Finite Group Databasebeta

← #11920398

Subgroups — #11920398

Order 1152.

540 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 56 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
1C2×Q8×C22⋊C9C211yesmaximal, characteristic, complemented
2#10593087C211yesmaximal, characteristic
3#10589519C212yesmaximal, complemented
4#10589518C212yesmaximal, complemented
5#10589511C211yesmaximal, characteristic, complemented
6#84059—31nomaximal
7Q8×C22⋊C9—21no—
8Q8×C22⋊C9C2212yes—
9C2×C4×C22⋊C9—21no—
10C2×C4×C22⋊C9C2211yescharacteristic
11#63694—21no—
12C4⋊((C22⋊C9)⋊C2)C2211yescharacteristic
13C4×(C22⋊C9)⋊C2—21no—
14(C22⋊C9)⋊Q8C2211yescharacteristic
15#63505—41nomaximal
16(C22⋊C9)⋊C8C2212yes—
17C23×C3×Q8D311yescharacteristic
18C2×#942—31no—
19C2×#942—31no—
20#5791—31no—
21#5228—32no—
22#5228—32no—
23#5227—32no—
24#5227—32no—
25#5225—31no—
26#5225—31no—
27#5207—31no—
28C22×C22⋊C9D411yescharacteristic
29C2×(C22⋊C9)⋊C2—21no—
30C2×Q8×C9—41no—
31#3697—41no—
32C4×C22⋊C9—22no—
33C4×C22⋊C9D411yescharacteristic
34C4×C22⋊C9C2311yescharacteristic
35(C22⋊C9)⋊C4—21no—
36#3675—42no—
37#3674—42no—
38#3667—41no—
39#3004—91noSylow
40C22×C3×Q8—21no—
41C22×C3×Q8—31no—
42C22×C3×Q8—31no—
43C22×C3×Q8—32no—
44C22×C3×Q8—61no—
45C22×C3×Q8—61no—
46C22×C3×Q8C2×D312yes—
47C23×C3×C4—21no—
48C23×C3×C4C2×D311yescharacteristic
49C2×#235—31no—
50#953—61no—
51C2×#216—32no—
52C2×#216—32no—
53#942—62no—
54#942—62no—
55#942—62no—
56#942—61no—
57#942—61no—
58C2×#215—32no—
59C2×#215—32no—
60#930—31no—
61#930—31no—
62#929—61no—
63C4×#67—61no—
64C4×#67—61no—
65#926—31no—
66#924—31no—
67#924—31no—
68C2×#208—31no—
69C2×#208—31no—
70#835—32no—
71#835—32no—
72#835—32no—
73#835—32no—
74#830—32no—
75#830—32no—
76C2×C22⋊C9—21no—
77C2×C22⋊C9C2×D411yescharacteristic
78(C22⋊C9)⋊C2—41no—
79Q8×C9—42no—
80Q8×C9—81no—
81C2×C4×C9—41no—
82C2×C4×C9—81no—
83#611—81no—
84D36—41no—
85C4×D9—81no—
86Dic18—41no—
87C9⋊C8—42no—
88C23×Q8D911yescharacteristic, complemented
89C2×#137—91no—
90C2×#137—91no—
91#525—91no—
92#451—92no—
93#451—92no—
94#448—92no—
95#448—92no—
96#419—91no—
97#419—91no—
98#407—91no—
99C24×C3#6711yescharacteristic
100C2×C3×Q8—32no—
101C2×C3×Q8—32no—
102C2×C3×Q8—32no—
103C2×C3×Q8—32no—
104C2×C3×Q8—32no—
105C2×C3×Q8—31no—
106C2×C3×Q8—61no—
107C2×C3×Q8—61no—
108C2×C3×Q8—61no—
109C2×C3×Q8—62no—
110C2×C3×Q8—61no—
111C2×C3×Q8—61no—
112C2×C3×Q8—62no—
113C2×C3×Q8—64no—
114C2×C3×Q8S411yescharacteristic
115C22×C3×C4—22no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—31no—
120C22×C3×C4—61no—
121C22×C3×C4—62no—
122C22×C3×C4—61no—
123C22×C3×C4#6711yescharacteristic
124C22×C3×C4C22×D311yescharacteristic
125C2×#67—61no—
