Finite Group Databasebeta

← #11792284

Subgroups — #11792284

Order 1152.

789 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 789 conjugacy classes of subgroups — too many to draw. Only its 124 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
2C2×#63827C212yesmaximal, complemented
3C3×#4973C211yesmaximal, characteristic, complemented
4C2×#63569C212yesmaximal, complemented
5#10589396C211yesmaximal, characteristic, complemented
6#71881—31nomaximal
7#69694—31nomaximal
8C2×C3×#235C2212yes—
9C22×C3×Dic6C2211yescharacteristic
10C2×C32×C4⋊C4C2211yescharacteristic
11C2×#3748C2212yes—
12C3×#930—22no—
13C2×C3×C4×Dic3C2211yescharacteristic
14C3×#897—22no—
15C3×#895—22no—
16C2×#3718—22no—
17#63827—24no—
18#63827—24no—
19C2×#3715—22no—
20C3×C3⋊#95—22no—
21#63569—24no—
22C3×#537—31no—
23C2×#951—32no—
24C2×#942—32no—
25#5510D311yescharacteristic
26C3×#391—31no—
27C2×#837—32no—
28#4973D311yescharacteristic, complemented
29C2×#805—32no—
30#4530—31no—
31#4530—31no—
32#4530—61no—
33C22×C32×C4D412yes—
34C22×C32×C4C2311yescharacteristic
35C2×C3×#67—22no—
36C22×C3×Dic3—21no—
37C22×C3×Dic3—21no—
38C3×#235—22no—
39C3×#235—22no—
40C3×#235—22no—
41C3×#235—22no—
42C2×C3×D12—22no—
43C2×C3×D12D412yes—
44C2×C3×C4×D3—22no—
45C2×C3×Dic6—22no—
46C2×C3×Dic6—22no—
47C2×C3×Dic6—21no—
48C2×C3×Dic6D412yes—
49C2×C3×Dic6D412yescomplemented
50C32×C4⋊C4—22no—
51#3748—22no—
52#3748D414yescomplemented
53C2×C3⋊(C3⋊C8)—22no—
54C3×(C2×D3)⋊C4—24no—
55C3×C4⋊Dic3—22no—
56C3×C4×Dic3—22no—
57#3718—28no—
58#3715—28no—
59#2007—91noSylow
60C2×C3×#41—32no—
61C22×C3×Q8—31no—
62C2×#235C2×D312yes—
63C22×Dic6C2×D311yescharacteristic
64C3×C22⋊Q8—62no—
65C3×#121—62no—
66C3×#121—62no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4C2×D311yescharacteristic
69C2×C3×C4⋊C4C2×D311yescharacteristic
70C2×C3×C42—31no—
71#951—64no—
72#951—64no—
73C2×#216—62no—
74C2×#216—62no—
75#942—64no—
76#942—64no—
77C2×#215—62no—
78C2×#214—62no—
79#930—22no—
80C2×C4×Dic3C2×D311yescharacteristic
81C2×#208—32no—
82C2×#208—32no—
83C2×#208—62no—
84#897—22no—
85#895—22no—
86C3×#95—62no—
87#837—64no—
88C3⋊#95—22no—
89#805—64no—
90C23×C32C2×D411yescharacteristic
91C22×C3×D3—22no—
92C2×C32×C4—22no—
93C2×C32×C4—42no—
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D412yes—
96C2×C32×C4C2×D412yes—
97C2×C32×C4#4111yescharacteristic
98C3×#67—42no—
99C3×#67—42no—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—41no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—41no—
107C3×D12—22no—
108C3×D12—22no—
109C3×D12—82no—
110C3×C4×D3—42no—
111C3×C4×D3—42no—
112C3×Dic6—24no—
113C3×Dic6—22no—
114C3×Dic6—22no—
115C3×Dic6—42no—
116C3⋊(C3⋊C8)—24no—
117C2×#137—92no—
118#537—31no—
119C2×#104—92no—
120#391—31no—
121#343—91no—
122C3×#41—62no—
123C3×#41—62no—
124C3×#41—62no—
125C3×#41—62no—
126C2×C3×Q8—32no—
127C2×C3×Q8—32no—
128C2×C3×Q8—61no—
129C2×C3×Q8—62no—
130C2×C3×Q8—62no—
131C2×C3×D4—32no—
132C2×C3×D4—62no—
133C2×C3×D4—62no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—61no—
137C22×C3×C4—61no—
138C22×C3×C4—62no—
139C22×C3×C4#6712yes—
140C22×C3×C4#6712yes—
141C22×C3×C4C22×D311yescharacteristic
142C22×C3×C4C22×D311yescharacteristic
143C2×#67—22no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146#235—22no—
147#235—22no—
148#235—22no—
149#235—22no—
150C2×D12—22no—
151C2×D12#6712yes—
152C2×C4×D3—22no—
153C2×Dic6—22no—
154C2×Dic6—22no—
155C2×Dic6—21no—
156C2×Dic6D1212yescomplemented
157C2×Dic6#6712yes—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4—62no—
162C3×C22⋊C4—64no—
163C3×C42—62no—
164#216—68no—
165#216—68no—
166#215—68no—
167#214—68no—
168(C2×D3)⋊C4—24no—
169C4⋊Dic3—22no—
170C4×Dic3—22no—
171#208—34no—
172#208—34no—
173#208—64no—
174#208—62no—
175#208—62no—
176#208—122no—
177C2×C3⋊C8—62no—
178C2×C3⋊C8—62no—
179C2×C3⋊C8—122no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32#12011yescharacteristic
183C22×C32#12111yescharacteristic
184C22×C32#12711yescharacteristic
185C2×C3×D3—42no—
186C2×C3×D3—42no—
187C2×C3×D3—42no—
188C32×C4—42no—
189C32×C4#12012yes—
190C32×C4#12112yes—
191C3×Dic3—42no—
192C3×Dic3—42no—
193C3×Dic3—42no—
194C2×#41—32no—
195C22×Q8—31no—
196#137—184no—
