Finite Group Databasebeta

← #11778709

Subgroups — #11778709

Order 1152.

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

normal subgroups only

This group has 521 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
2C3×#4949C211yesmaximal, characteristic
3#10591132C212yesmaximal, complemented
4#10589391C211yesmaximal, characteristic
5#10589153C212yesmaximal, complemented
6#71891—31nomaximal
7#69716—31nomaximal
8C2×C32×C22⋊C4C2211yescharacteristic
9C2×C3×#217—21no—
10C2×C3×#217—21no—
11C2×C3×C4⋊Dic3C2211yescharacteristic
12C3×#924—21no—
13C3×#924C2212yes—
14C2×C3×Dic3⋊C4—21no—
15C3×#878—22no—
16C3×#878—22no—
17#63657C2212yes—
18C3×C3⋊#95C2211yescharacteristic
19#63576—22no—
20#63565—22no—
21C3×#543—31no—
22#5483D311yescharacteristic
23C3×#399—31no—
24#5044—32no—
25#4970—32no—
26#4949D311yescharacteristic
27#4538—32no—
28#4525—31no—
29#4525—31no—
30#4525—61no—
31#4452—32no—
32C24×C32D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C22×C3×Dic3—21no—
36C22×C3×Dic3—21no—
37C22×C3×Dic3—21no—
38C22×C3×Dic3—21no—
39C2×C3×Dic6—21no—
40C2×C3×Dic6D412yes—
41C32×C22⋊C4—22no—
42C2×C3⋊(C3⋊C8)—22no—
43C3×#217—22no—
44C3×#217—22no—
45C3×#217—42no—
46C3×C4⋊Dic3—21no—
47C3×C4⋊Dic3—22no—
48C3×C4⋊Dic3D412yes—
49C3×Dic3⋊C4—22no—
50C3×Dic3⋊C4—22no—
51C3×Dic3⋊C4—42no—
52#1614—91noSylow
53C3×C22⋊Q8—32no—
54C3×C22⋊Q8—62no—
55C3×C22⋊Q8—62no—
56C3×C22⋊Q8—61no—
57C2×C3×C4⋊C4—31no—
58C2×C3×C4⋊C4—61no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—61no—
61C2×C3×C22⋊C4—61no—
62C2×C3×C22⋊C4C2×D311yescharacteristic
63C2×C3×C22⋊C4C2×D311yescharacteristic
64C2×#217—21no—
65C2×#217—21no—
66C2×C4⋊Dic3C2×D311yescharacteristic
67#924—21no—
68#924C2×D312yes—
69C2×Dic3⋊C4—21no—
70#878—22no—
71#878—22no—
72C3×#95—31no—
73#835—62no—
74C3⋊#95C2×D311yescharacteristic
75#830—32no—
76#830—32no—
77#830—62no—
78#810—62no—
79#807—62no—
80#807—62no—
81C23×C32—21no—
82C23×C32—21no—
83C23×C32—41no—
84C23×C32—41no—
85C23×C32Q1612yes—
86C23×C32C2×D411yescharacteristic
87C2×C32×C4—21no—
88C2×C32×C4—41no—
89C2×C32×C4—41no—
90C2×C32×C4C2×D412yes—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—81no—
100C2×C3×Dic3—81no—
101C3×Dic6—42no—
102C3×Dic6—81no—
103C3⋊(C3⋊C8)—42no—
104#543—31no—
105#484—92no—
106#399—31no—
107#328—92no—
108#323—91no—
109C24×C3—21no—
110C24×C3#6711yescharacteristic
111C24×C3#6711yescharacteristic
112C2×C3×Q8—32no—
113C2×C3×Q8—61no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—61no—
117C22×C3×C4—61no—
118C22×C3×C4—61no—
119C22×C3×C4—61no—
120C22×C3×C4#6711yescharacteristic
121C22×C3×C4#6711yescharacteristic
122C22×C3×C4C22×D311yescharacteristic
123C22×C3×C4C22×D311yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C2×Dic6—21no—
129C2×Dic6#6712yes—
130C3×C4⋊C4—32no—
131C3×C4⋊C4—62no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—62no—
134C3×C4⋊C4—62no—
135C3×C4⋊C4—122no—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—42no—
139C3×C22⋊C4—62no—
140C3×C22⋊C4—62no—
141C3×C22⋊C4—122no—
142#217—22no—
143#217—22no—
144#217—42no—
145C4⋊Dic3—21no—
146C4⋊Dic3—22no—
147C4⋊Dic3#6712yes—
148Dic3⋊C4—22no—
149Dic3⋊C4—22no—
150Dic3⋊C4—42no—
151C2×C3⋊C8—62no—
152C2×C3⋊C8—62no—
153C2×C3⋊C8—122no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—42no—
158C22×C32—41no—
159C22×C32—41no—
160C22×C32—81no—
161C22×C32#12011yescharacteristic
162C22×C32C2×Q1611yescharacteristic
163C22×C32#13711yescharacteristic
164C32×C4—22no—
165C32×C4—41no—
166C3×Dic3—42no—
167C3×Dic3—42no—
168C3×Dic3—42no—
169C3×Dic3—81no—
170C22⋊Q8—32no—
171C22⋊Q8—62no—
172C22⋊Q8—62no—