126C2×#67—61no—
127C2×#67—61no—
128#235—61no—
129#235—61no—
130#235—61no—
131#235—61no—
132C2×D12—31no—
133C2×C4×D3—61no—
134C2×Dic6—31no—
135#217—61no—
136#217—61no—
137#217—61no—
138#216—64no—
139#216—62no—
140#216—62no—
141#215—64no—
142#215—62no—
143#215—62no—
144(C2×D3)⋊C4—61no—
145(C2×D3)⋊C4—61no—
146C4⋊Dic3—31no—
147C4⋊Dic3—31no—
148Dic3⋊C4—61no—
149Dic3⋊C4—61no—
150C4×Dic3—61no—
151#208—61no—
152#208—61no—
153#208—61no—
154#208—61no—
155C2×C3⋊C8—32no—
156C2×C3⋊C8—32no—
157C22×C9—41no—
158C2×D9—81no—
159C22⋊C9#13711yescharacteristic, complemented
160C4×C9—41no—
161C4×C9—41no—
162C4×C9—82no—
163Dic9—81no—
164C2×#41—91no—
165C22×Q8—21no—
166C22×Q8—31no—
167C22×Q8—31no—
168C22×Q8—32no—
169C22×Q8—61no—
170C22×Q8—61no—
171C22×Q8C2×D912yes—
172C23×C4—21no—
173C23×C4C2×D911yescharacteristic
174#137—182no—
175#137—182no—
176#137—182no—
177#137—181no—
178#137—181no—
179C2×Q16—92no—
180C2×Q16—92no—
181C2×SD16—92no—
182C2×SD16—92no—
183C2×M16—91no—
184C2×M16—91no—
185#123—181no—
186C22⋊Q8—91no—
187C22⋊Q8—91no—
188#121—91no—
189#121—91no—
190#120—181no—
191C4×D4—181no—
192C4×D4—181no—
193#117—91no—
194#103—92no—
195#103—92no—
196#103—92no—
197#103—92no—
198C22⋊C8—92no—
199C22⋊C8—92no—
200C23×C3—21no—
201C23×C3—31no—
202C23×C3—31no—
203C23×C3—61no—
204C23×C3C2×#6711yescharacteristic
205C22×D3—61no—
206C3×Q8—21no—
207C3×Q8—34no—
208C3×Q8—32no—
209C3×Q8—62no—
210C3×Q8—62no—
211C3×Q8—62no—
212C3×Q8—61no—
213C3×Q8C2×S412yes—
214C2×C3×C4—21no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—61no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—62no—
228C2×C3×C4C2×S411yescharacteristic
229#67—121no—
230#67—121no—
231#67—122no—
232#67—121no—
233#67—121no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237D12—61no—
238D12—61no—
239C4×D3—121no—
240C4×D3—121no—
241Dic6—61no—
242Dic6—61no—
243C3⋊C8—62no—
244C3⋊C8—62no—
245C2×C9—41no—
246C2×C9—81no—
247D9—161no—
248C24#61111yescharacteristic
249#41—181no—
250#41—181no—
251#41—181no—
252#41—181no—
253C2×Q8—32no—
254C2×Q8—32no—
255C2×Q8—32no—
256C2×Q8—32no—
257C2×Q8—32no—
258C2×Q8—31no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—62no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×Q8—62no—
266C2×Q8—64no—
267C2×Q8—91no—
268C2×Q8(C22⋊C9)⋊C211yescharacteristic, complemented
269C2×D4—91no—
270C2×D4—181no—
271C2×D4—181no—
272C2×D4—181no—
273C22×C4—22no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—31no—
277C22×C4—31no—
278C22×C4—61no—
279C22×C4—62no—
280C22×C4—61no—
281C22×C4—181no—
282C22×C4#61111yescharacteristic
283C22×C4C22×D911yescharacteristic
284Q16—184no—
285Q16—182no—
286Q16—182no—
287SD16—184no—
