Finite Group Databasebeta

← #11792664

Subgroups — #11792664

Order 1152.

779 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 779 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C3×#5748C211yesmaximal, characteristic, complemented
2#10594795C212yesmaximal, complemented
3#10592828C211yesmaximal, characteristic, complemented
4#10591071C212yesmaximal, complemented
5C3×C3⋊#338C211yesmaximal, characteristic, complemented
6#72121—31nomaximal
7#68727—31nomaximal
8C2×#3828C2211yescharacteristic
9C2×C3×#235C2212yes—
10C2×C32×M16C2211yescharacteristic
11C22×C3⋊(C3⋊C8)C2211yescharacteristic
12C2×C3×#208C2212yes—
13C3×#909—22no—
14C3×#908—22no—
15C3×#904—22no—
16#63833—22no—
17#63831—22no—
18#63830—22no—
19#63829—24no—
20#63825—22no—
21#63647—22no—
22(C2×D3)⋊(C3⋊C8)—24no—
23C3×#575—31no—
24#5822—32no—
25#5748—32no—
26#5748D311yescharacteristic, complemented
27#5234—32no—
28#5129—32no—
29#5129—31no—
30#5129—31no—
31#5129—61no—
32C3×#338—31no—
33C3⋊#338D311yescharacteristic, complemented
34C22×C32×C4C2311yescharacteristic
35C2×#638—21no—
36#3828—22no—
37#3828—22no—
38C2×C4⋊(C3⋊D3)D411yescharacteristic
39C2×C4×C3⋊D3—21no—
40C2×#634D411yescharacteristic
41C2×C3×#67—22no—
42C3×#235—22no—
43C3×#235—22no—
44C3×#235—22no—
45C3×#235—22no—
46C2×C3×D12D412yes—
47C2×C3×C4×D3—22no—
48C2×C3×Dic6D412yes—
49C32×M16—22no—
50C2×C32×C8D412yes—
51C2×C3⋊(C3⋊C8)—21no—
52C2×C3⋊(C3⋊C8)—22no—
53C3×#208—22no—
54C3×#208—22no—
55C2×C3×C3⋊C8D414yes—
56C3×Dic12—22no—
57C3×D24—22no—
58C3×#204—24no—
59#3719—24no—
60#3717—24no—
61#3716—24no—
62#3714—24no—
63#2017—91noSylow
64C2×C3×#41—32no—
65C2×#235—31no—
66C2×#235—31no—
67C2×#235—61no—
68C2×#235C2×D312yes—
69C3×#137—62no—
70C3×AGL(1,ℤ/8)—62no—
71C3×#135—62no—
72C2×C3×M16—21no—
73C2×C3×M16—32no—
74C2×C3×M16C2×D311yescharacteristic
75C2×C3×M16C2×D311yescharacteristic
76#951—62no—
77#950—64no—
78#949—62no—
79#942—62no—
80#932—62no—
81C2×#208C2×D312yes—
82C22×C3⋊C8—31no—
83C22×C3⋊C8—31no—
84C22×C3⋊C8—61no—
85#909—22no—
86#909—62no—
87#909—62no—
88#908—22no—
89#908—62no—
90#908—62no—
91#904—22no—
92#904—64no—
93#830—64no—
94(C2×D3)⋊C8—64no—
95(C2×D3)⋊C8—64no—
96(C2×D3)⋊C8—62no—
97(C2×D3)⋊C8—62no—
98C23×C32#4111yescharacteristic
99C22×C3⋊D3—21no—
100C22×C3×D3—22no—
101C2×C32×C4C2×D412yes—
102C2×C32×C4C2×D411yescharacteristic
103C2×C32×C4C2×D411yescharacteristic
104C2×C32×C4C2×D412yes—
105#638—42no—
106C2×C3⋊Dic3—21no—
107C4⋊(C3⋊D3)—22no—
108C4×C3⋊D3—41no—
109C4×C3⋊D3—41no—
110#634—22no—
111C3×#67—42no—
112C3×#67—42no—
113C2×C3×Dic3—22no—
114C3×D12—22no—
115C3×D12—22no—
116C3×C4×D3—42no—
117C3×C4×D3—42no—
118C3×Dic6—22no—
119C3×Dic6—22no—
120C32×C8—22no—
121C3⋊(C3⋊C8)—41no—
122C3×C3⋊C8—24no—
123#575—31no—
124#575—92no—
125#408—92no—
126#408—91no—
127#338—31no—
128C3×#41—62no—
129C3×#41—62no—
130C3×#41—62no—
131C3×#41—62no—
132C2×C3×Q8—32no—
133C2×C3×D4—32no—
