Finite Group Databasebeta

← #11780201

Subgroups — #11780201

Order 1152.

522 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 104 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×Q8×D12C211yesmaximal, characteristic
2#10592730C211yesmaximal, characteristic, complemented
3#10590311C211yesmaximal, characteristic, complemented
4#10590272C211yesmaximal, characteristic
5C3×#4535C211yesmaximal, characteristic, complemented
6#10589092C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic
8#71916—31nomaximal
9#68273—31nomaximal
10C2×C3×D3×Q8—21no—
11C32×C4×Q8C2211yescharacteristic
12C2×#3756—21no—
13#64078C2211yescharacteristic
14C3×#897—21no—
15C3×#897—22no—
16C3×C4×D12—21no—
17C3×C4×D12C2211yescharacteristic
18C3×#869—21no—
19C3×#869C2211yescharacteristic
20#63637—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4×C3⋊C8C2211yescharacteristic
24#63575—21no—
25#63575—22no—
26#63563—21no—
27C3×D4×Q8—31no—
28Q8×D12D311yescharacteristic
29#5078—31no—
30#5031—31no—
31#5031—31no—
32#5031—61no—
33#5007—31no—
34#4846—31no—
35#4682—31no—
36C3×Q8⋊C8—31no—
37#4547—31no—
38#4535D311yescharacteristic, complemented
39D4⋊(C3⋊C8)—31no—
40#4486—31no—
41#4459—31no—
42C2×C32×Q8D411yescharacteristic
43C2×#634—21no—
44C3×D3×Q8—42no—
45C3×D3×Q8—42no—
46C2×C3×D12D411yescharacteristic
47C2×C3×C4×D3—22no—
48C2×C3×C4×D3—21no—
49C2×C3×Dic6—22no—
50C2×C3×Dic6—21no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4D411yescharacteristic
53C32×C42—21no—
54C32×C42C2311yescharacteristic
55#3756—42no—
56C4⋊(C3⋊Dic3)—21no—
57C4⋊(C3⋊Dic3)—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C3×(C2×D3)⋊C4—22no—
60C3×(C2×D3)⋊C4—21no—
61C3×C4⋊Dic3—22no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3D412yes—
64C3×C4⋊Dic3D411yescharacteristic
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67#1649—91noSylow
68C22×C3×Q8—61no—
69C2×D3×Q8—21no—
70C3×C4⋊Q8—31no—
71C3×C4⋊Q8—61no—
72C3×C22⋊Q8—62no—
73C3×C22⋊Q8—61no—
74C3×C4×Q8—21no—
75C3×C4×Q8C2×D311yescharacteristic
76C3×C4×Q8C2×D311yescharacteristic
77C3×C4×D4—31no—
78C3×C4×D4—61no—
79C2×#216—62no—
80C2×#216—61no—
81C2×#216—61no—
82#897—21no—
83#897—22no—
84C4×D12—21no—
85C4×D12C2×D311yescharacteristic
86#869—21no—
87#869—31no—
88#869—31no—
89#869—61no—
90#869C2×D311yescharacteristic
91C3×C4⋊C8—31no—
92C3×C4×C8—31no—
93#832—61no—
94#821—61no—
95#818—62no—
96#818—61no—
97#810—62no—
98#810—61no—
99#810—61no—
100#807—61no—
101#807—62no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—31no—
104C4⋊(C3⋊C8)—61no—
105C4⋊(C3⋊C8)C2×D311yescharacteristic
106C4×C3⋊C8C2×D311yescharacteristic
107C22×C3×D3—21no—
108C32×Q8—21no—
109C32×Q8D812yes—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C3⋊Dic3—41no—
116#634—41no—
117#634—41no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—22no—
120C3×D12—21no—
121C3×D12Q1612yescomplemented
122C3×C4×D3—44no—
123C3×C4×D3—41no—
124C3×C4×D3—41no—
125C3×Dic6—44no—
126C3×Dic6—41no—
127C3×Dic6—41no—
