Finite Group Databasebeta

← #10690250

Subgroups — #10690250

Order 768.

696 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74004C211yesmaximal, characteristic
2C2×#4674C211yesmaximal, characteristic
3#69924C211yesmaximal, characteristic, complemented
4C2×#4468C211yesmaximal, characteristic
5#69501C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9028—31nomaximal, Sylow
9C22×(C2×D3)⋊C4—21no—
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C22×C4⋊Dic3C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5212C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×#830C2211yescharacteristic
18#4990—22no—
19#4955—21no—
20Dic3×C22⋊C4—21no—
21#4674—22no—
22#4674C414yescomplemented
23#4670—22no—
24#4551—22no—
25#4529—22no—
26#4468—22no—
27C2×#397—31no—
28#2312—31no—
29#1823—31no—
30#1823—31no—
31C2×#323—31no—
32#1294—31no—
33#1294D311yescharacteristic, complemented
34C24×D3D411yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—21no—
38C23×Dic3D411yescharacteristic
39C22×C3×C8—21no—
40C2×C3×C4⋊C4D412yes—
41C2×C3×C22⋊C4—21no—
42C2×#217—21no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—24no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—42no—
51C2×(C2×D3)⋊C4—42no—
52C2×(C2×D3)⋊C4C2×C412yes—
53C2×(C2×D3)⋊C4D412yes—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3C2×C412yes—
60C2×C4⋊Dic3D412yes—
61C2×C4×Dic3—22no—
62C22×C3⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95C2×C412yes—
68C3⋊#95—22no—
69C3⋊#95—21no—
70C3⋊#95—24no—
71#830—21no—
72#830—21no—
73C22×C4⋊C4—31no—
74C22×C22⋊C4—31no—
75C22×C22⋊C4—61no—
76C2×C22⋊C8—31no—
77C2×C22⋊C8C2×D311yescharacteristic
78#397—34no—
79#397—62no—
80#388—62no—
81#388—62no—
82#380—31no—
83#380C2×D311yescharacteristic
84#379—61no—
85C4×C22⋊C4—61no—
86C2×#95—61no—
87C2×#95C2×D311yescharacteristic
88#340—62no—
89#340—62no—
90#323—62no—
91C24×C3C2×D411yescharacteristic
92C23×D3—21no—
93C23×D3—21no—
94C23×D3—21no—
95C23×D3—21no—
96C23×D3—21no—
97C23×D3—21no—
98C23×D3—42no—
99C23×D3—42no—
100C23×D3C22⋊C412yes—
101C22×C3×C4—22no—
102C22×C3×C4—22no—
103C22×C3×C4—42no—
104C22×C3×C4—42no—
105C22×C3×C4—42no—
106C22×C3×C4—41no—
107C22×C3×C4—41no—
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×C3×C4#4111yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3—22no—
117C22×Dic3—22no—
118C22×Dic3—42no—
119C22×Dic3—41no—
120C22×Dic3—41no—
121C22×Dic3—41no—
122C22×Dic3—41no—
123C22×Dic3—41no—
124C22×Dic3—41no—
125C22×Dic3C22⋊C412yes—
126C2×C3×C8—42no—
127C2×C3×C8—41no—
128C3×C4⋊C4—42no—
129C3×C22⋊C4—41no—
130C3×C22⋊C4—41no—
131#217—41no—
132#217—41no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—48no—
138(C2×D3)⋊C4—82no—
139(C2×D3)⋊C4—82no—
140C4⋊Dic3—22no—
141C4⋊Dic3—22no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—42no—
145C4×Dic3—82no—
146C2×C3⋊C8—42no—
147C2×C3⋊C8—41no—
148C25—31no—
149C23×C4—21no—
150C23×C4—31no—
151C23×C4—61no—
152C23×C4C22×D311yescharacteristic
153C22×C8—21no—
154C22×C8—61no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—32no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4D1212yes—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—32no—
165C2×C22⋊C4—32no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—61no—
168C2×C22⋊C4—61no—
169C2×C22⋊C4—61no—
170C2×C22⋊C4—64no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174C2×C22⋊C4—122no—
175C2×C22⋊C4—122no—
176C2×C42—62no—
177C22⋊C8—21no—
178C22⋊C8—21no—
179C22⋊C8—61no—
180C22⋊C8—61no—
181#95—21no—
182#95—21no—
183#95—64no—
184#95—62no—
185#95—61no—
186#95C4×D312yes—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3C2×C22⋊C411yescharacteristic
190C23×C3#12011yescharacteristic
191C23×C3#12311yescharacteristic
192C23×C3#13511yescharacteristic
193C23×C3#13511yescharacteristic
194C23×C3AGL(1,ℤ/8)11yescharacteristic
195C23×C3AGL(1,ℤ/8)11yescharacteristic
196C22×D3—22no—
197C22×D3—22no—
198C22×D3—22no—
199C22×D3—42no—
200C22×D3—42no—
201C22×D3—44no—
202C22×D3—44no—
203C22×D3—41no—
204C22×D3—41no—
205C22×D3—42no—
206C22×D3—42no—
207C22×D3—41no—
