Finite Group Databasebeta

← #10798398

Subgroups — #10798398

Order 768.

862 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#204C211yesmaximal, characteristic
2#73761C211yesmaximal, characteristic
3#73616C211yesmaximal, characteristic, complemented
4C3×C4×#107C211yesmaximal, characteristic, complemented
5#70642C212yesmaximal, complemented
6#70623C212yesmaximal
7#69751C211yesmaximal, characteristic, complemented
8#69475C211yesmaximal, characteristic, complemented
9#69474C211yesmaximal, characteristic
10#68909C212yesmaximal
11#68816C212yesmaximal, complemented
12#21136—31nomaximal, Sylow
13C22×#204C2211yescharacteristic
14C2×#887C2212yes—
15C2×C4×D12C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C2×C3×#107C2212yes—
18C2×C3×C4×C8C2211yescharacteristic
19C3×C4×C4⋊C4C2211yescharacteristic
20C3×C4×C4⋊C4C2211yescharacteristic
21C2×#821C2211yescharacteristic
22C2×#817C2211yescharacteristic
23C2×#816C2211yescharacteristic
24#4988C2212yes—
25Dic3×C4⋊C4C2212yes—
26#4972C2212yes—
27C2×#810C2212yes—
28C2×#809C2212yes—
29#4940—22no—
30#4932C2211yescharacteristic
31C4×Dic3⋊C4—22no—
32C2×#803C2211yescharacteristic
33C4×#204C418yescomplemented
34#4667—21no—
35#4664—21no—
36#4649—21no—
37#4648—21no—
38C3×#340C2212yes—
39#4550C2212yes—
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×C4×SD16—31no—
44#2319—31no—
45#2276—31no—
46#1925—32no—
47#1919—32no—
48#1880—32no—
49#1863—32no—
50C4×#107D311yescharacteristic, complemented
51#1763—31no—
52#1762—31no—
53#1746—31no—
54C22×D12C2311yescharacteristic
55C22×C4×D3—21no—
56C22×Dic6C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C42—21no—
62C2×C3×C42—21no—
63C2×C3×C42C2311yescharacteristic
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×C4⋊Dic3C2311yescharacteristic
68C2×C4⋊Dic3C2311yescharacteristic
69C2×Dic3⋊C4—21no—
70C2×Dic3⋊C4—22no—
71C2×Dic3⋊C4—22no—
72C2×C4×Dic3—21no—
73C2×C4×Dic3—22no—
74C2×C4×Dic3—22no—
75C2×C4×Dic3C2312yes—
76C22×C3⋊C8C2312yes—
77C2×#204—24no—
78C2×#204C2×C414yes—
79#887—24no—
80#887—24no—
81C4×D12—22no—
82C4×D12C2×C414yes—
83C4×Dic6—22no—
84C4×Dic6C2×C414yes—
85C3×#107—24no—
86C3×C4×C8C2×C414yes—
87C3×#95—21no—
88C3×#95—21no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95D411yescharacteristic
92C3⋊#95D411yescharacteristic
93C3⋊#95D412yes—
94#821C2×C414yes—
95#817C2×C414yes—
96#816C2×C414yes—
97#810—24no—
98#809—24no—
99#803—22no—
100C22×SD16—31no—
101C2×C4×Q8—32no—
102C2×C4×Q8—31no—
103C2×C4×D4—31no—
104C4×SD16—38no—
105C2×#107C2×D312yes—
106C2×#106—31no—
107C2×#103—31no—
108C2×#103—32no—
109C2×#102—31no—
110C2×#102—32no—
111C2×#97—31no—
112C2×C4×C8C2×D311yescharacteristic
113#390—32no—
114#389—32no—
115#388—62no—
116#381—61no—
117#380—61no—
118C4×C4⋊C4—32no—
119C4×C4⋊C4—61no—
120C4×C4⋊C4—62no—
121C4×C4⋊C4C2×D311yescharacteristic
122C4×C4⋊C4C2×D311yescharacteristic
123#376—31no—
124#376—61no—
125#340—32no—
126#340—32no—
127#340C2×D312yes—
128#336—32no—
129#336—32no—
130C23×D3—21no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2411yescharacteristic
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139C22×Dic3C2×D412yes—
140C22×Dic3C2×D412yes—
141C2×D12—22no—
142C2×D12—22no—
143C2×D12—21no—
144C2×D12C22×C412yes—
145C2×C4×D3—24no—
146C2×C4×D3—82no—
147C2×Dic6—21no—
148C2×Dic6—22no—
149C2×Dic6C22×C412yes—
150C2×Dic6#4114yes—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8C22×C412yes—
154C2×C3×C8C22×C412yes—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4#4114yes—
159C3×C42—81no—
160C3×C42—81no—
161C3×C42C22×C412yes—
162C3×C42#4112yes—
163(C2×D3)⋊C4—24no—
164(C2×D3)⋊C4—82no—
165(C2×D3)⋊C4—82no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C4⋊Dic3C22×C412yes—
169C4⋊Dic3C22×C412yes—
170Dic3⋊C4—28no—
171Dic3⋊C4—24no—
172Dic3⋊C4—82no—
173C4×Dic3—28no—
174C4×Dic3—24no—
175C4×Dic3—22no—
176C4×Dic3—22no—
177C4×Dic3—82no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181#204—28no—
182C22×Q8—31no—
183C22×D4—31no—
184C23×C4—61no—
185C2×SD16—34no—
186C2×SD16—64no—
187C22×C8—32no—
188C22×C8C22×D311yescharacteristic
189C22×C8C22×D311yescharacteristic
190C4×Q8—34no—
191C4×Q8—64no—
192C4×Q8—64no—
193C4×Q8—62no—
194C4×D4—34no—
195C4×D4—62no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—31no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4C22×D312yes—
