Finite Group Databasebeta

← #10897658

Subgroups — #10897658

Order 768.

654 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2956C211yesmaximal, characteristic, complemented
2#78519C211yesmaximal, characteristic, complemented
3#72121C211yesmaximal, characteristic
4#72115C211yesmaximal, characteristic, complemented
5#72026C212yesmaximal, complemented
6#72020C212yesmaximal, complemented
7#70738C211yesmaximal, characteristic, complemented
8#70726C211yesmaximal, characteristic, complemented
9#70391C212yesmaximal, complemented
10#70371C212yesmaximal, complemented
11#69871C211yesmaximal, characteristic, complemented
12#29386—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C3×#568C2212yes—
15#5822C2212yes—
16C4×#235C2211yescharacteristic
17C3×#444—22no—
18C3×#443—22no—
19C3×#430C2211yescharacteristic
20C2×C3×#104C2212yes—
21C3×C4×M16C2211yescharacteristic
22#5240C2212yes—
23#5234C2212yes—
24#5233C2212yes—
25#5139C2212yes—
26#5129C2211yescharacteristic
27#5129C2211yescharacteristic
28#5119C2211yescharacteristic
29#5119C2211yescharacteristic
30#5119C2212yes—
31C2×C4×C3⋊C8C2211yescharacteristic
32#4709—22no—
33#4685—22no—
34C3⋊#338C2211yescharacteristic
35#4558C2212yes—
36#4557C2211yescharacteristic
37#4556C2212yes—
38#4556C2212yes—
39#4523C2212yes—
40#2956D311yescharacteristic, complemented
41#2922—31no—
42#2103—31no—
43#2098—31no—
44#2053—31no—
45#2017—31no—
46#1937—32no—
47#1906—32no—
48#1881—32no—
49#1868—32no—
50#1841—31no—
51C2×C3×#41C2312yes—
52C2×#235C2311yescharacteristic
53C3×#137—22no—
54C3×AGL(1,ℤ/8)—22no—
55C3×#135—22no—
56C3×#131—24no—
57C3×#131—42no—
58C2×C3×M16—42no—
59C2×C3×M16C2312yes—
60C2×C3×M16C2311yescharacteristic
61C2×C3×M16C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63#951—22no—
64#950—42no—
65#949—22no—
66#942—22no—
67#932—22no—
68C4×#67—22no—
69#926C2312yes—
70#926C2311yescharacteristic
71C2×#208C2312yes—
72C22×C3⋊C8C2311yescharacteristic
73C22×C3⋊C8C2311yescharacteristic
74C4×D12—21no—
75#872—21no—
76C42×D3—21no—
77C4×Dic6—21no—
78C3×#108—22no—
79C3×#104—22no—
80C3×#104—22no—
81C3×#97D412yes—
82C3×C4×C8D412yes—
83#835—22no—
84#834—24no—
85#832—22no—
86#830—42no—
87#825—42no—
88(C2×D3)⋊C8—22no—
89(C2×D3)⋊C8—22no—
90Dic3⋊C8—22no—
91Dic3⋊C8—22no—
92Dic3⋊C8—42no—
93C4×C3⋊C8—21no—
94C4×C3⋊C8—21no—
95#575—32no—
96#575C2×D311yescharacteristic
97#568C2×D312yes—
98C4×#41—31no—
99#444—22no—
100#443—22no—
101#433—62no—
102#431—62no—
103#430C2×D311yescharacteristic
104#424—31no—
105#424—31no—
106#424—32no—
107#421—32no—
108C2×#104C2×D312yes—
109#416—32no—
110#410—32no—
111#408—31no—
112#408—31no—
113#408—32no—
114C4×M16C2×D311yescharacteristic
115C2×C4×C8—31no—
116#344—32no—
117#344—32no—
118#341—32no—
119#339—32no—
120#338—31no—
121H(3,ℤ/4)—31no—
122C3×#41—22no—
123C3×#41—22no—
124C3×#41—22no—
125C3×#41—22no—
126C2×C3×Q8#4112yes—
127C2×C3×D4—22no—
128C2×C3×D4#4112yes—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4C2411yescharacteristic
132C2×#67—21no—
133#235—22no—
134#235—22no—
135C2×D12#4111yescharacteristic
136C2×C4×D3—22no—
137C2×C4×D3—21no—
138C2×Dic6#4111yescharacteristic
139C3×Q16—22no—
140C3×SD16—24no—
141C3×D8—22no—
142C3×M16—22no—
143C3×M16—22no—
144C3×M16—22no—
145C3×M16—22no—
146C3×M16—44no—
147C2×C3×C8—22no—
148C2×C3×C8—44no—
149C2×C3×C8C2×D412yes—
150C2×C3×C8C2×D412yes—
151C3×C42—21no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154#217—22no—
155#217#4112yes—
156#216—42no—
157#215—42no—
158#214—42no—
159#213—42no—
160(C2×D3)⋊C4—22no—
161C4⋊Dic3—21no—
162C4⋊Dic3—22no—
163Dic3⋊C4—22no—
164C4×Dic3—22no—
165C4×Dic3—21no—
166C4×Dic3—22no—
167#208—22no—
168#208—22no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×#41—31no—
175C2×#41C22×D312yes—
176#137—22no—
177#137—62no—
178#137—62no—
179AGL(1,ℤ/8)—22no—
180AGL(1,ℤ/8)—62no—
181AGL(1,ℤ/8)—62no—
182#135—22no—
183#135—122no—
184#131—24no—
185#131—42no—
186C2×M16—32no—
187C2×M16—42no—
188C2×M16C22×D312yes—
189C2×M16C22×D311yescharacteristic
190C2×M16C22×D311yescharacteristic