197#137—184no—
198C2×Q16—182no—
199C2×SD16—182no—
200C2×M16—92no—
201C22⋊Q8—62no—
202#121—62no—
203#121—62no—
204C2×C4⋊C4D3211yescharacteristic
205C2×C42—31no—
206#104—184no—
207#95—62no—
208C23×C3—21no—
209C23×C3—62no—
210C23×C3C2×#6711yescharacteristic
211C23×C3C2×#6711yescharacteristic
212C22×D3—22no—
213C3×Q8—62no—
214C3×Q8—62no—
215C3×Q8—64no—
216C3×Q8—122no—
217C3×D4—62no—
218C3×D4—62no—
219C3×D4—122no—
220C3×D4—122no—
221C3×D4—242no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—62no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—82no—
235C2×C3×C4—121no—
236C2×C3×C4—121no—
237C2×C3×C4—121no—
238C2×C3×C4—121no—
239C2×C3×C4—122no—
240C2×C3×C4—122no—
241C2×C3×C4C2×D1211yescharacteristic
242C2×C3×C4#23511yescharacteristic
243C2×C3×C4D3×D411yescharacteristic
244C2×C3×C4D3×D412yes—
245C2×C3×C4D3×D412yes—
246C2×C3×C4D3×D412yes—
247C2×C3×C4#23711yescharacteristic
248C2×C3×C4C2×#6712yes—
249#67—42no—
250#67—42no—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—22no—
254C2×Dic3—41no—
255C2×Dic3—41no—
256C2×Dic3—41no—
257C2×Dic3—41no—
258D12—22no—
259D12—22no—
260D12—82no—
261C4×D3—42no—
262C4×D3—42no—
263Dic6—24no—
264Dic6—22no—
265Dic6—22no—
266Dic6—42no—
267C3⋊C8—64no—
268C3⋊C8—64no—
269C3⋊C8—124no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32#39211yescharacteristic
273C2×C32#45511yescharacteristic
274C2×C32#45611yescharacteristic
275C3×D3—82no—
276#41—62no—
277#41—62no—
278#41—62no—
279#41—62no—
280C2×Q8—32no—
281C2×Q8—32no—
282C2×Q8—61no—
283C2×Q8—62no—
284C2×Q8—62no—
285C2×D4—32no—
286C2×D4—62no—
287C2×D4—62no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—62no—
291C22×C4#62512yes—
292C22×C4C2×D3211yescharacteristic
293Q16—188no—
294SD16—188no—
295M16—94no—
296M16—182no—
297C2×C8—182no—
298C4⋊C4—22no—
299C4⋊C4—62no—
300C22⋊C4—64no—
301C42—62no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—122no—
312C22×C3—122no—
313C22×C3—122no—
314C22×C3#93011yescharacteristic
315C22×C3#93711yescharacteristic
316C22×C3#93811yescharacteristic
317C22×C3#94011yescharacteristic
318C22×C3#94011yescharacteristic
319C22×C3#95311yescharacteristic
320C2×D3—42no—
321C2×D3—42no—
322C2×D3—42no—
323C3×C4—22no—
324C3×C4—22no—
325C3×C4—42no—
326C3×C4—42no—
327C3×C4—82no—
328C3×C4—122no—
329C3×C4—122no—
330C3×C4—122no—
331C3×C4#88212yes—
332C3×C4#88412yes—
333C3×C4#93712yes—
334C3×C4#93912yes—
335Dic3—42no—
336Dic3—42no—
337Dic3—42no—
338C32#200711yescharacteristic, complemented, Sylow
339C23—62no—
340C23C2×#62511yescharacteristic
341Q8—62no—
342Q8—64no—
343Q8—62no—
344Q8—122no—
345D4—62no—
346D4—62no—
347D4—122no—
348D4—122no—
349D4—242no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—62no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4#379511yescharacteristic
362C2×C4D3×D1211yescharacteristic
363C2×C4#380212yes—
364C2×C4D3×#6712yes—
365C8—184no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—242no—
376C2×C3#496011yescharacteristic
377C2×C3#506111yescharacteristic
378C2×C3#509511yescharacteristic
379C2×C3#522311yescharacteristic
380C2×C3#524211yescharacteristic
381C2×C3#524611yescharacteristic
382D3—82no—
383C22—21no—
384C22—21no—
385C22—122no—
386C22—122no—
387C22—122no—
388C22#6390811yescharacteristic
389C22#6391411yescharacteristic
390C22#6397611yescharacteristic
391C4—42no—
392C4—122no—
393C4—122no—
394C4—122no—
395C4#6388512yes—
396C4#6392212yes—
397C3—21no—
398C3#6969411yescharacteristic, complemented
399C3#7188111yescharacteristic, complemented
400C2—21no—
401C2—21no—
402C2—242no—
403C2#1059138911yescharacteristic
404C2#1059151211yescharacteristic
405C2#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#2007
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#63827C2semidirect
3. C3×#4973C2semidirect
4. C2×#63569C2semidirect
5. #10589396C2semidirect
28. #4973D3semidirect
49. C2×C3×Dic6D4semidirect
52. #3748D4semidirect
156. C2×Dic6D12semidirect
338. C32#2007semidirect
398. C3#69694semidirect
399. C3#71881semidirect

Maximal subgroups

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

subgroupindex
C3×#55102
C2×#638272
C3×#49732
C2×#635692
#105893962
#718813
#696943