Finite Group Databasebeta

← #10690225

Subgroups — #10690225

Order 768.

616 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73869C211yesmaximal, characteristic
2C2×#4659C211yesmaximal, characteristic
3#69925C211yesmaximal, characteristic
4C2×#4469C211yesmaximal, characteristic
5#69502C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#9023—31nomaximal, Sylow
9C22×C4⋊Dic3C2211yescharacteristic
10C22×Dic3⋊C4C2211yescharacteristic
11C2×#877—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14C2×C3×#95C2211yescharacteristic
15#5212C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95—21no—
18C2×#830C2211yescharacteristic
19#4976—22no—
20#4942—21no—
21#4659—22no—
22#4659C414yescomplemented
23#4647—22no—
24#4551—22no—
25#4529—22no—
26#4469—22no—
27C2×#400—31no—
28#2316—31no—
29#1824—31no—
30#1824—31no—
31C2×#324—31no—
32#1294—31no—
33#1294D311yescharacteristic
34C23×C3×C4—21no—
35C23×C3×C4C2311yescharacteristic
36C23×Dic3—21no—
37C23×Dic3D411yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×C3×C8—21no—
40C2×C3×C4⋊C4D412yes—
41C2×C3×C22⋊C4—21no—
42C2×#217—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—21no—
48C2×C4⋊Dic3C2×C412yes—
49C2×C4⋊Dic3D412yes—
50C2×Dic3⋊C4—21no—
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—22no—
55C2×Dic3⋊C4—22no—
56C2×Dic3⋊C4C2×C412yes—
57C2×Dic3⋊C4D412yes—
58C2×C4×Dic3—22no—
59C22×C3⋊C8—21no—
60#877—42no—
61#877—42no—
62C3×C22⋊C8—21no—
63C3×C22⋊C8—21no—
64C3×#95—21no—
65C3×#95—21no—
66C3×#95C2×C412yes—
67C3⋊#95—22no—
68C3⋊#95—21no—
69C3⋊#95—24no—
70C3⋊#95—24no—
71#830—21no—
72#830—21no—
73C2×#117—61no—
74C22×C4⋊C4—31no—
75C22×C4⋊C4—31no—
76C2×C22⋊C8—31no—
77C2×C22⋊C8C2×D311yescharacteristic
78#400—34no—
79#400—62no—
80#389—62no—
81#389—62no—
82#380—31no—
83#380—61no—
84#380C2×D311yescharacteristic
85C2×#95—61no—
86C2×#95—61no—
87C2×#95C2×D311yescharacteristic
88#340—62no—
89#340—62no—
90#324—62no—
91C24×C3C2×D411yescharacteristic
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C22×C3×C4#4111yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—21no—
107C22×Dic3—21no—
108C22×Dic3—21no—
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—22no—
112C22×Dic3—22no—
113C22×Dic3—42no—
114C22×Dic3—44no—
115C22×Dic3—41no—
116C22×Dic3—41no—
117C22×Dic3C22⋊C412yes—
118C22×Dic3C22⋊C412yes—
119C22×Dic3SD1612yes—
120C22×Dic3Q1612yes—
121C2×C3×C8—42no—
122C2×C3×C8—41no—
123C3×C4⋊C4—42no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126#217—41no—
127#217—41no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132C4⋊Dic3—42no—
133Dic3⋊C4—22no—
134Dic3⋊C4—22no—
135Dic3⋊C4—22no—
136Dic3⋊C4—22no—
137Dic3⋊C4—44no—
138Dic3⋊C4—82no—
139C4×Dic3—44no—
140C2×C3⋊C8—42no—
141C2×C3⋊C8—41no—
142C23×C4—21no—
143C23×C4—31no—
144C23×C4—31no—
145C23×C4—61no—
146C23×C4C22×D311yescharacteristic
147C22×C8—21no—
148C22×C8—61no—
149#117—122no—
150#117—122no—
151C2×C4⋊C4—32no—
152C2×C4⋊C4—32no—
153C2×C4⋊C4—32no—
154C2×C4⋊C4—32no—
155C2×C4⋊C4—61no—
156C2×C4⋊C4—61no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—62no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—62no—
166C2×C4⋊C4D1212yes—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—61no—
169C2×C42—62no—
170C22⋊C8—21no—
171C22⋊C8—21no—
172C22⋊C8—61no—
173C22⋊C8—61no—
174#95—21no—
175#95—21no—
176#95—64no—
177#95—64no—
178#95—62no—
179#95—61no—
180#95C4×D312yes—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3C2×C22⋊C411yescharacteristic
184C23×C3#12011yescharacteristic
185C23×C3#12311yescharacteristic
186C23×C3C2×SD1611yescharacteristic
187C23×C3C2×Q1611yescharacteristic
