Finite Group Databasebeta

← #13166622

Subgroups — #13166622

Order 1728.

752 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 72 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×#89028C211yesmaximal, characteristic
2C2×#88925C211yesmaximal, characteristic, complemented
3C2×D3×#622C211yesmaximal, characteristic, complemented
4#11723112C211yesmaximal, characteristic, complemented
5C3×C22⋊#622C211yesmaximal, characteristic, complemented
6#11722726C211yesmaximal, characteristic, complemented
7#11722726C211yesmaximal, characteristic, complemented
8#10593822—31nomaximal
9C2×#6312C411yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C2×C32×#67C411yescharacteristic
12C2×#6293C411yescharacteristic
13C2×#6291C411yescharacteristic
14C2×#6289C2211yescharacteristic
15#89028—22no—
16#89028—22no—
17C2×D3×C3⋊Dic3C2211yescharacteristic
18C2×C3⋊#622C2211yescharacteristic
19C2×C3⋊#622C2211yescharacteristic
20C2×C3×#622C2211yescharacteristic
21C2×C3×#622C2211yescharacteristic
22#88925—22no—
23D3×#622—22no—
24C2×#3830—31no—
25C2×#3804—62no—
26C2×#3792—31no—
27C22×#622—31no—
28C22⋊#622—31no—
29C22⋊#622D311yescharacteristic, complemented
30#63778—31no—
31#63778—31no—
32C23×C33C2×C411yescharacteristic
33C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
34C22×C32×D3C2×C411yescharacteristic
35#6312—22no—
36C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3C2×C411yescharacteristic
40C2×C3×C3⋊Dic3C2311yescharacteristic
41C32×#67—22no—
42C2×C32×Dic3C2×C411yescharacteristic
43#6293—22no—
44#6291—22no—
45#6289—21no—
46#6289—21no—
47D3×C3⋊Dic3D412yes—
48D3×C3⋊Dic3D412yes—
49C3⋊#622—21no—
50C3⋊#622—21no—
51C3×#622—21no—
52C3×#622—21no—
53#4727—91nomaximal
54C2×C32×D4—31no—
55C2×#638—31no—
56C2×#638—31no—
57C2×#638—62no—
58C2×#638—122no—
59C22×C3⋊Dic3—31no—
60C22×C3⋊Dic3C2×D311yescharacteristic
61#3830—62no—
62#3830—62no—
63C2×C4×C3⋊D3—31no—
64C2×#634—31no—
65C2×C3×#67—22no—
66C22×C3×Dic3—62no—
67C2×#627—62no—
68C2×#626—62no—
69C2×#624—62no—
70#3804—68no—
71#3804—68no—
72C2×D3×Dic3—62no—
73#3792—62no—
74C2×#622—31no—
75C2×#622—31no—
76C2×#622—62no—
77C2×#622C2×D311yescharacteristic
78C2×#622C2×D311yescharacteristic
79C22×C33—21no—
80C22×C33—21no—
81C22×C33—21no—
82C22×C33C22×C411yescharacteristic
83C2×C3⋊(C3⋊D3)—21no—
84C2×C3⋊(C3⋊D3)—21no—
85C2×C3⋊(C3⋊D3)—21no—
86C2×C32×D3—21no—
87C2×C32×D3—21no—
88C2×C32×D3C22⋊C412yes—
89C3⋊(C3⋊Dic3)C22⋊C412yes—
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3C2×D411yescharacteristic
92C3×C3⋊Dic3C2×D411yescharacteristic
93C32×Dic3—21no—
94C2×#237—182no—
95C2×#235—91no—
96C2×ΓL(1,𝔽25)—91no—
97C2×D3×C8—91no—
98C3×C22⋊C8—91no—
99#830—91no—
100(C2×D3)⋊C8—91no—
101(C2×D3)⋊C8—91no—
102C23×C32—22no—
103C23×C32—31no—
104C23×C32—42no—
105C23×C32C4×D311yescharacteristic
106C22×C3⋊D3—31no—
107C22×C3⋊D3—62no—
108C22×C3⋊D3—122no—
109C22×C3×D3—22no—
110C32×D4—62no—
