Finite Group Databasebeta

← #11778337

Subgroups — #11778337

Order 1152.

603 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 92 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×#5510C211yesmaximal, characteristic
2C3×#4979C211yesmaximal, characteristic, complemented
3#10591171C212yesmaximal, complemented
4#10589394C211yesmaximal, characteristic, complemented
5#10589149C212yesmaximal, complemented
6#71880—31nomaximal
7#69706—31nomaximal
8C2×C3×#235—21no—
9C22×C3×Dic6C2211yescharacteristic
10C2×C32×C4⋊C4C2211yescharacteristic
11C3×#930—21no—
12C3×#930C2212yes—
13C2×C3×Dic3⋊C4—21no—
14C3×#897—22no—
15C3×#895—22no—
16C2×#3715—22no—
17#63657C2212yes—
18C3×C3⋊#95—21no—
19C3×C3⋊#95C2211yescharacteristic
20#63565—22no—
21C3×#537—31no—
22#5510D311yescharacteristic
23C3×#395—31no—
24#5227—32no—
25#5040—32no—
26#4979D311yescharacteristic, complemented
27#4537—32no—
28#4528—31no—
29#4528—31no—
30#4528—61no—
31#4451—32no—
32C22×C32×C4D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C2×C3×#67—22no—
36C22×C3×Dic3—21no—
37C22×C3×Dic3—21no—
38C3×#235—42no—
39C3×#235—42no—
40C2×C3×D12—21no—
41C2×C3×D12—21no—
42C2×C3×D12D412yes—
43C2×C3×C4×D3—22no—
44C2×C3×Dic6—22no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6—42no—
47C32×C4⋊C4—22no—
48C2×C3⋊(C3⋊C8)—22no—
49C3×(C2×D3)⋊C4—22no—
50C3×(C2×D3)⋊C4—22no—
51C3×C4⋊Dic3—21no—
52C3×C4⋊Dic3D412yes—
53C3×Dic3⋊C4—42no—
54#3715—44no—
55#1604—91noSylow
56C2×C3×#41—61no—
57C22×C3×Q8—31no—
58C2×#235—21no—
59C22×Dic6C2×D311yescharacteristic
60C3×C22⋊Q8—62no—
61C3×#121—32no—
62C3×#121—62no—
63C3×#121—61no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—61no—
66C2×C3×C4⋊C4C2×D311yescharacteristic
67C2×C3×C4⋊C4C2×D311yescharacteristic
68C2×#215—62no—
69C2×#214—62no—
70#930—21no—
71#930C2×D312yes—
72C2×Dic3⋊C4—21no—
73#897—22no—
74#895—22no—
75C3×#95—31no—
76C3×#95—61no—
77#835—62no—
78C3⋊#95—21no—
79C3⋊#95C2×D311yescharacteristic
80#830—32no—
81#830—32no—
82#830—62no—
83#810—62no—
84C23×C32C2×D411yescharacteristic
85C22×C3×D3—22no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4SD1612yescomplemented
92C2×C32×C4C2×D412yes—
93C3×#67—42no—
94C3×#67—42no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3—41no—
102C3×D12—42no—
103C3×D12—42no—
104C3×D12—81no—
105C3×C4×D3—44no—
106C3×Dic6—24no—
107C3×Dic6—42no—
108C3×Dic6—42no—
109C3⋊(C3⋊C8)—42no—
110#537—31no—
111#448—92no—
112#395—31no—
113#327—92no—
114#324—91no—
115C3×#41—122no—
116C3×#41—122no—
117C2×C3×Q8—61no—
118C2×C3×Q8—62no—
119C2×C3×Q8—122no—
120C2×C3×D4—32no—
121C2×C3×D4—61no—
122C2×C3×D4—61no—
123C2×C3×D4—62no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—62no—
128C22×C3×C4—61no—
129C22×C3×C4—61no—
130C22×C3×C4#6711yescharacteristic
131C22×C3×C4#6711yescharacteristic
132C22×C3×C4#6711yescharacteristic
133C22×C3×C4#6711yescharacteristic
134C22×C3×C4C22×D311yescharacteristic
135C22×C3×C4C22×D311yescharacteristic
136C2×#67—22no—
137C22×Dic3—21no—
138C22×Dic3—21no—
139#235—42no—
140#235—42no—
141C2×D12—21no—
142C2×D12—21no—
143C2×D12#6712yes—
144C2×C4×D3—22no—
145C2×Dic6—22no—
146C2×Dic6—21no—
147C2×Dic6—42no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4—122no—
154C3×C22⋊C4—62no—
155C3×C22⋊C4—62no—
156#215—124no—
157#214—124no—
158(C2×D3)⋊C4—22no—
159(C2×D3)⋊C4—22no—
160C4⋊Dic3—21no—
161C4⋊Dic3#6712yes—
162Dic3⋊C4—42no—
163C2×C3⋊C8—62no—
164C2×C3⋊C8—62no—
165C2×C3⋊C8—122no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32#12011yescharacteristic
169C22×C32C2×SD1611yescharacteristic
170C22×C32AGL(1,ℤ/8)11yescharacteristic
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—42no—
174C32×C4—22no—
175C32×C4—22no—
176C32×C4—41no—
177C3×Dic3—42no—
178C3×Dic3—42no—
179C3×Dic3—42no—
180C2×#41—61no—
