Finite Group Databasebeta

← #10692955

Subgroups — #10692955

Order 768.

578 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78601C211yesmaximal, characteristic
2#74028C211yesmaximal, characteristic
3#69925C211yesmaximal, characteristic, complemented
4#69530C211yesmaximal, characteristic
5#69502C211yesmaximal, characteristic, complemented
6C3×#1289C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9469—31nomaximal, Sylow
9C2×C4×#67C2211yescharacteristic
10C22×C4⋊Dic3C2211yescharacteristic
11C2×C3×C22⋊C8C2211yescharacteristic
12C3×#380C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5212C2211yescharacteristic
15C2×#830C2211yescharacteristic
16C8×#67—22no—
17#4987—21no—
18#4984—21no—
19#4958—21no—
20C2×#806—21no—
21#4950—21no—
22#67⋊C8—22no—
23D3×C22⋊C8—21no—
24#4718—21no—
25#4677C412yescomplemented
26#4659C412yescomplemented
27#4551—22no—
28#4529—22no—
29#4527—21no—
30#2972—31no—
31#2487—31no—
32#1824—31no—
33#1824—31no—
34#1799—31no—
35#1289—31no—
36#1289D311yescharacteristic, complemented
37C23×C3×C4C2311yescharacteristic
38C22×#67D411yescharacteristic
39C23×Dic3—21no—
40C22×C4×D3D411yescharacteristic
41C22×C3×C8D412yes—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×#217—21no—
47C2×#217D411yescharacteristic
48C4×#67—21no—
49C4×#67—21no—
50C4×#67—21no—
51C4×#67—21no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4C2×C411yescharacteristic
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—22no—
61C2×C4⋊Dic3C2×C412yes—
62C2×Dic3⋊C4C2×C411yescharacteristic
63C2×C4×Dic3D411yescharacteristic
64C22×C3⋊C8D412yes—
65C2×D3×C8—22no—
66C3×C22⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#95—21no—
69C3×#95C2×C412yes—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—21no—
73#830—21no—
74#830—21no—
75(C2×D3)⋊C8—22no—
76Dic3⋊C8—22no—
77C8×Dic3—22no—
78#806—42no—
79C2×C4×D4—31no—
80C22×C4⋊C4—31no—
81C8×D4—62no—
82C8×D4—62no—
83#431—61no—
84C2×#99—61no—
85C2×C22⋊C8—31no—
86C2×C22⋊C8—61no—
87C2×C22⋊C8C2×D311yescharacteristic
88#400—32no—
89#399—32no—
90#389—61no—
91#386—61no—
92#385—61no—
93#380—31no—
94#380—61no—
95#380C2×D311yescharacteristic
96#380C2×D311yescharacteristic
97#342—61no—
98#340—62no—
99#340—62no—
100C24×C3C2×D411yescharacteristic
101C23×D3C22⋊C411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4—42no—
104C22×C3×C4—41no—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×#67—21no—
112C2×#67—21no—
113C2×#67—21no—
114C2×#67—21no—
115C2×#67SD1612yes—
116C2×#67Q1612yes—
117C22×Dic3—22no—
118C22×Dic3—41no—
119C22×Dic3—41no—
120C22×Dic3—41no—
121C22×Dic3—82no—
122C22×Dic3C22⋊C411yescharacteristic
123C22×Dic3C22⋊C411yescharacteristic
124C22×Dic3C22⋊C411yescharacteristic
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—42no—
128C2×C3×C8—22no—
129C2×C3×C8—42no—
130C3×C4⋊C4—42no—
131C3×C4⋊C4—42no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136#217—21no—
137#217—21no—
138#217—41no—
139#217—41no—
140(C2×D3)⋊C4—21no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—82no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—41no—
146C4⋊Dic3—41no—
147C4⋊Dic3—42no—
148C4⋊Dic3—82no—
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—42no—
155D3×C8—82no—
156C22×D4—31no—
157C23×C4—31no—
158C23×C4—61no—
159C23×C4C22×D311yescharacteristic
160C22×C8—32no—
161C22×C8—62no—
162C22×C8D1212yes—
163C4×D4—61no—
164C4×D4—61no—
165C4×D4—61no—
166C4×D4—61no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—32no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—62no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—31no—
181C2×C22⋊C4—31no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—62no—
184C2×C42—31no—
185C4⋊C8—62no—
186#99—122no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—62no—
190C22⋊C8—61no—
191C22⋊C8—61no—
192C4×C8—62no—
193#95—21no—
194#95—62no—
195#95—62no—
196#95—61no—
197#95C4×D312yes—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3C2×C22⋊C411yescharacteristic
201C23×C3#12111yescharacteristic