191C22×C8—31no—
192C22×C8—31no—
193C4×Q8—61no—
194C4×D4—61no—
195C4×D4—62no—
196#117—31no—
197#117—32no—
198#117—61no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#108—22no—
202C4⋊C8—62no—
203C4⋊C8—62no—
204C4⋊C8—122no—
205#104—22no—
206#104—22no—
207#104—122no—
208#103—62no—
209#102—62no—
210#99—64no—
211C22⋊C8—62no—
212C22⋊C8—62no—
213C22⋊C8—122no—
214#97#6712yes—
215C4×C8—61no—
216C4×C8—61no—
217C4×C8#6712yes—
218C23×C3—22no—
219C23×C3C2×#4111yescharacteristic
220C22×D3—21no—
221C3×Q8—22no—
222C3×Q8—22no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—42no—
226C3×D4—42no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—22no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4C22×D411yescharacteristic
234C2×C3×C4C2×#4112yes—
235C2×C3×C4C2×#4111yescharacteristic
236C2×C3×C4H(5,𝔽2)12yes—
237#67—42no—
238C2×Dic3—21no—
239C2×Dic3—22no—
240C2×Dic3—42no—
241D12—22no—
242C4×D3—44no—
243C4×D3—41no—
244C4×D3—41no—
245Dic6—22no—
246C3×C8—22no—
247C3×C8—22no—
248C3×C8—42no—
249C3⋊C8—41no—
250C3⋊C8—41no—
251C3⋊C8—42no—
252#41—22no—
253#41—22no—
254#41—22no—
255#41—22no—
256#41—62no—
257#41—62no—
258C2×Q8—31no—
259C2×Q8#23512yes—
260C2×D4—22no—
261C2×D4—31no—
262C2×D4—61no—
263C2×D4#23512yes—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—62no—
267C22×C4—61no—
268C22×C4C23×D311yescharacteristic
269Q16—22no—
270Q16—122no—
271SD16—24no—
272SD16—122no—
273SD16—122no—
274D8—22no—
275D8—122no—
276M16—22no—
277M16—22no—
278M16—22no—
279M16—22no—
280M16—44no—
281M16—62no—
282M16—62no—
283C2×C8—22no—
284C2×C8—44no—
285C2×C8—62no—
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8C2×#6712yes—
291C2×C8C2×#6712yes—
292C4⋊C4—62no—
293C4⋊C4—61no—
294C4⋊C4—62no—
295C22⋊C4—32no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C42—21no—
299C42—62no—
300C42—61no—
301C42—62no—
302C42C2×#6711yescharacteristic
303C42C2×#6711yescharacteristic
304C22×C3—41no—
305C22×C3—42no—
306C22×C3—82no—
307C22×C3#52511yescharacteristic
308C22×C3#55312yes—
309C2×D3—42no—
310C2×D3—41no—
311C3×C4—22no—
312C3×C4—41no—
313C3×C4—42no—
314C3×C4#52511yescharacteristic
315C3×C4#53511yescharacteristic
316C3×C4#55112yes—
317Dic3—42no—
318Dic3—41no—
319Dic3—82no—
320C23—22no—
321C23—61no—
322C23D3×#4111yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—62no—
326D4—22no—
327D4—22no—
328D4—42no—
329D4—42no—
330D4—62no—
331D4—122no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—62no—
339C2×C4—61no—
340C2×C4—124no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—122no—
344C2×C4C2×#23512yes—
345C2×C4#100412yes—
346C2×C4D3×#4111yescharacteristic
347C2×C4C22×#6711yescharacteristic
348C8—22no—
349C8—22no—
350C8—42no—
351C8—121no—
352C8—121no—
353C8—122no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3—82no—
357C2×C3#259811yescharacteristic
358D3—81no—
359C22—41no—
360C22—42no—
361C22—82no—
362C22—122no—
363C22—121no—
364C22#554012yes—
365C22#582811yescharacteristic
366C4—22no—
367C4—41no—
368C4—42no—
369C4—122no—
370C4—121no—
371C4—242no—
372C4#555612yes—
373C4#579111yescharacteristic
374C4#580511yescharacteristic
375C3#2938611yescharacteristic, complemented, Sylow
376C2—21no—
377C2—22no—
378C2—82no—
379C2—241no—
380C2#8241811yescharacteristic
subgroupG/H
centerC4#5805
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2956C2
socleC2×C3#2598
prime radical#72121C2
hypercenter#2956D3
hypocenterC3#29386
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2956

Complements

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

NH ≅ G/Ntype
1. C3×#2956C2semidirect
2. #78519C2semidirect
4. #72115C2semidirect
5. #72026C2semidirect
6. #72020C2semidirect
7. #70738C2semidirect
8. #70726C2semidirect
9. #70391C2semidirect
10. #70371C2semidirect
11. #69871C2semidirect
40. #2956D3semidirect
375. C3#29386semidirect

Maximal subgroups

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

subgroupindex
C3×#29562
#785192
#721212
#721152
#720262
#720202
#707382
#707262
#703912
#703712
#698712
#293863