202C2×C4⋊C4C22×D312yes—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—32no—
209C2×C42—62no—
210C2×C42—61no—
211C2×C42—62no—
212C2×C42C22×D311yescharacteristic
213#107—24no—
214#106—34no—
215#103—34no—
216#103—64no—
217#102—34no—
218#102—64no—
219#97—62no—
220C4×C8C4×D314yes—
221#95—21no—
222#95—21no—
223#95—31no—
224#95—31no—
225#95—32no—
226#95—62no—
227#95—62no—
228C23×C3C22×D411yescharacteristic
229C22×D3—22no—
230C22×D3—42no—
231C22×D3—41no—
232C22×D3—42no—
233C22×D3—41no—
234C2×C3×C4—21no—
235C2×C3×C4—22no—
236C2×C3×C4—21no—
237C2×C3×C4—24no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4C4×D412yes—
244C2×C3×C4C4×D412yes—
245C2×C3×C4C23×C411yescharacteristic
246C2×C3×C4C2×#4112yes—
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×Dic3—22no—
250C2×Dic3—24no—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—22no—
258C2×Dic3—22no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262D12—22no—
263D12—22no—
264D12—42no—
265C4×D3—84no—
266Dic6—22no—
267Dic6—24no—
268Dic6#11714yes—
269C3×C8—22no—
270C3×C8—22no—
271C3⋊C8—42no—
272C24—61no—
273C2×Q8—32no—
274C2×Q8—34no—
275C2×Q8—62no—
276C2×Q8—61no—
277C2×D4—32no—
278C2×D4—62no—
279C2×D4—61no—
280C2×D4—62no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—32no—
284C22×C4—32no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—64no—
289C22×C4—242no—
290C22×C4D3×D411yescharacteristic
291C22×C4D3×D411yescharacteristic
292C22×C4C23×D311yescharacteristic
293SD16—68no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8C2×C4×D312yes—
300C2×C8C2×C4×D312yes—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—68no—
307C4⋊C4—64no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—242no—
311C4⋊C4#23514yes—
312C22⋊C4—64no—
313C22⋊C4—242no—
314C22⋊C4—242no—
315C42—68no—
316C42—64no—
317C42—62no—
318C42—62no—
319C42—81no—
320C42—81no—
321C42—242no—
322C42C2×C4×D312yes—
323C42#23912yes—
324C22×C3C2×C4×D411yescharacteristic
325C22×C3#52512yes—
326C22×C3#57611yescharacteristic
327C22×C3#57712yes—
328C22×C3#57811yescharacteristic
329C2×D3—42no—
330C2×D3—42no—
331C2×D3—42no—
332C2×D3—82no—
333C2×D3—82no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—44no—
337C3×C4—82no—
338C3×C4C2×#11711yescharacteristic
339C3×C4C4×#4111yescharacteristic
340C3×C4#52912yes—
341Dic3—24no—
342Dic3—42no—
343Dic3—42no—
344Dic3—42no—
345Dic3—42no—
346C23—62no—
347C23—121no—
348C23—121no—
349C23—122no—
350C23—122no—
351C23C2×D3×D411yescharacteristic
352Q8—34no—
353Q8—62no—
354Q8—64no—
355D4—62no—
356D4—62no—
357D4—122no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—62no—
368C2×C4—64no—
369C2×C4—62no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—122no—
380C2×C4—244no—
381C2×C4#89312yes—
382C2×C4#89312yes—
383C2×C4C22×C4×D311yescharacteristic
384C2×C4C2×#23512yes—
385C2×C4C2×#23911yescharacteristic
386C2×C4D3×#4112yes—
387C8—22no—
388C8—22no—
389C8—122no—
390C2×C3#229011yescharacteristic
391C2×C3#294911yescharacteristic
392C2×C3#295111yescharacteristic
393C2×C3#311912yes—
394C2×C3#312112yes—
395D3—82no—
396C22—122no—
397C22—122no—
398C22—122no—
399C22—242no—
400C22—242no—
401C22C2×#89311yescharacteristic
402C22#554712yes—
403C22#575711yescharacteristic
404C22#576611yescharacteristic
405C22#577012yes—
406C4—22no—
407C4—22no—
408C4—44no—
409C4—64no—
410C4—82no—
411C4—122no—
412C4—122no—
413C4—122no—
414C4—122no—
415C4C4×#23511yescharacteristic
416C4D3×#11711yescharacteristic
417C4#558012yes—
418C3#2113611yescharacteristic, complemented, Sylow
419C2—242no—
420C2#7375411yescharacteristic
421C2#7913611yescharacteristic
422C2#7916811yescharacteristic
423C2#7959512yes—
424C2#7975012yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#107C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenterC4×#107D3
hypocenterC3#21136
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
3. #73616C2semidirect
4. C3×C4×#107C2semidirect
5. #70642C2semidirect
7. #69751C2semidirect
8. #69475C2semidirect
11. #68816C2semidirect
33. C4×#204C4semidirect
50. C4×#107D3semidirect
418. C3#21136semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#2042
#737612
#736162
C3×C4×#1072
#706422
#706232
#697512
#694752
#694742
#689092
#688162
#211363