128C3⋊(C3⋊C8)—41no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—41no—
131D4×Q8—31no—
132#500—91no—
133#474—91no—
134#462—91no—
135#332—91no—
136Q8⋊C8—31no—
137D4⋊C8—91no—
138C2×C3×Q8—21no—
139C2×C3×Q8—62no—
140C2×C3×Q8—61no—
141C2×C3×Q8—122no—
142C2×C3×Q8—122no—
143C2×C3×Q8D1211yescharacteristic
144C2×C3×Q8#6711yescharacteristic
145C2×C3×D4—31no—
146C22×C3×C4—62no—
147C22×C3×C4—61no—
148D3×Q8—42no—
149D3×Q8—42no—
150C2×D12#6711yescharacteristic
151C2×C4×D3—22no—
152C2×C4×D3—21no—
153C2×Dic6—22no—
154C2×Dic6—21no—
155C2×Dic6—62no—
156C2×Dic6—61no—
157C2×Dic6—61no—
158C2×C3×C8—61no—
159C2×C3×C8—61no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—32no—
165C3×C4⋊C4—41no—
166C3×C4⋊C4—62no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4D1211yescharacteristic
169C3×C4⋊C4#6711yescharacteristic
170C3×C22⋊C4—62no—
171C3×C22⋊C4—61no—
172C3×C42—21no—
173C3×C42—21no—
174C3×C42—21no—
175C3×C42—41no—
176C3×C42C22×D311yescharacteristic
177C3×C42C22×D311yescharacteristic
178#216—124no—
179#216—122no—
180#216—122no—
181(C2×D3)⋊C4—22no—
182(C2×D3)⋊C4—21no—
183C4⋊Dic3—22no—
184C4⋊Dic3—21no—
185C4⋊Dic3—61no—
186C4⋊Dic3—61no—
187C4⋊Dic3—61no—
188C4⋊Dic3—61no—
189C4⋊Dic3—121no—
190C4⋊Dic3—121no—
191C4⋊Dic3#6711yescharacteristic
192C4⋊Dic3#6712yes—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—62no—
196C2×C3⋊C8—61no—
197C2×C3⋊C8—61no—
198C22×C32#12011yescharacteristic
199C2×C3×D3—22no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C32×C4—22no—
203C32×C4—21no—
204C32×C4—41no—
205C32×C4C2×D811yescharacteristic
206C32×C4C2×Q1611yescharacteristic
207C32×C4AGL(1,ℤ/8)11yescharacteristic
208C32×C4#13711yescharacteristic
209C3⋊Dic3—81no—
210C3×Dic3—41no—
211C3×Dic3—42no—
212C22×Q8—61no—
213C2×Q16—181no—
214C4⋊Q8—31no—
215C4⋊Q8—61no—
216C4⋊Q8—91no—
217C22⋊Q8—62no—
218C22⋊Q8—61no—
219C4×Q8D3211yescharacteristic
220C4×D4—31no—
221C4×D4—61no—
222#107—181no—
223#107—182no—
224C4⋊C8—31no—
225C4⋊C8—91no—
226#103—181no—
227#102—181no—
228C4×C8—31no—
229C23×C3—61no—
230C22×D3—21no—
231C3×Q8—21no—
232C3×Q8—21no—
233C3×Q8—22no—
234C3×Q8—41no—
235C3×Q8—124no—
236C3×Q8—121no—
237C3×Q8—121no—
238C3×Q8D2412yes—
239C3×Q8#21312yes—
240C3×D4—32no—
241C3×D4—61no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—62no—
252C2×C3×C4—61no—
253C2×C3×C4—124no—
254C2×C3×C4—121no—
255C2×C3×C4—121no—
256C2×C3×C4C2×D1211yescharacteristic
257C2×C3×C4D3×D411yescharacteristic
258C2×C3×C4D3×D411yescharacteristic
259C2×C3×C4C2×#6711yescharacteristic
260C2×C3×C4C2×#6711yescharacteristic
261C2×C3×C4C2×#6711yescharacteristic
262C2×Dic3—21no—
263C2×Dic3—22no—
264C2×Dic3—122no—
265C2×Dic3—121no—
266C2×Dic3—121no—
267D12—21no—
268D12#21612yescomplemented
269C4×D3—44no—
270C4×D3—41no—
271C4×D3—41no—
272Dic6—44no—
273Dic6—41no—
274Dic6—41no—
275Dic6—122no—
276Dic6—122no—
277Dic6—121no—
278Dic6—121no—
279Dic6—121no—
280Dic6—121no—
281C3×C8—62no—
282C3×C8—121no—
283C3⋊C8—22no—
284C3⋊C8—41no—
285C3⋊C8—122no—
286C3⋊C8—121no—
287C3⋊C8—121no—
288C2×C32#44711yescharacteristic