188C23×C3AGL(1,ℤ/8)11yescharacteristic
189C23×C3#13711yescharacteristic
190C2×C3×C4—22no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—82no—
203C2×C3×C4—82no—
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×Dic3—22no—
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—22no—
214C2×Dic3—22no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—44no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—81no—
223C2×Dic3—82no—
224C2×Dic3—82no—
225C2×Dic3#10314yes—
226C3×C8—81no—
227C3⋊C8—81no—
228C24D3×D411yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—32no—
232C22×C4—32no—
233C22×C4—32no—
234C22×C4—32no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4—62no—
247C22×C4—62no—
248C22×C4—122no—
249C22×C4—124no—
250C22×C4—121no—
251C22×C4—121no—
252C22×C4C2×C4×D311yescharacteristic
253C22×C4C2×D1211yescharacteristic
254C22×C4#23511yescharacteristic
255C22×C4D3×D411yescharacteristic
256C22×C4D3×D411yescharacteristic
257C22×C4#23911yescharacteristic
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—122no—
261C2×C8—121no—
262C4⋊C4—42no—
263C4⋊C4—62no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—62no—
267C4⋊C4—62no—
268C4⋊C4—62no—
269C4⋊C4—122no—
270C4⋊C4—124no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—242no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C42—124no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#38611yescharacteristic
288C22×C3C2×#10311yescharacteristic
289C22×C3#41711yescharacteristic
290C22×C3#45011yescharacteristic
291C22×C3#45111yescharacteristic
292C22×C3#45412yes—
293C22×C3#45812yes—
294C22×C3#48311yescharacteristic
295C22×C3#48411yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—82no—
301C3×C4—82no—
302Dic3—24no—
303Dic3—42no—
304Dic3—82no—
305Dic3—82no—
306C23—22no—
307C23—22no—
308C23D3×C22⋊C411yescharacteristic
309C23#88211yescharacteristic
310C23#89411yescharacteristic
311C23#91111yescharacteristic
312C23D3×SD1611yescharacteristic
313C23#91511yescharacteristic
314C23D3×Q1611yescharacteristic
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—34no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—62no—
331C2×C4—62no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—124no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—241no—
345C2×C4—242no—
346C2×C4—242no—
347C2×C4C4×D1211yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4#89311yescharacteristic
350C2×C4#89511yescharacteristic
351C8—81no—
352C8—241no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3#161612yes—
358C2×C3#161712yes—
359C2×C3#187211yescharacteristic
360C2×C3#188811yescharacteristic
361C2×C3#189011yescharacteristic
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22#466511yescharacteristic
371C22#474212yes—
372C22#477111yescharacteristic
373C22#477611yescharacteristic
374C22D3×#10311yescharacteristic
375C22#480911yescharacteristic
376C22#482712yes—
377C22#486211yescharacteristic
378C22#488411yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—64no—
384C4—82no—
385C4—82no—
386C4—122no—
387C4—242no—
388C4—242no—
389C3#902311yescharacteristic, complemented, Sylow
390C2—21no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2#6829912yes—
395C2#6833112yes—
396C2#6876611yescharacteristic
397C2#6887811yescharacteristic
398C2#6896311yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9023
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
22. #4659C4semidirect
389. C3#9023semidirect

Maximal subgroups

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

subgroupindex
#738692
C2×#46592
#699252
C2×#44692
#695022
C3×#12942
#676122
#90233