Finite Group Databasebeta

← #11778255

Subgroups — #11778255

Order 1152.

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

normal subgroups only

This group has 613 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×#5483C211yesmaximal, characteristic
2#10592646C211yesmaximal, characteristic
3#10591190C212yesmaximal, complemented
4C3×#4525C211yesmaximal, characteristic
5#10589152C212yesmaximal, complemented
6#71893—31nomaximal
7#68115—31nomaximal
8C22×#634C2211yescharacteristic
9C2×C32×C22⋊C4C2211yescharacteristic
10C2×#3757—21no—
11C2×C3×#217—21no—
12C3×#924—21no—
13C3×#924C2212yes—
14C3×#878—22no—
15C3×#878—22no—
16C2×#3719—22no—
17#63658—21no—
18#63658C2211yescharacteristic
19C3×#830C2212yes—
20#63567—22no—
21C3×#543—31no—
22#5483D311yescharacteristic
23#5050—32no—
24#4947—31no—
25#4947—31no—
26#4947—61no—
27#4739—32no—
28C3×#323—31no—
29#4538—32no—
30#4525D311yescharacteristic
31#4467—32no—
32C24×C32D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C22×C3⋊Dic3—21no—
36C22×C3⋊Dic3—21no—
37C2×#634—22no—
38C2×#634—42no—
39C22×C3×Dic3—21no—
40C22×C3×Dic3—21no—
41C2×C3×Dic6—21no—
42C2×C3×Dic6D412yes—
43C32×C22⋊C4—22no—
44#3757—42no—
45C3×#217—22no—
46C3×#217—22no—
47C3×C4⋊Dic3—21no—
48C3×C4⋊Dic3D412yes—
49C3×Dic3⋊C4—22no—
50C3×Dic3⋊C4—22no—
51C2×C3×C3⋊C8—22no—
52#3719—44no—
53#1602—91noSylow
54C22×Dic6—31no—
55C22×Dic6—31no—
56C22×Dic6—61no—
57C3×C22⋊Q8—32no—
58C3×C22⋊Q8—62no—
59C3×C22⋊Q8—62no—
60C3×C22⋊Q8—61no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—61no—
63C2×C3×C22⋊C4C2×D311yescharacteristic
64C2×C3×C22⋊C4C2×D311yescharacteristic
65C2×#217—21no—
66C2×#217—62no—
67C2×#217—61no—
68C2×#217—61no—
69C2×#216—62no—
70#924—21no—
71#924C2×D312yes—
72C2×Dic12—62no—
73#878—22no—
74#878—22no—
75C3×C22⋊C8—32no—
76C3⋊#95—31no—
77C3⋊#95—31no—
78C3⋊#95—61no—
79C3⋊#95—62no—
80C3⋊#95—61no—
81C3⋊#95—61no—
82#830C2×D312yes—
83#816—62no—
84#810—62no—
85C23×C32—21no—
86C23×C32—21no—
87C23×C32—41no—
88C23×C32—41no—
89C23×C32Q1612yescomplemented
90C23×C32C2×D411yescharacteristic
91C2×C32×C4—21no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4C2×D412yes—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—41no—
97C2×C3⋊Dic3—41no—
98C2×C3⋊Dic3—41no—
99C2×C3⋊Dic3—41no—
100C2×C3⋊Dic3—41no—
101#634—24no—
102#634—42no—
103C2×C3×Dic3—22no—
104C2×C3×Dic3—22no—
105C2×C3×Dic3—81no—
106C2×C3×Dic3—81no—
107C3×Dic6—42no—
108C3×Dic6—81no—
109C3×C3⋊C8—42no—
110#543—31no—
111#451—92no—
112#393—91no—
113#328—92no—
114#323—31no—
115C24×C3—21no—
116C24×C3D1211yescharacteristic
117C24×C3#6711yescharacteristic
118C2×C3×Q8—32no—
119C2×C3×Q8—61no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—61no—
123C22×C3×C4—61no—
124C22×C3×C4D1211yescharacteristic
125C22×C3×C4#6711yescharacteristic
126C22×C3×C4C22×D311yescharacteristic
127C22×C3×C4C22×D311yescharacteristic
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—62no—
131C22×Dic3—61no—
132C22×Dic3—61no—
133C22×Dic3—62no—
134C22×Dic3—61no—
135C22×Dic3—61no—
136C2×Dic6—21no—
137C2×Dic6—62no—
138C2×Dic6—62no—
139C2×Dic6—122no—
140C2×Dic6—122no—
141C2×Dic6—122no—
142C2×Dic6—124no—
143C2×Dic6#6712yes—
144C2×C3×C8—62no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—62no—
147C3×C4⋊C4—61no—
148C3×C4⋊C4—62no—
149C3×C22⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—42no—
152C3×C22⋊C4—62no—
153C3×C22⋊C4—62no—
154#217—22no—
155#217—22no—
156#217—122no—
157#217—122no—
158#217—124no—
159#216—124no—
160C4⋊Dic3—21no—
161C4⋊Dic3#6712yes—
162Dic3⋊C4—22no—
163Dic3⋊C4—22no—
164C2×C3⋊C8—22no—
165Dic12—124no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—22no—
169C22×C32—42no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—81no—
173C22×C32#12011yescharacteristic
174C22×C32C2×Q1611yescharacteristic
175C22×C32#13711yescharacteristic
176C32×C4—22no—
177C32×C4—41no—
178C3⋊Dic3—42no—
179C3⋊Dic3—81no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C22×Q8—91no—
183C2×Q16—182no—
184C22⋊Q8—32no—
185C22⋊Q8—62no—
186C22⋊Q8—62no—
187C22⋊Q8—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—181no—