173C22⋊Q8—61no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—61no—
176C2×C22⋊C4—61no—
177C2×C22⋊C4—61no—
178C2×C22⋊C4D3211yescharacteristic
179#107—182no—
180#103—182no—
181C22⋊C8—92no—
182#95—31no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—22no—
188C23×C3—21no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—81no—
196C23×C3—81no—
197C23×C3#21612yes—
198C23×C3#21612yes—
199C23×C3C2×#6711yescharacteristic
200C23×C3C2×#6711yescharacteristic
201C3×Q8—122no—
202C3×Q8—241no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—62no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×C3×C4—121no—
217C2×C3×C4—121no—
218C2×C3×C4—121no—
219C2×C3×C4—121no—
220C2×C3×C4—121no—
221C2×C3×C4—241no—
222C2×C3×C4—241no—
223C2×C3×C4D3×D412yes—
224C2×C3×C4C2×#6712yes—
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—81no—
234C2×Dic3—81no—
235Dic6—42no—
236Dic6—81no—
237C3⋊C8—122no—
238C3⋊C8—122no—
239C3⋊C8—242no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—42no—
243C2×C32#45111yescharacteristic
244C2×C32#45311yescharacteristic
245C2×C32#45811yescharacteristic
246C24#62511yescharacteristic
247C2×Q8—32no—
248C2×Q8—61no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4#62511yescharacteristic
254C22×C4C2×D3211yescharacteristic
255C2×C8—182no—
256C4⋊C4—32no—
257C4⋊C4—62no—
258C4⋊C4—61no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—122no—
262C22⋊C4—22no—
263C22⋊C4—62no—
264C22⋊C4—62no—
265C22⋊C4—122no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—82no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—82no—
290C22×C3#93711yescharacteristic
291C22×C3#94211yescharacteristic
292C22×C3C2×#21611yescharacteristic
293C22×C3C2×#21611yescharacteristic
294C22×C3#95111yescharacteristic
295C22×C3#95311yescharacteristic
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—42no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—81no—
302C3×C4—122no—
303C3×C4—122no—
304C3×C4—122no—
305C3×C4—241no—
306Dic3—42no—
307Dic3—42no—
308Dic3—42no—
309Dic3—81no—
310C32#161411yescharacteristic, complemented, Sylow
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23#371812yes—
316C23C2×#62511yescharacteristic
317Q8—122no—
318Q8—241no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—121no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4D3×#6712yes—
333C8—362no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—42no—
344C2×C3—42no—
345C2×C3—82no—
346C2×C3#505011yescharacteristic
347C2×C3#505311yescharacteristic
348C2×C3#507711yescharacteristic
349C2×C3#522811yescharacteristic
350C2×C3#524111yescharacteristic
351C2×C3#524311yescharacteristic
352C22—21no—
353C22—21no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22#6382711yescharacteristic
360C22C2×#371811yescharacteristic
361C22#6397611yescharacteristic
362C4—22no—
363C4—41no—
364C4—122no—
365C4—122no—
366C4—122no—
367C4—241no—
368C3—21no—
369C3#6971611yescharacteristic, complemented
370C3#7189111yescharacteristic, complemented
371C2—21no—
372C2—21no—
373C2—42no—
374C2#1059118511yescharacteristic
375C2#1059150911yescharacteristic
376C2#1059153311yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C22⋊C4C22
socleC22×C32#120
prime radicalC24×C32D4
hypercenterC2×C22⋊C4D32
hypocenterC32#1614
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. #10591132C2semidirect
5. #10589153C2semidirect
310. C32#1614semidirect
369. C3#69716semidirect
370. C3#71891semidirect

Maximal subgroups

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

subgroupindex
C3×#54832
C3×#49492
#105911322
#105893912
#105891532
#718913
#697163