181C22×Q8—31no—
182C2×SD16—182no—
183C22⋊Q8—62no—
184#121—32no—
185#121—62no—
186#121—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4D3211yescharacteristic
189#103—182no—
190C22⋊C8—92no—
191#95—31no—
192#95—61no—
193C23×C3—21no—
194C23×C3—62no—
195C23×C3C2×#6711yescharacteristic
196C23×C3C2×#6711yescharacteristic
197C22×D3—22no—
198C3×Q8—64no—
199C3×Q8—122no—
200C3×Q8—122no—
201C3×D4—122no—
202C3×D4—122no—
203C3×D4—122no—
204C3×D4—122no—
205C3×D4—241no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—124no—
227C2×C3×C4—121no—
228C2×C3×C4—121no—
229C2×C3×C4—121no—
230C2×C3×C4—121no—
231C2×C3×C4#21412yescomplemented
232C2×C3×C4#21512yescomplemented
233C2×C3×C4D3×D412yes—
234C2×C3×C4C2×#6712yes—
235#67—42no—
236#67—42no—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244D12—42no—
245D12—42no—
246D12—81no—
247C4×D3—44no—
248Dic6—24no—
249Dic6—42no—
250Dic6—42no—
251C3⋊C8—122no—
252C3⋊C8—122no—
253C3⋊C8—242no—
254C2×C32—21no—
255C2×C32—21no—
256C2×C32#45011yescharacteristic
257C2×C32#45611yescharacteristic
258C2×C32B(4,𝔽2)11yescharacteristic
259C3×D3—82no—
260#41—122no—
261#41—122no—
262C2×Q8—61no—
263C2×Q8—62no—
264C2×Q8—122no—
265C2×D4—32no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—62no—
269C22×C4—62no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4#62511yescharacteristic
273C22×C4#62511yescharacteristic
274C22×C4C2×D3211yescharacteristic
275SD16—364no—
276C2×C8—182no—
277C4⋊C4—22no—
278C4⋊C4—32no—
279C4⋊C4—61no—
280C4⋊C4—122no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—122no—
293C22×C3—122no—
294C22×C3—122no—
295C22×C3#93211yescharacteristic
296C22×C3C2×#21411yescharacteristic
297C22×C3#93711yescharacteristic
298C22×C3C2×#21511yescharacteristic
299C22×C3#94911yescharacteristic
300C22×C3#95311yescharacteristic
301C2×D3—42no—
302C2×D3—42no—
303C2×D3—42no—
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—81no—
313C3×C4—122no—
314C3×C4—122no—
315C3×C4—122no—
316Dic3—42no—
317Dic3—42no—
318Dic3—42no—
319C32#160411yescharacteristic, complemented, Sylow
320C23—62no—
321C23C2×#62511yescharacteristic
322Q8—64no—
323Q8—122no—
324Q8—122no—
325D4—122no—
326D4—122no—
327D4—122no—
328D4—122no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—124no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4#371512yescomplemented
344C2×C4D3×#6712yes—
345C8—362no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—242no—
356C2×C3#503211yescharacteristic
357C2×C3#504611yescharacteristic
358C2×C3#509511yescharacteristic
359C2×C3#521811yescharacteristic
360C2×C3#524411yescharacteristic
361C2×C3#524611yescharacteristic
362D3—82no—
363C22—21no—
364C22—21no—
365C22—122no—
366C22—122no—
367C22—122no—
368C22#6382311yescharacteristic
369C22C2×#371511yescharacteristic
370C22#6397611yescharacteristic
371C4—22no—
372C4—22no—
373C4—41no—
374C4—122no—
375C4—122no—
376C4—122no—
377C3—21no—
378C3#6970611yescharacteristic, complemented
379C3#7188011yescharacteristic, complemented
380C2—21no—
381C2—21no—
382C2—242no—
383C2#1059116911yescharacteristic
384C2#1059153011yescharacteristic
385C2#1059164811yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32#120
prime radicalC3×#5510C2
hypercenterC2×C4⋊C4D32
hypocenterC32#1604
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#4979C2semidirect
3. #10591171C2semidirect
4. #10589394C2semidirect
5. #10589149C2semidirect
26. #4979D3semidirect
91. C2×C32×C4SD16semidirect
231. C2×C3×C4#214semidirect
232. C2×C3×C4#215semidirect
319. C32#1604semidirect
343. C2×C4#3715semidirect
378. C3#69706semidirect
379. C3#71880semidirect

Maximal subgroups

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

subgroupindex
C3×#55102
C3×#49792
#105911712
#105893942
#105891492
#718803
#697063