111C2×C32×C4—31no—
112#638—62no—
113#638—62no—
114#638—68no—
115#638—128no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—32no—
120C2×C3⋊Dic3—32no—
121C2×C3⋊Dic3—62no—
122C2×C3⋊Dic3—122no—
123C2×C3⋊Dic3C4×D311yescharacteristic
124C2×C3⋊Dic3C22×D311yescharacteristic
125C4×C3⋊D3—61no—
126C4×C3⋊D3—61no—
127#634—62no—
128C3×#67—28no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3—64no—
134#627—68no—
135#626—68no—
136#624—64no—
137#624—64no—
138D3×Dic3—64no—
139D3×Dic3—64no—
140#622—21no—
141#622—21no—
142#622—61no—
143#622—61no—
144#408—271noSylow
145C2×C33—41no—
146C2×C33C2×C22⋊C411yescharacteristic
147C2×C33#13111yescharacteristic
148C2×C33#13111yescharacteristic
149C3⋊(C3⋊D3)—41no—
150C32×D3—22no—
151C2×C3×D4—62no—
152C22×C3×C4—91no—
153C2×#67—91no—
154C2×#67—182no—
155C2×#67—182no—
156C2×#67—362no—
157C2×#67#15711yescharacteristic
158C22×Dic3—62no—
159C22×Dic3—182no—
160#237—188no—
161#237—188no—
162#235—182no—
163#235—182no—
164C2×D12—91no—
165C2×C4×D3—91no—
166C2×C4×D3—91no—
167C2×C4×D3—182no—
168C2×Dic6—91no—
169C2×Dic6—182no—
170C2×C3×C8—91no—
171C2×C3×C8—91no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—91no—
174ΓL(1,𝔽25)—182no—
175D3×C8—182no—
176C22×C32—21no—
177C22×C32—24no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—32no—
182C22×C32—44no—
183C22×C32—42no—
184C22×C32—42no—
185C22×C32—42no—
186C22×C32—61no—
187C22×C32—61no—
188C22×C32—82no—
189C22×C32—82no—
190C22×C32C2×C4×D311yescharacteristic
191C2×C3⋊D3—61no—
192C2×C3⋊D3—64no—
193C2×C3⋊D3—61no—
194C2×C3⋊D3—61no—
195C2×C3⋊D3—124no—
196C2×C3⋊D3—122no—
197C2×C3⋊D3—122no—
198C2×C3⋊D3—242no—
199C2×C3⋊D3—242no—
200C2×C3×D3—24no—
201C2×C3×D3—42no—
202C2×C3×D3—42no—
203C32×C4—61no—
204C3⋊Dic3—21no—
205C3⋊Dic3—32no—
206C3⋊Dic3—64no—
207C3⋊Dic3—124no—
208C3⋊Dic3D3×D411yescharacteristic
209C3⋊Dic3D3×D411yescharacteristic
210C3×Dic3—24no—
211C3×Dic3—62no—
212C3×Dic3—62no—
213C3×Dic3—64no—
214C2×#41—271no—
215C2×M16—271no—
216C22×C8—271no—
217C22⋊C8—91no—
218C22⋊C8—271no—
219C22⋊C8—271no—
220C22⋊C8—271no—
221C33#40811yescharacteristic, complemented, Sylow
222C23×C3—22no—
223C23×C3—42no—
224C23×C3—62no—
225C23×C3C2×#15711yescharacteristic
226C22×D3—91no—
227C22×D3—182no—
228C22×D3—362no—
229C22×D3C2×#15711yescharacteristic
230C3×D4—68no—
231C2×C3×C4—62no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236#67—22no—
237#67—188no—
238#67—182no—
239#67—188no—
240#67—368no—
241C2×Dic3—62no—
242C2×Dic3—62no—
243C2×Dic3—64no—
244C2×Dic3—64no—
245C2×Dic3—91no—
246C2×Dic3—184no—
247C2×Dic3—184no—
248C2×Dic3—182no—
249C2×Dic3—362no—
250C2×Dic3C2×#15711yescharacteristic
251D12—182no—
252C4×D3—92no—
253C4×D3—92no—
254C4×D3—181no—
255C4×D3—181no—
256C4×D3—184no—
257C4×D3—184no—
258Dic6—182no—
259Dic6—188no—
260C3×C8—181no—
261C3×C8—181no—
262C3⋊C8—181no—
263C3⋊C8—181no—
264C2×C32—22no—
265C2×C32—22no—
266C2×C32—22no—
267C2×C32—42no—
