Finite Group Databasebeta

← C3×#85535 #11922821

Subgroups — C3×#85535 #11922821

Order 1152.

639 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3×2−1+4⋊C3C211yesmaximal, characteristic
2C22×C3×#226C211yesmaximal, characteristic, complemented
3C3×#5425C212yesmaximal, complemented
4C3×#5420C211yesmaximal, characteristic, complemented
5C3×#5419C212yesmaximal, complemented
6#85535C311yesmaximal, characteristic, direct product factor
7C3×#3011—31nomaximal
8C3×2−1+4⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C22×C3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3×#226—22no—
13C2×C3×#226C2212yes—
14C3×#935—41nomaximal
15C3×#859C2212yes—
16C2×C3×2−1+4D311yescharacteristic
17C2×2−1+4⋊C3—21no—
18C2×2−1+4⋊C3C2×C311yescharacteristic
19C22×#226C2×C311yescharacteristic, complemented
20C2×C3×#137—31no—
21C22×C3×Q16—31no—
22C3×#536—31no—
23#5425C2×C312yescomplemented
24#5420C2×C311yescharacteristic, complemented
25#5419C2×C312yescomplemented
26C3×#452—32no—
27C3×#452—32no—
28C3×#451—32no—
29C3×#451—32no—
30C3×#419—31no—
31C3×#416—31no—
32C3×#408—31no—
33C2×C32×D4—41no—
34C22×C3×Dic3—41no—
35C3×#41⋊C3—21no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)D412yes—
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39C3×#226—22no—
40C3×#217—42no—
41C3×#217—41no—
42C3×Dic3⋊C4—42no—
43#3011—31noSylow
44C3×2−1+4—22no—
45C3×2−1+4—62no—
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41—31no—
49C2×C3×#41C2×D311yescharacteristic
50C22×C3×Q8—31no—
51C22×C3×Q8—31no—
52C22×C3×Q8C2×D311yescharacteristic
53C22×C3×Q8C2×D311yescharacteristic
542−1+4⋊C3—24no—
552−1+4⋊C3—22no—
56C2×#41⋊C3—21no—
57C2×#41⋊C3C22×C311yescharacteristic
58C22×SL(2,𝔽3)—21no—
59C22×SL(2,𝔽3)—21no—
60C22×SL(2,𝔽3)C22×C311yescharacteristic
61C22×SL(2,𝔽3)C22×C311yescharacteristic
62C2×#226—22no—
63C2×#226C22×C312yes—
64C3×#137—64no—
65C2×C3×Q16—32no—
66C2×C3×Q16—32no—
67C2×C3×Q16—32no—
68C2×C3×Q16—62no—
69C2×C3×Q16—62no—
70C2×C3×SD16—32no—
71C2×C3×M16—31no—
72C22×C3×C8—31no—
73C3×C4⋊Q8—61no—
74C3×#124—61no—
75C3×#123—61no—
76C3×#123—61no—
77C3×C22⋊Q8—32no—
78C3×C22⋊Q8—32no—
79C3×#117—31no—
80#935—41no—
81#859C22×C312yes—
82C3×#103—32no—
83C3×#103—32no—
84C3×#103—32no—
85C3×#102—32no—
86C3×C22⋊C8—32no—
87C3×C22⋊C8—32no—
88C23×C32—41no—
89C23×C32—41no—
90C32×D4—82no—
91C2×C32×C4—41no—
92C2×C3×Dic3—42no—
93C2×C3×Dic3—42no—
94C2×C3×Dic3—82no—
95C3×SL(2,𝔽3)C2×D411yescharacteristic
96C2×2−1+4C3×D311yescharacteristic
97C2×#137—31no—
98C22×Q16—31no—
99#536—31no—
100#452—32no—
101#452—32no—
102#451—32no—
103#451—32no—
104#419—31no—
105#416—31no—
106#408—31no—
107C3×#41—21no—
108C3×#41—34no—
109C3×#41—62no—
110C3×#41—62no—
111C3×#41—62no—
112C3×#41—62no—
113C3×#41—62no—
114C3×#41—61no—
115C2×C3×Q8—21no—
116C2×C3×Q8—31no—