202C23×C3#12111yescharacteristic
203C23×C3C2×SD1611yescharacteristic
204C23×C3C2×Q1611yescharacteristic
205C23×C3#13511yescharacteristic
206C23×C3#13511yescharacteristic
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—41no—
210C22×D3—41no—
211C22×D3#10312yes—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—81no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12311yescharacteristic
228#67—22no—
229#67—22no—
230#67—22no—
231#67—22no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—82no—
242C2×Dic3—82no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246C2×Dic3#10312yes—
247C4×D3—42no—
248C4×D3—81no—
249C3×C8—42no—
250C3⋊C8—42no—
251C24—31no—
252C24D3×D411yescharacteristic
253C2×D4—32no—
254C2×D4—32no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C22×C4—22no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—62no—
268C22×C4—122no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—242no—
273C22×C4C2×C4×D311yescharacteristic
274C22×C4C2×D1211yescharacteristic
275C22×C4#23511yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4#23911yescharacteristic
279C2×C8—22no—
280C2×C8—42no—
281C2×C8—62no—
282C2×C8—122no—
283C2×C8—242no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C4⋊C4—122no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C4⋊C4—242no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—242no—
305C42—61no—
306C42—61no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3#38611yescharacteristic
318C22×C3C2×#10311yescharacteristic
319C22×C3#41611yescharacteristic
320C22×C3#46511yescharacteristic
321C22×C3#46511yescharacteristic
322C22×C3#46711yescharacteristic
323C22×C3#46711yescharacteristic
324C2×D3—22no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—42no—
328C2×D3—81no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—82no—
333C3×C4—161no—
334C3×C4#45812yes—
335Dic3—22no—
336Dic3—41no—
337Dic3—41no—
338Dic3—82no—
339Dic3—161no—
340C23—32no—
341C23—41no—
342C23—41no—
343C23—61no—
344C23—61no—
345C23—121no—
346C23—121no—
347C23D3×C22⋊C411yescharacteristic
348C23#88411yescharacteristic
349C23#89511yescharacteristic
350C23#91211yescharacteristic
351C23D3×SD1611yescharacteristic
352C23#91611yescharacteristic
353C23D3×Q1611yescharacteristic
354D4—62no—
355D4—62no—
356D4—62no—
357D4—62no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—32no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—241no—
382C2×C4—242no—
383C2×C4—242no—
384C2×C4—241no—
385C2×C4—241no—
386C2×C4—241no—
387C2×C4C4×D1211yescharacteristic
388C2×C4#88211yescharacteristic
389C2×C4#89311yescharacteristic
390C2×C4#89411yescharacteristic
391C8—42no—
392C8—122no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3#189011yescharacteristic
400C2×C3#189211yescharacteristic
401C2×C3#191011yescharacteristic
402D3—42no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—62no—
414C22—122no—
415C22—121no—
416C22—121no—
417C22—241no—
418C22#466511yescharacteristic
419C22#477211yescharacteristic
420C22#477811yescharacteristic
421C22D3×#10311yescharacteristic
422C22#481311yescharacteristic
423C22#486411yescharacteristic
424C22#488711yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—62no—
429C4—82no—
430C4—121no—
431C4—121no—
432C4—161no—
433C4—242no—
434C4—481no—
435C4#474212yes—
436C3#946911yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—122no—
444C2#6871311yescharacteristic
445C2#6888211yescharacteristic
446C2#6896311yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#386
prime radicalC22×#67D4
cosocleC23×C3×C4C23
hypercenter#1289D3
hypocenterC3#9469
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
3. #69925C2semidirect
5. #69502C2semidirect
6. C3×#1289C2semidirect
7. #67625C2semidirect
25. #4677C4semidirect
26. #4659C4semidirect
36. #1289D3semidirect
436. C3#9469semidirect

Maximal subgroups

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

subgroupindex
#786012
#740282
#699252
#695302
#695022
C3×#12892
#676252
#94693