268C2×C32—42no—
269C2×C32—42no—
270C2×C32—41no—
271C2×C32—44no—
272C2×C32—62no—
273C2×C32—84no—
274C2×C32D3×C22⋊C411yescharacteristic
275C2×C32#90111yescharacteristic
276C2×C32#90711yescharacteristic
277C3⋊D3—121no—
278C3⋊D3—124no—
279C3⋊D3—244no—
280C3×D3—44no—
281#41—542no—
282#41—542no—
283C2×Q8—271no—
284C2×D4—31no—
285C2×D4—271no—
286C2×D4—271no—
287C22×C4—91no—
288C22×C4—271no—
289C22×C4—271no—
290M16—542no—
291C2×C8—91no—
292C2×C8—91no—
293C2×C8—271no—
294C2×C8—271no—
295C2×C8—542no—
296C22×C3—24no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—44no—
304C22×C3—42no—
305C22×C3—64no—
306C22×C3—82no—
307C22×C3—82no—
308C22×C3—122no—
309C22×C3—122no—
310C22×C3C22×#15711yescharacteristic
311C2×D3—21no—
312C2×D3—21no—
313C2×D3—184no—
314C2×D3—181no—
315C2×D3—181no—
316C2×D3—181no—
317C2×D3—362no—
318C2×D3—362no—
319C2×D3—364no—
320C2×D3—722no—
321C2×D3—722no—
322C2×D3C22⋊#15712yes—
323C3×C4—64no—
324C3×C4—91no—
325C3×C4—91no—
326C3×C4—181no—
327Dic3—21no—
328Dic3—62no—
329Dic3—62no—
330Dic3—64no—
331Dic3—92no—
332Dic3—184no—
333Dic3—364no—
334C32—22no—
335C32—42no—
336C32#472711yescharacteristic, complemented
337C23—31no—
338C23—271no—
339C23D3×#15711yescharacteristic
340Q8—542no—
341D4—62no—
342D4—542no—
343D4—542no—
344C2×C4—31no—
345C2×C4—91no—
346C2×C4—91no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—272no—
350C2×C4—272no—
351C2×C4—271no—
352C2×C4—541no—
353C2×C4—541no—
354C8—181no—
355C8—181no—
356C8—541no—
357C8—541no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—22no—
361C2×C3—44no—
362C2×C3—41no—
363C2×C3—42no—
364C2×C3—42no—
365C2×C3—42no—
366C2×C3—84no—
367C2×C3—124no—
368C2×C3#6428711yescharacteristic
369C2×C3#6428711yescharacteristic
370C2×C3C2×C22⋊#15711yescharacteristic
371D3—22no—
372D3—364no—
373D3—361no—
374D3—724no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—32no—
379C22—61no—
380C22—61no—
381C22—541no—
382C22—541no—
383C22—541no—
384C22C2×D3×#15711yescharacteristic
385C4—61no—
386C4—91no—
387C4—91no—
388C4—181no—
389C4—272no—
390C3—22no—
391C3—42no—
392C3#1059382211yescharacteristic, complemented
393C2—41no—
394C2—62no—
395C2—1081no—
396C2#1172426611yescharacteristic
397C2#1172426711yescharacteristic
398C2D3×C22⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC23×C33C2×C4
prime radicalC2×#89028C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D3×#157
hypocenterC33#408
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88925C2semidirect
3. C2×D3×#622C2semidirect
4. #11723112C2semidirect
5. C3×C22⋊#622C2semidirect
6. #11722726C2semidirect
7. #11722726C2semidirect
29. C22⋊#622D3semidirect
221. C33#408semidirect
336. C32#4727semidirect
392. C3#10593822semidirect

Maximal subgroups

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

subgroupindex
C2×#890282
C2×#889252
C2×D3×#6222
#117231122
C3×C22⋊#6222
#117227262
#105938223
#47279