134C2×C3×D4—62no—
135C22×C3×C4—21no—
136C22×C3×C4—62no—
137C22×C3×C4C22×D311yescharacteristic
138C22×C3×C4C22×D311yescharacteristic
139C2×#67—22no—
140C2×#67—62no—
141C2×#67—61no—
142C2×#67—61no—
143#235—22no—
144#235—22no—
145#235—22no—
146#235—22no—
147#235—62no—
148#235—62no—
149#235—62no—
150#235—62no—
151#235—122no—
152#235—122no—
153C2×D12—31no—
154C2×D12—31no—
155C2×D12—61no—
156C2×D12#6712yes—
157C2×C4×D3—22no—
158C2×C4×D3—62no—
159C2×C4×D3—61no—
160C2×C4×D3—61no—
161C2×Dic6—31no—
162C2×Dic6—31no—
163C2×Dic6—61no—
164C2×Dic6#6712yes—
165C3×Q16—62no—
166C3×SD16—64no—
167C3×D8—62no—
168C3×M16—22no—
169C3×M16—22no—
170C3×M16—42no—
171C3×M16—62no—
172C3×M16—62no—
173C2×C3×C8—22no—
174C2×C3×C8—34no—
175C2×C3×C8D1212yes—
176C2×C3×C8#6712yes—
177#216—64no—
178#215—64no—
179#214—64no—
180#213—64no—
181#208—22no—
182#208—22no—
183C2×C3⋊C8—62no—
184C2×C3⋊C8—61no—
185C2×C3⋊C8—62no—
186C2×C3⋊C8—62no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—122no—
189C2×C3⋊C8D1214yes—
190Dic12—22no—
191Dic12—64no—
192D24—22no—
193D24—64no—
194#204—24no—
195#204—64no—
196#204—64no—
197C22×C32—41no—
198C22×C32#12111yescharacteristic
199C22×C32#12112yes—
200C2×C3⋊D3—42no—
201C2×C3⋊D3—41no—
202C2×C3×D3—44no—
203C2×C3×D3—42no—
204C32×C4#12012yes—
205C32×C4#12011yescharacteristic
206C32×C4#12711yescharacteristic
207C3⋊Dic3—41no—
208C3×Dic3—42no—
209C2×#41—32no—
210C2×#41—91no—
211#137—62no—
212#137—182no—
213#137—182no—
214AGL(1,ℤ/8)—62no—
215AGL(1,ℤ/8)—182no—
216AGL(1,ℤ/8)—182no—
217#135—62no—
218#135—184no—
219C2×M16—32no—
220C2×M16D3211yescharacteristic
221C22×C8—91no—
222C22⋊C8—184no—
223C22⋊C8—182no—
224C23×C3—21no—
225C23×C3—62no—
226C23×C3#23711yescharacteristic
227C23×C3#23911yescharacteristic
228C22×D3—22no—
229C22×D3—62no—
230C22×D3—61no—
231C22×D3—61no—
232C3×Q8—62no—
233C3×Q8—62no—
234C3×D4—62no—
235C3×D4—62no—
236C3×D4—122no—
237C3×D4—122no—
238C2×C3×C4—22no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—62no—
243C2×C3×C4—122no—
244C2×C3×C4—122no—
245C2×C3×C4C2×D1212yes—
246C2×C3×C4C2×D1211yescharacteristic
247C2×C3×C4D3×D412yes—
248C2×C3×C4D3×D411yescharacteristic
249C2×C3×C4D3×D411yescharacteristic
250C2×C3×C4D3×D412yes—
251C2×C3×C4C2×#6711yescharacteristic
252C2×C3×C4C2×#6712yes—
253#67—42no—
254#67—42no—
255#67—124no—
256#67—122no—
257#67—122no—
258C2×Dic3—22no—
259C2×Dic3—62no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262D12—22no—
263D12—22no—
264D12—62no—
265D12—62no—
266D12—122no—
267C4×D3—42no—
268C4×D3—42no—
269C4×D3—122no—
270C4×D3—122no—
271C4×D3—121no—
272C4×D3—121no—
273C4×D3—121no—
274C4×D3—121no—
275Dic6—22no—
276Dic6—22no—
277Dic6—62no—
278Dic6—62no—
279Dic6—122no—
280C3×C8—22no—
281C3×C8—22no—
282C3×C8—42no—
283C3×C8—64no—
284C3⋊C8—24no—
285C3⋊C8—122no—
286C3⋊C8—121no—
287C3⋊C8—121no—
288C2×C32—21no—
289C2×C32—22no—
290C2×C32#39211yescharacteristic