117C2×C3×Q8—31no—
118C2×C3×Q8—31no—
119C2×C3×Q8—32no—
120C2×C3×Q8—32no—
121C2×C3×Q8—61no—
122C2×C3×Q8—62no—
123C2×C3×Q8—62no—
124C2×C3×Q8—62no—
125C2×C3×Q8—62no—
126C2×C3×Q8—61no—
127C2×C3×Q8#6712yes—
128C2×C3×Q8C22×D311yescharacteristic
129C2×C3×D4—31no—
130C2×C3×D4—31no—
131C2×C3×D4—31no—
132C2×C3×D4—41no—
133C2×C3×D4—81no—
134C2×C3×D4S411yescharacteristic
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—31no—
138C22×C3×C4—61no—
139C22×C3×C4—61no—
140C22×Dic3—41no—
141#41⋊C3—22no—
142#41⋊C3—21no—
143C2×SL(2,𝔽3)—22no—
144C2×SL(2,𝔽3)—21no—
145C2×SL(2,𝔽3)—22no—
146C2×SL(2,𝔽3)—21no—
147C2×SL(2,𝔽3)C3×D412yes—
148C2×SL(2,𝔽3)C23×C311yescharacteristic
149#226—22no—
150C3×Q16—64no—
151C3×Q16—62no—
152C3×Q16—64no—
153C3×Q16—62no—
154C3×SD16—64no—
155C3×M16—62no—
156C2×C3×C8—32no—
157C2×C3×C8—32no—
158C2×C3×C8—62no—
159C3×C4⋊C4—32no—
160C3×C4⋊C4—62no—
161C3×C4⋊C4—62no—
162C3×C22⋊C4—62no—
163C3×C22⋊C4—61no—
164C3×C22⋊C4—62no—
165C3×C42—61no—
166#217—42no—
167#217—41no—
168Dic3⋊C4—42no—
169C22×C32—42no—
170C22×C32—41no—
171C22×C32—81no—
172C22×C32—82no—
173C22×C32—82no—
174C32×C4—81no—
175C3×Dic3—82no—
176C3×Dic3—82no—
1772−1+4—22no—
1782−1+4—62no—
179C2×#41—31no—
180C2×#41—31no—
181C2×#41—31no—
182C2×#41C2×C3×D311yescharacteristic
183C22×Q8—31no—
184C22×Q8—31no—
185C22×Q8C2×C3×D311yescharacteristic
186C22×Q8C2×C3×D311yescharacteristic
187#137—64no—
188C2×Q16—32no—
189C2×Q16—32no—
190C2×Q16—32no—
191C2×Q16—62no—
192C2×Q16—62no—
193C2×SD16—32no—
194C2×M16—31no—
195C22×C8—31no—
196C4⋊Q8—61no—
197#124—61no—
198#123—61no—
199#123—61no—
200C22⋊Q8—32no—
201C22⋊Q8—32no—
202#117—31no—
203#103—32no—
204#103—32no—
205#103—32no—
206#102—32no—
207C22⋊C8—32no—
208C22⋊C8—32no—
209C23×C3—31no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—81no—
213C23×C3—81no—
214C23×C3C2×S411yescharacteristic
215C23×C3C2×S411yescharacteristic
216C3×Q8—32no—
217C3×Q8—31no—
218C3×Q8—32no—
219C3×Q8—62no—
220C3×Q8—61no—
221C3×Q8—61no—
222C3×Q8—62no—
223C3×Q8—62no—
224C3×Q8—122no—
225C3×Q8C2×#6711yescharacteristic
226C3×D4—22no—
227C3×D4—32no—
228C3×D4—62no—
229C3×D4—61no—
230C3×D4—62no—
231C3×D4—82no—
232C3×D4—84no—
233C2×C3×C4—32no—
234C2×C3×C4—32no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—41no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—64no—
244C2×C3×C4—62no—
245C2×C3×C4—62no—
246C2×C3×C4—81no—
247C2×C3×C4—122no—
248C2×C3×C4C2×S411yescharacteristic
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—82no—
252SL(2,𝔽3)—21no—
253SL(2,𝔽3)C2×C3×D411yescharacteristic
254C3×C8—62no—
255C3×C8—62no—
256C2×C32—42no—
257C2×C32—41no—
258C2×C32—82no—
259C2×C32—161no—
260#41—21no—
261#41—34no—
262#41—62no—
263#41—62no—
264#41—62no—
265#41—62no—
266#41—62no—
267#41—61no—
268C2×Q8—21no—
269C2×Q8—31no—
270C2×Q8—31no—