190C2×C22⋊C4D3211yescharacteristic
191#103—182no—
192C22⋊C8—32no—
193#95—91no—
194#95—181no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—22no—
200C23×C3—21no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3Dic1212yescomplemented
210C23×C3#21612yescomplemented
211C23×C3C2×D1211yescharacteristic
212C23×C3C2×#6711yescharacteristic
213C3×Q8—122no—
214C3×Q8—241no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—241no—
228C2×C3×C4—241no—
229C2×C3×C4D3×D412yes—
230C2×C3×C4C2×#6712yes—
231C2×Dic3—22no—
232C2×Dic3—22no—
233C2×Dic3—64no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—81no—
237C2×Dic3—81no—
238C2×Dic3—122no—
239C2×Dic3—121no—
240C2×Dic3—121no—
241C2×Dic3—122no—
242C2×Dic3—121no—
243C2×Dic3—121no—
244C2×Dic3—122no—
245C2×Dic3—121no—
246C2×Dic3—121no—
247C2×Dic3—122no—
248C2×Dic3—121no—
249C2×Dic3—121no—
250C2×Dic3—122no—
251C2×Dic3—121no—
252C2×Dic3—121no—
253Dic6—42no—
254Dic6—64no—
255Dic6—64no—
256Dic6—81no—
257Dic6—124no—
258Dic6—122no—
259Dic6—122no—
260Dic6—124no—
261C3×C8—122no—
262C3⋊C8—42no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—42no—
266C2×C32#45111yescharacteristic
267C2×C32#45511yescharacteristic
268C2×C32B(4,𝔽2)11yescharacteristic
269C24#62611yescharacteristic
270C2×Q8—32no—
271C2×Q8—61no—
272C2×Q8—182no—
273C2×Q8—362no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—181no—
277C22×C4—181no—
278C22×C4#62611yescharacteristic
279C22×C4C2×D3211yescharacteristic
280Q16—364no—
281C2×C8—62no—
282C4⋊C4—32no—
283C4⋊C4—62no—
284C4⋊C4—61no—
285C4⋊C4—62no—
286C22⋊C4—22no—
287C22⋊C4—62no—
288C22⋊C4—62no—
289C22⋊C4—362no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—42no—
304C22×C3—44no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—82no—
314C22×C3#88211yescharacteristic
315C22×C3C2×Dic1211yescharacteristic
316C22×C3#90911yescharacteristic
317C22×C3#94211yescharacteristic
318C22×C3C2×#21611yescharacteristic
319C22×C3#95311yescharacteristic
320C3×C4—22no—
321C3×C4—22no—
322C3×C4—42no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—81no—
326C3×C4—122no—
327C3×C4—122no—
328Dic3—42no—
329Dic3—42no—
330Dic3—124no—
331Dic3—122no—
332Dic3—122no—
333Dic3—242no—
334Dic3—241no—
335Dic3—241no—
336C32#160211yescharacteristic, complemented, Sylow
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23#371912yescomplemented
342C23C2×#62611yescharacteristic
343Q8—122no—
344Q8—184no—
345Q8—241no—
346Q8—362no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—62no—
351C2×C4—62no—
352C2×C4—182no—
353C2×C4—241no—
354C2×C4—241no—
355C2×C4—361no—
356C2×C4—361no—
357C2×C4—361no—
358C2×C4—361no—
359C2×C4—361no—
360C2×C4D3×#6712yes—
361C8—122no—
362C2×C3—22no—
363C2×C3—22no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—42no—
372C2×C3—42no—
373C2×C3—82no—
374C2×C3#473911yescharacteristic
375C2×C3#474111yescharacteristic
376C2×C3#474611yescharacteristic
377C2×C3#522811yescharacteristic
378C2×C3#524211yescharacteristic
379C2×C3#524411yescharacteristic
380C22—21no—
381C22—21no—
382C22—22no—
383C22—42no—
384C22—41no—
385C22—41no—
386C22—81no—
387C22#6383311yescharacteristic
388C22C2×#371911yescharacteristic
389C22#6397711yescharacteristic
390C4—22no—
391C4—41no—
392C4—122no—
393C4—122no—
394C4—362no—
395C4—721no—
396C3—21no—
397C3#6811511yescharacteristic, complemented
398C3#7189311yescharacteristic, complemented
399C2—21no—
400C2—21no—
401C2—42no—
402C2#1059118611yescharacteristic
403C2#1059151311yescharacteristic
404C2#1059153111yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C22⋊C4C22
socleC22×C32#120
prime radicalC24×C32D4
hypercenterC2×C22⋊C4D32
hypocenterC32#1602
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #10591190C2semidirect
5. #10589152C2semidirect
89. C23×C32Q16semidirect
209. C23×C3Dic12semidirect
210. C23×C3#216semidirect
336. C32#1602semidirect
341. C23#3719semidirect
397. C3#68115semidirect
398. C3#71893semidirect

Maximal subgroups

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

subgroupindex
C3×#54832
#105926462
#105911902
C3×#45252
#105891522
#718933
#681153