291C3⋊D3—81no—
292C3×D3—82no—
293#41—62no—
294#41—62no—
295#41—62no—
296#41—62no—
297#41—182no—
298#41—182no—
299C2×Q8—32no—
300C2×Q8—91no—
301C2×D4—32no—
302C2×D4—62no—
303C2×D4—91no—
304C2×D4—181no—
305C22×C4—62no—
306C22×C4—181no—
307C22×C4C2×D3211yescharacteristic
308Q16—62no—
309Q16—184no—
310SD16—64no—
311SD16—184no—
312SD16—184no—
313D8—62no—
314D8—184no—
315M16—22no—
316M16—62no—
317M16—62no—
318C2×C8—34no—
319C2×C8—182no—
320C2×C8—181no—
321C2×C8#62612yes—
322C22×C3—22no—
323C22×C3—21no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3—124no—
328C22×C3—122no—
329C22×C3#89512yes—
330C22×C3#89511yescharacteristic
331C22×C3#93811yescharacteristic
332C22×C3#93912yes—
333C2×D3—44no—
334C2×D3—42no—
335C2×D3—122no—
336C2×D3—124no—
337C2×D3—122no—
338C2×D3—122no—
339C2×D3—121no—
340C2×D3—121no—
341C3×C4—21no—
342C3×C4—22no—
343C3×C4—21no—
344C3×C4—122no—
345C3×C4#87411yescharacteristic
346C3×C4#88212yes—
347C3×C4#88211yescharacteristic
348C3×C4#93712yes—
349C3×C4#94011yescharacteristic
350C3×C4#95311yescharacteristic
351Dic3—42no—
352Dic3—122no—
353Dic3—121no—
354Dic3—121no—
355C32#201711yescharacteristic, complemented, Sylow
356C23—62no—
357C23—181no—
358C23#379611yescharacteristic
359Q8—62no—
360Q8—62no—
361Q8—182no—
362D4—62no—
363D4—62no—
364D4—122no—
365D4—122no—
366D4—182no—
367D4—362no—
368C2×C4—62no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—181no—
372C2×C4—361no—
373C2×C4—361no—
374C2×C4D3×D1212yes—
375C2×C4#380211yescharacteristic
376C2×C4C2×#62611yescharacteristic
377C2×C4D3×#6712yes—
378C8—22no—
379C8—64no—
380C8—361no—
381C2×C3—22no—
382C2×C3—21no—
383C2×C3—22no—
384C2×C3—22no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—42no—
388C2×C3—242no—
389C2×C3#467111yescharacteristic
390C2×C3#522311yescharacteristic
391D3—82no—
392D3—242no—
393D3—241no—
394D3—241no—
395C22—41no—
396C22—124no—
397C22—122no—
398C22—362no—
399C22—361no—
400C22#6388612yes—
401C22#6391611yescharacteristic
402C4—122no—
403C4—361no—
404C4#6391111yescharacteristic
405C4#6392212yes—
406C4#6397711yescharacteristic
407C3—21no—
408C3#6872711yescharacteristic, complemented
409C3#7212111yescharacteristic, complemented
410C2—22no—
411C2—21no—
412C2—242no—
413C2—721no—
414C2#1059139111yescharacteristic
subgroupG/H
centerC4#63911
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC2×C32#392
hypercenterC2×M16D32
hypocenterC32#2017
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C3×#5748C2semidirect
2. #10594795C2semidirect
3. #10592828C2semidirect
4. #10591071C2semidirect
5. C3×C3⋊#338C2semidirect
26. #5748D3semidirect
33. C3⋊#338D3semidirect
355. C32#2017semidirect
408. C3#68727semidirect
409. C3#72121semidirect

Maximal subgroups

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

subgroupindex
C3×#57482
#105947952
#105928282
#105910712
C3×C3⋊#3382
#721213
#687273