271C2×Q8—31no—
272C2×Q8—32no—
273C2×Q8—32no—
274C2×Q8—61no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—62no—
278C2×Q8—62no—
279C2×Q8—61no—
280C2×Q8C3×#6712yes—
281C2×Q8C22×C3×D311yescharacteristic
282C2×D4—31no—
283C2×D4—31no—
284C2×D4—31no—
285C2×D4C3×S411yescharacteristic
286C22×C4—31no—
287C22×C4—31no—
288C22×C4—31no—
289C22×C4—61no—
290C22×C4—61no—
291Q16—64no—
292Q16—62no—
293Q16—64no—
294Q16—62no—
295SD16—64no—
296M16—62no—
297C2×C8—32no—
298C2×C8—32no—
299C2×C8—62no—
300C4⋊C4—32no—
301C4⋊C4—62no—
302C4⋊C4—62no—
303C22⋊C4—62no—
304C22⋊C4—61no—
305C22⋊C4—62no—
306C42—61no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—21no—
310C22×C3—42no—
311C22×C3—41no—
312C22×C3—62no—
313C22×C3—61no—
314C22×C3—82no—
315C22×C3—81no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3—82no—
319C22×C3—81no—
320C22×C3—162no—
321C22×C3—162no—
322C22×C3GL(2,ℤ/4)12yes—
323C22×C3C22×S411yescharacteristic
324C3×C4—21no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—32no—
328C3×C4—61no—
329C3×C4—81no—
330C3×C4—82no—
331C3×C4—122no—
332C3×C4—122no—
333Dic3—82no—
334Dic3—82no—
335C32—41noSylow
336C23—31no—
337C23C2×C3×S411yescharacteristic
338C23C2×C3×S411yescharacteristic
339Q8—32no—
340Q8—31no—
341Q8—32no—
342Q8—62no—
343Q8—61no—
344Q8—61no—
345Q8—62no—
346Q8—62no—
347Q8—122no—
348Q8C2×C3×#6711yescharacteristic
349D4—22no—
350D4—32no—
351D4—62no—
352D4—61no—
353D4—62no—
354C2×C4—32no—
355C2×C4—32no—
356C2×C4—31no—
357C2×C4—31no—
358C2×C4—31no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—64no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—122no—
367C2×C4C2×C3×S411yescharacteristic
368C8—62no—
369C8—62no—
370C2×C3—22no—
371C2×C3—41no—
372C2×C3—42no—
373C2×C3—41no—
374C2×C3—82no—
375C2×C3—82no—
376C2×C3—81no—
377C2×C3—121no—
378C2×C3—162no—
379C2×C3—161no—
380C2×C3—162no—
381C2×C3#592712yes—
382C2×C3C2×GL(2,ℤ/4)11yescharacteristic
383C22—22no—
384C22—22no—
385C22—21no—
386C22—62no—
387C22—61no—
388C22C3×GL(2,ℤ/4)12yes—
389C22C22×C3×S411yescharacteristic
390C4—21no—
391C4—31no—
392C4—31no—
393C4—32no—
394C4—61no—
395C4—122no—
396C4—122no—
397C3—41no—
398C3—81no—
399C3#8553511yescharacteristic, direct product factor
400C2—22no—
401C2—41no—
402C2—121no—
403C2C3×#592712yes—
404C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC2×C3×2−1+4D3
prime radicalC2×C3×2−1+4⋊C3C2
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#226C2semidirect
3. C3×#5425C2semidirect
4. C3×#5420C2semidirect
5. C3×#5419C2semidirect
6. #85535C3direct (N × H)
19. C22×#226C2×C3semidirect
23. #5425C2×C3semidirect
24. #5420C2×C3semidirect
25. #5419C2×C3semidirect
399. C3#85535direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C22×C3×#2262
C3×#54252
C3×#54202
C3×#54192
#855353
C3×#30113
C3×#9354