Finite Group Databasebeta

← #10897665

Subgroups — #10897665

Order 768.

654 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79258C211yesmaximal, characteristic, complemented
2#78648C211yesmaximal, characteristic, complemented
3C3×#2098C211yesmaximal, characteristic, complemented
4#70726C211yesmaximal, characteristic, complemented
5#70546C212yesmaximal, complemented
6#70535C212yesmaximal, complemented
7#70416C212yesmaximal, complemented
8#70395C212yesmaximal, complemented
9#70013C211yesmaximal, characteristic
10#68698C211yesmaximal, characteristic, complemented
11#68696C211yesmaximal, characteristic, complemented
12#29386—31nomaximal, Sylow
13Dic3×#41C2211yescharacteristic
14#5822C2212yes—
15#5748C2211yescharacteristic
16#5745C2212yes—
17C3×#430C2211yescharacteristic
18C3×#424C2212yes—
19C3×#421C2212yes—
20C3×#408C2211yescharacteristic
21#5234C2211yescharacteristic
22C2×#825C2212yes—
23#5119C2211yescharacteristic
24Dic3×M16C2211yescharacteristic
25#5110C2212yes—
26#5102C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#5006C2212yes—
29#5002C2212yes—
30#4895—22no—
31#4894—22no—
32#4720—22no—
33#4718—22no—
34C3×H(3,ℤ/4)C2211yescharacteristic
35C3⋊#338C2211yescharacteristic
36#4558C2212yes—
37#4556C2212yes—
38#4533C2212yes—
39#4532C2212yes—
40#2956—31no—
41#2922—31no—
42#2103—31no—
43#2098D311yescharacteristic, complemented
44#2053—31no—
45#2017—31no—
46#1937—32no—
47#1906—32no—
48#1881—32no—
49#1868—32no—
50#1841—31no—
51C2×C3×#41C2311yescharacteristic
52C2×#235C2312yes—
53C2×C3×M16C2312yes—
54C2×C3×M16C2311yescharacteristic
55C22×C3×C8C2311yescharacteristic
56C3×#117C2312yes—
57#951—22no—
58#950—42no—
59#949—22no—
60Q8×Dic3—21no—
61#942—22no—
62D4×Dic3—21no—
63D4×Dic3—22no—
64#932—22no—
65#926—21no—
66#926C2311yescharacteristic
67C2×C4×Dic3—21no—
68C2×C4×Dic3C2311yescharacteristic
69C2×#208C2312yes—
70C2×#208C2311yescharacteristic
71C22×C3⋊C8C2311yescharacteristic
72#909—22no—
73#908—22no—
74#907—24no—
75D3×M16—42no—
76#904—22no—
77#901—42no—
78C3×#108—22no—
79C3×C4⋊C8—42no—
80C3×#104—42no—
81C3×C22⋊C8—22no—
82#830—22no—
83#825—22no—
84#825—22no—
85(C2×D3)⋊C8—42no—
86#815D412yes—
87Dic3⋊C8—22no—
88Dic3⋊C8—22no—
89C8×Dic3—21no—
90C8×Dic3—21no—
91C8×Dic3D412yes—
92#810—22no—
93#809—22no—
94#806—24no—
95#575—32no—
96#575—31no—
97#568—32no—
98C4×#41—31no—
99#444—62no—
100#443—62no—
101#433—62no—
102#431—62no—
103#430C2×D311yescharacteristic
104#424—31no—
105#424—31no—
106#424C2×D312yes—
107#421C2×D312yes—
108C2×#104—32no—
109#416—32no—
110#410—32no—
111#408—31no—
112#408—32no—
113#408C2×D311yescharacteristic
114C4×M16—31no—
115C2×C4×C8—31no—
116#344—32no—
117#344—32no—
118#341—32no—
119#339—32no—
120#338—31no—
121H(3,ℤ/4)C2×D311yescharacteristic
122C3×#41—22no—
123C3×#41—22no—
124C2×C3×Q8#4111yescharacteristic
125C2×C3×D4—21no—
126C2×C3×D4#4111yescharacteristic
127C22×C3×C4—21no—
128C22×C3×C4C2411yescharacteristic
129C2×#67—22no—
130C22×Dic3—22no—
131C22×Dic3—21no—
132#235—22no—
133#235—22no—
134#235—22no—
135#235—22no—
136C2×D12#4112yes—
137C2×C4×D3—22no—
138C2×Dic6#4112yes—
139C3×M16—22no—
140C3×M16—22no—
141C3×M16—22no—
142C2×C3×C8—21no—
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8C2×D412yes—
146C3×C4⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C42—22no—
149#217—22no—
150#217#4112yes—
151#216—42no—
152#215—42no—
153#214—42no—
154#213—42no—
155C4⋊Dic3—22no—
156C4⋊Dic3—21no—
157C4×Dic3—22no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3C2×D411yescharacteristic
161C4×Dic3C2×D411yescharacteristic
162#208—22no—
163#208—22no—
164#208—22no—
165C2×C3⋊C8—21no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8C2×D412yes—
169Dic12—22no—
170D24—22no—
171#204—24no—
172ΓL(1,𝔽25)—44no—
173D3×C8—44no—
174C2×#41—32no—
175C2×#41C22×D311yescharacteristic
176#137—62no—
177#137—62no—
178#137—62no—
179AGL(1,ℤ/8)—62no—
180AGL(1,ℤ/8)—62no—
181AGL(1,ℤ/8)—62no—
182#135—62no—
183#135—122no—
184#131—64no—
185#131—122no—
186C2×M16—31no—
187C2×M16—32no—
188C2×M16—122no—