208C22×D3—41no—
209C22×D3—81no—
210C22×D3—81no—
211C22×D3—82no—
212C22×D3—82no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—82no—
226C2×C3×C4—82no—
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×Dic3—22no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—81no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C2×Dic3—81no—
245C3×C8—81no—
246C3⋊C8—81no—
247C24—32no—
248C24—61no—
249C24—61no—
250C24—61no—
251C24—61no—
252C24—61no—
253C24—61no—
254C24—122no—
255C24—122no—
256C24D3×D411yescharacteristic
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—32no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—122no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—121no—
273C22×C4—121no—
274C22×C4—121no—
275C22×C4—121no—
276C22×C4C2×C4×D311yescharacteristic
277C22×C4C2×D1211yescharacteristic
278C22×C4#23511yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4#23911yescharacteristic
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—122no—
285C2×C8—121no—
286C4⋊C4—42no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—122no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—128no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C22⋊C4—242no—
302C22⋊C4—242no—
303C42—242no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3#38611yescharacteristic
313C22×C3#41611yescharacteristic
314C22×C3#41811yescharacteristic
315C22×C3#44911yescharacteristic
316C22×C3#44911yescharacteristic
317C22×C3#45512yes—
318C22×C3#45812yes—
319C22×C3#48211yescharacteristic
320C22×C3#48211yescharacteristic
321C2×D3—44no—
322C2×D3—42no—
323C2×D3—42no—
324C2×D3—42no—
325C2×D3—82no—
326C2×D3—82no—
327C2×D3—82no—
328C2×D3—82no—
329C2×D3—82no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—82no—
335C3×C4—82no—
336Dic3—42no—
337Dic3—82no—
338Dic3—82no—
339C23—22no—
340C23—22no—
341C23—62no—
342C23—62no—
343C23—62no—
344C23—122no—
345C23—122no—
346C23—124no—
347C23—124no—
348C23—121no—
349C23—121no—
350C23—122no—
351C23—121no—
352C23—121no—
353C23—122no—
354C23—241no—
355C23—241no—
356C23—242no—
357C23—242no—
358C23D3×C22⋊C411yescharacteristic
359C23#88211yescharacteristic
360C23#89411yescharacteristic
361C23#91111yescharacteristic
362C23#91411yescharacteristic
363C23#91611yescharacteristic
364C23#91911yescharacteristic
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—62no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—241no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4—241no—
392C2×C4—241no—
393C2×C4C4×D1211yescharacteristic
394C2×C4#88411yescharacteristic
395C2×C4#89311yescharacteristic
396C2×C4#89511yescharacteristic
397C8—81no—
398C8—241no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3#161212yes—
404C2×C3#161212yes—
405C2×C3#187511yescharacteristic
406C2×C3#188511yescharacteristic
407C2×C3#189011yescharacteristic
408D3—82no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—124no—
418C22—122no—
419C22—122no—
420C22—122no—
421C22—242no—
422C22—242no—
423C22—242no—
424C22—242no—
425C22—242no—
426C22#466511yescharacteristic
427C22#474212yes—
428C22#477011yescharacteristic
429C22#478111yescharacteristic
430C22#480411yescharacteristic
431C22#481011yescharacteristic
432C22#482512yes—
433C22#486611yescharacteristic
434C22#488511yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—82no—
440C4—82no—
441C4—122no—
442C4—242no—
443C4—242no—
444C3#902811yescharacteristic, complemented, Sylow
445C2—21no—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—242no—
450C2#6828912yes—
451C2#6832112yes—
452C2#6875911yescharacteristic
453C2#6888111yescharacteristic
454C2#6896311yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9028
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
3. #69924C2semidirect
5. #69501C2semidirect
6. C3×#1294C2semidirect
7. #67612C2semidirect
22. #4674C4semidirect
33. #1294D3semidirect
444. C3#9028semidirect

Maximal subgroups

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

subgroupindex
#740042
C2×#46742
#699242
C2×#44682
#695012
C3×#12942
#676122
#90283