289C2×C32#45111yescharacteristic
290C2×C32#45611yescharacteristic
291C3×D3—42no—
292C2×Q8—62no—
293C2×Q8—61no—
294C2×Q8—122no—
295C2×Q8—122no—
296C2×Q8—181no—
297C2×Q8#62611yescharacteristic
298C2×D4—31no—
299C22×C4—62no—
300C22×C4—61no—
301Q16—362no—
302C2×C8—61no—
303C2×C8—61no—
304C2×C8—181no—
305C4⋊C4—21no—
306C4⋊C4—31no—
307C4⋊C4—32no—
308C4⋊C4—62no—
309C4⋊C4—61no—
310C4⋊C4—181no—
311C4⋊C4—181no—
312C4⋊C4#62611yescharacteristic
313C22⋊C4—62no—
314C22⋊C4—61no—
315C42—21no—
316C42C2×D3211yescharacteristic
317C22×C3—21no—
318C22×C3—62no—
319C22×C3—121no—
320C22×C3—121no—
321C22×C3#88211yescharacteristic
322C22×C3#95311yescharacteristic
323C2×D3—22no—
324C2×D3—41no—
325C2×D3—41no—
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—42no—
336C3×C4—41no—
337C3×C4—41no—
338C3×C4—81no—
339C3×C4—122no—
340C3×C4—121no—
341C3×C4C2×D2411yescharacteristic
342C3×C4#90811yescharacteristic
343C3×C4#91511yescharacteristic
344C3×C4D3×Q1611yescharacteristic
345C3×C4C2×#21311yescharacteristic
346C3×C4#93211yescharacteristic
347C3×C4#94211yescharacteristic
348C3×C4C2×#21611yescharacteristic
349Dic3—41no—
350Dic3—42no—
351Dic3—242no—
352Dic3—241no—
353Dic3—241no—
354C32#164911yescharacteristic, complemented, Sylow
355C23—61no—
356Q8—21no—
357Q8—124no—
358Q8—121no—
359Q8—121no—
360Q8—361no—
361Q8—361no—
362Q8#371412yes—
363D4—32no—
364D4—61no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—62no—
368C2×C4—61no—
369C2×C4—124no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—361no—
373C2×C4C2×#62611yescharacteristic
374C2×C4D3×#6711yescharacteristic
375C2×C4D3×#6711yescharacteristic
376C8—62no—
377C8—121no—
378C8—361no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—122no—
383C2×C3#473211yescharacteristic
384C2×C3#480911yescharacteristic
385C2×C3#482611yescharacteristic
386C2×C3#521711yescharacteristic
387C2×C3#522811yescharacteristic
388C2×C3#524611yescharacteristic
389D3—42no—
390C22—62no—
391C22—121no—
392C22—121no—
393C22#6397711yescharacteristic
394C4—22no—
395C4—21no—
396C4—41no—
397C4—122no—
398C4—121no—
399C4—721no—
400C4C2×#371411yescharacteristic
401C4#6383011yescharacteristic
402C4D3×#21611yescharacteristic
403C4#6394611yescharacteristic
404C3—21no—
405C3#6827311yescharacteristic, complemented
406C3#7191611yescharacteristic, complemented
407C2—122no—
408C2#1059116211yescharacteristic
409C2#1059159511yescharacteristic
410C2#1059168911yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×Q8C22
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC4×Q8D32
hypocenterC32#1649
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. #10592730C2semidirect
3. #10590311C2semidirect
5. C3×#4535C2semidirect
6. #10589092C2semidirect
38. #4535D3semidirect
121. C3×D12Q16semidirect
268. D12#216semidirect
354. C32#1649semidirect
405. C3#68273semidirect
406. C3#71916semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×D122
#105927302
#105903112
#105902722
C3×#45352
#105890922
#105890812
#719163
#682733