C42—21no—
56C2×C3×C42—21no—
57C2×#217—21no—
58C2×#217—21no—
59C2×#216—22no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3C2×C411yescharacteristic
62#924—21no—
63#924—21no—
64#924—21no—
65#924—21no—
66C2×Dic3⋊C4—21no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95D411yescharacteristic
76C3×#95D411yescharacteristic
77#835—22no—
78C3⋊#95—21no—
79#830—21no—
80#830—21no—
81#830—21no—
82#830C2×C412yes—
83#810—22no—
84#807—22no—
85C2×C22⋊Q8—31no—
86C2×C22⋊Q8C2×D311yescharacteristic
87#484—32no—
88#484—61no—
89#467—62no—
90#451—32no—
91#451—61no—
92C2×C22⋊C8—31no—
93C2×C22⋊C8—31no—
94#391—21no—
95#387—21no—
96#387—21no—
97#385—21no—
98#380—31no—
99C4×C4⋊C4—21no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4C2×D311yescharacteristic
102C2×#95C2×D311yescharacteristic
103#342—61no—
104#342—61no—
105#328—62no—
106#328—61no—
107#328—61no—
108C24×C3C2×D411yescharacteristic
109C2×C3×Q8—21no—
110C2×C3×Q8—81no—
111C2×C3×Q8C22⋊C412yes—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—42no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—81no—
122C22×C3×C4—81no—
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—41no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—81no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—42no—
139C3×C4⋊C4—81no—
140C3×C4⋊C4C22⋊C412yes—
141C3×C22⋊C4—22no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4Q1612yescomplemented
147C3×C42—42no—
148C3×C42—81no—
149C3×C42—81no—
150#217—41no—
151#217—41no—
152#217—41no—
153#217—41no—
154#216—82no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—81no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—81no—
163C2×C3⋊C8—81no—
164C22×Q8—31no—
165C22×Q8#6711yescharacteristic
166C23×C4—21no—
167C23×C4C22×D311yescharacteristic
168C2×Q16—62no—
169C22×C8—61no—
170C22×C8—61no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C22⋊Q8C4×D312yes—
178C22⋊Q8D1212yes—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4#6711yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4#6711yescharacteristic
190C2×C22⋊C4#6711yescharacteristic
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194#107—62no—
195#103—62no—
196#103—62no—
197C22⋊C8—32no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—61no—
208#95#6711yescharacteristic
209#95#6711yescharacteristic
210C23×C3—41no—
211C23×C3—41no—
212C23×C3C4×D411yescharacteristic
213C23×C3#12011yescharacteristic
214C23×C3#12111yescharacteristic
215C23×C3C2×Q1611yescharacteristic
216C23×C3#13511yescharacteristic
217C23×C3#13711yescharacteristic
218C23×C3#13711yescharacteristic
219C3×Q8—42no—
220C3×Q8—81no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—82no—
246C2×C3×C4—82no—
247C2×C3×C4C2×C22⋊C411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—81no—
254C2×Dic3—81no—
255C2×Dic3—81no—
256C2×Dic3—81no—
257C2×Dic3—81no—
258Dic6—42no—
259Dic6—81no—
260C3⋊C8—42no—
261C3⋊C8—81no—
262C24C2×#6711yescharacteristic
263C2×Q8—21no—
264C2×Q8—61no—
265C2×Q8—61no—
266C2×Q8—81no—
267C2×Q8—241no—
268C2×Q8(C2×D3)⋊C412yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—121no—
283C22×C4C2×C4×D311yescharacteristic
284C22×C4C2×D1211yescharacteristic
285C22×C4#23511yescharacteristic
286C22×C4#23511yescharacteristic
287C22×C4C2×#6711yescharacteristic
288C22×C4C2×#6711yescharacteristic
289Q16—242no—
290C2×C8—62no—
291C2×C8—121no—
292C2×C8—241no—
293C2×C8—241no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—81no—
301C4⋊C4—121no—
302C4⋊C4—121no—
303C4⋊C4—241no—
304C4⋊C4(C2×D3)⋊C412yes—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4#21612yescomplemented
315C42—42no—
316C42—81no—
317C42—81no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3#38611yescharacteristic
329C22×C3C4×Q1611yescharacteristic
330C22×C3#44111yescharacteristic
331C22×C3#45111yescharacteristic
332C22×C3#45211yescharacteristic
333C22×C3#46211yescharacteristic
334C22×C3#46411yescharacteristic
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4—42no—
342C3×C4—42no—
343C3×C4—81no—
344C3×C4—81no—
345Dic3—81no—
346Dic3—81no—
347Dic3—161no—
348C23—41no—
349C23—41no—
350C23C4×#6711yescharacteristic
351C23#93011yescharacteristic
352C23#94211yescharacteristic
353C23C2×#21611yescharacteristic
354C23#95011yescharacteristic
355C23#95111yescharacteristic
356C23#95311yescharacteristic
357Q8—42no—
358Q8—81no—
359Q8—122no—
360Q8—241no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4—241no—
394C2×C4C2×(C2×D3)⋊C411yescharacteristic
395C2×C4C4×#6711yescharacteristic
396C2×C4#92911yescharacteristic
397C2×C4#93011yescharacteristic
398C8—122no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3#187811yescharacteristic
407C2×C3#190611yescharacteristic
408C2×C3#191311yescharacteristic
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22#502811yescharacteristic
420C22C4×#21611yescharacteristic
421C22#503011yescharacteristic
422C22#503111yescharacteristic
423C22#521311yescharacteristic
424C22#522811yescharacteristic
425C22#523911yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—81no—
435C4—81no—
436C4—241no—
437C4—241no—
438C4—481no—
439C3#927611yescharacteristic, complemented, Sylow
440C2—21no—
441C2—21no—
442C2—21no—
443C2—21no—
444C2—21no—
445C2—21no—
446C2#7163211yescharacteristic
447C2#7164511yescharacteristic
448C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2368C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2368D3
hypocenterC3#9276
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2368

Complements

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

NH ≅ G/Ntype
24. #5050C4semidirect
25. #5044C4semidirect
146. C3×C22⋊C4Q16semidirect
314. C22⋊C4#216semidirect
439. C3#9276semidirect

Maximal subgroups

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

subgroupindex
C3×#23682
#827412
#716902
#694572
C2×#44522
#676172
#676122
#92763