288SD16—182no—
289SD16—182no—
290M16—181no—
291M16—181no—
292M16—181no—
293M16—181no—
294C2×C8—92no—
295C2×C8—92no—
296C4⋊C4—91no—
297C4⋊C4—91no—
298C4⋊C4—181no—
299C4⋊C4—181no—
300C22⋊C4—181no—
301C22⋊C4—181no—
302C22⋊C4—181no—
303C22⋊C4—181no—
304C22⋊C4—181no—
305C42—181no—
306C22×C3—31no—
307C22×C3—31no—
308C22×C3—32no—
309C22×C3—31no—
310C22×C3—61no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3#94211yescharacteristic
314C22×C3GL(2,ℤ/4)11yescharacteristic
315C2×D3—121no—
316C2×D3—121no—
317C2×D3—121no—
318C3×C4—22no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—62no—
322C3×C4GL(2,ℤ/4)11yescharacteristic
323C3×C4C22×S411yescharacteristic
324Dic3—121no—
325Dic3—121no—
326Dic3—121no—
327C9—41noSylow
328C23—21no—
329C23—31no—
330C23—31no—
331C23—61no—
332C23—181no—
333C23C2×#61111yescharacteristic
334Q8—21no—
335Q8—34no—
336Q8—32no—
337Q8—62no—
338Q8—62no—
339Q8—62no—
340Q8—61no—
341Q8—181no—
342Q8—181no—
343Q8C2×(C22⋊C9)⋊C212yes—
344D4—181no—
345D4—181no—
346D4—361no—
347D4—361no—
348D4—362no—
349D4—361no—
350D4—361no—
351C2×C4—21no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—31no—
355C2×C4—31no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—62no—
360C2×C4—62no—
361C2×C4—61no—
362C2×C4—62no—
363C2×C4—61no—
364C2×C4—62no—
365C2×C4—181no—
366C2×C4—181no—
367C2×C4—181no—
368C2×C4—361no—
369C2×C4—361no—
370C2×C4C2×(C22⋊C9)⋊C211yescharacteristic
371C8—182no—
372C8—182no—
373C2×C3—21no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—61no—
377C2×C3C2×GL(2,ℤ/4)11yescharacteristic
378D3—241no—
379C22—31no—
380C22—31no—
381C22—32no—
382C22—31no—
383C22—61no—
384C22—61no—
385C22—61no—
386C22—361no—
387C22—361no—
388C22—361no—
389C22#6350511yescharacteristic, complemented
390C22#6369411yescharacteristic
391C4—22no—
392C4—31no—
393C4—31no—
394C4—62no—
395C4—361no—
396C4—361no—
397C4—361no—
398C4#6369411yescharacteristic
399C4C22×(C22⋊C9)⋊C211yescharacteristic
400C3#8549911yescharacteristic
401C2—21no—
402C2—31no—
403C2—31no—
404C2—61no—
405C2—721no—
406C2C2×#6369411yescharacteristic
subgroupG/H
centerC2C2×#63694
commutator, cosocleC4×C22⋊C9C23
FrattiniC3×C4C22×S4
FittingC23×C3×Q8D3
socleC23×C3C2×#67
prime radical#10593087C2
hypercenterC2×Q8(C22⋊C9)⋊C2
hypocenterC22⋊C9#137
LCSG ▷ C4×C22⋊C9 ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C22⋊C9C2semidirect
3. #10589519C2semidirect
4. #10589518C2semidirect
5. #10589511C2semidirect
88. C23×Q8D9semidirect
159. C22⋊C9#137semidirect
268. C2×Q8(C22⋊C9)⋊C2semidirect
389. C22#63505semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C22⋊C92
#105930872
#105895192
#105895182
#105895112
#840593
#635054