189C2×M16C22×D312yes—
190C2×M16C22×D311yescharacteristic
191C22×C8—31no—
192C22×C8C22×D311yescharacteristic
193C4×Q8—61no—
194C4×D4—62no—
195C4×D4—61no—
196#117—31no—
197#117—61no—
198#117C22×D312yes—
199C2×C42—31no—
200C2×C42—61no—
201#108—22no—
202C4⋊C8—42no—
203C4⋊C8—62no—
204C4⋊C8—62no—
205#104—42no—
206#104—62no—
207#104—62no—
208#103—62no—
209#102—62no—
210#99—64no—
211C22⋊C8—22no—
212C22⋊C8—62no—
213C22⋊C8—122no—
214#97—32no—
215C4×C8—32no—
216C4×C8—61no—
217C4×C8—61no—
218C23×C3—21no—
219C23×C3C2×#4111yescharacteristic
220C22×D3—22no—
221C3×Q8—22no—
222C3×D4—22no—
223C3×D4—42no—
224C2×C3×C4—21no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4C22×D411yescharacteristic
229C2×C3×C4C2×#4111yescharacteristic
230C2×C3×C4C2×#4112yes—
231C2×C3×C4H(5,𝔽2)12yes—
232#67—42no—
233#67—42no—
234C2×Dic3—21no—
235C2×Dic3—22no—
236C2×Dic3—21no—
237C2×Dic3—22no—
238C2×Dic3—42no—
239C2×Dic3—44no—
240D12—22no—
241D12—22no—
242C4×D3—42no—
243C4×D3—42no—
244Dic6—22no—
245Dic6—22no—
246C3×C8—22no—
247C3×C8—42no—
248C3×C8—41no—
249C3⋊C8—22no—
250C3⋊C8—41no—
251C3⋊C8—42no—
252#41—22no—
253#41—22no—
254#41—62no—
255#41—62no—
256#41—62no—
257#41—62no—
258C2×Q8—32no—
259C2×Q8#23911yescharacteristic
260C2×D4—21no—
261C2×D4—32no—
262C2×D4—62no—
263C2×D4#23911yescharacteristic
264C22×C4—21no—
265C22×C4—62no—
266C22×C4—61no—
267C22×C4—62no—
268C22×C4C23×D311yescharacteristic
269Q16—62no—
270Q16—122no—
271SD16—64no—
272SD16—122no—
273SD16—122no—
274D8—62no—
275D8—122no—
276M16—22no—
277M16—22no—
278M16—22no—
279M16—62no—
280M16—62no—
281M16—62no—
282M16—124no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—32no—
287C2×C8—61no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—124no—
291C2×C8D3×D412yes—
292C4⋊C4—22no—
293C4⋊C4—62no—
294C4⋊C4—61no—
295C22⋊C4—22no—
296C22⋊C4—32no—
297C22⋊C4—62no—
298C42—22no—
299C42—31no—
300C42—31no—
301C42—62no—
302C42—61no—
303C42—61no—
304C22×C3—41no—
305C22×C3—42no—
306C22×C3—41no—
307C22×C3#52511yescharacteristic
308C22×C3#55312yes—
309C2×D3—42no—
310C2×D3—82no—
311C3×C4—41no—
312C3×C4—82no—
313C3×C4#52511yescharacteristic
314C3×C4#53511yescharacteristic
315C3×C4#55112yes—
316Dic3—22no—
317Dic3—42no—
318Dic3—41no—
319Dic3—42no—
320C23—21no—
321C23—62no—
322C23D3×#4111yescharacteristic
323Q8—22no—
324Q8—62no—
325Q8—62no—
326D4—22no—
327D4—42no—
328D4—62no—
329D4—62no—
330D4—122no—
331D4—122no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—61no—
337C2×C4—62no—
338C2×C4—61no—
339C2×C4—62no—
340C2×C4—122no—
341C2×C4—124no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4C2×D3×D411yescharacteristic
345C2×C4#100412yes—
346C2×C4C2×#23911yescharacteristic
347C2×C4D3×#4112yes—
348C8—22no—
349C8—41no—
350C8—42no—
351C8—62no—
352C8—122no—
353C8—121no—
354C2×C3—21no—
355C2×C3—22no—
356C2×C3—81no—
357C2×C3#259811yescharacteristic
358D3—82no—
359C22—41no—
360C22—42no—
361C22—41no—
362C22—122no—
363C22—242no—
364C22#562711yescharacteristic
365C22#566112yes—
366C4—41no—
367C4—62no—
368C4—82no—
369C4—121no—
370C4—122no—
371C4—122no—
372C4#563712yes—
373C4#565611yescharacteristic
374C4#571511yescharacteristic
375C3#2938611yescharacteristic, complemented, Sylow
376C2—21no—
377C2—22no—
378C2—81no—
379C2—242no—
380C2#7804411yescharacteristic
subgroupG/H
centerC4#5715
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2098C2
socleC2×C3#2598
prime radical#70013C2
hypercenter#2098D3
hypocenterC3#29386
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2098

Complements

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

NH ≅ G/Ntype
1. #79258C2semidirect
2. #78648C2semidirect
3. C3×#2098C2semidirect
4. #70726C2semidirect
5. #70546C2semidirect
6. #70535C2semidirect
7. #70416C2semidirect
8. #70395C2semidirect
10. #68698C2semidirect
11. #68696C2semidirect
43. #2098D3semidirect
375. C3#29386semidirect

Maximal subgroups

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

subgroupindex
#792582
#786482
C3×#20982
#707262
#705462
#705352
#704162
#703952
#700132
#686982
#686962
#293863