Finite Group Databasebeta

← #13166358

Subgroups — #13166358

Order 1728.

736 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 74 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×#89020C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic, complemented
3C3×#64283C211yesmaximal, characteristic, complemented
4#11722730C212yesmaximal, complemented
5#11722729C212yesmaximal, complemented
6#10593808—31nomaximal
7C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×C4×D3C411yescharacteristic
10C2×#6293C2211yescharacteristic
11C2×#6292C411yescharacteristic
12C2×#6291C2211yescharacteristic
13C2×#6290C411yescharacteristic
14#89020—21no—
15#89020—21no—
16#89020—21no—
17#89020—21no—
18#88993C2212yes—
19#88989—21no—
20#88988—21no—
21C3×#3792C2212yes—
22C3×#3788—21no—
23C3×#3787—21no—
24C2×#3828—31no—
25C2×#3798—62no—
26#64283—31no—
27#64283D311yescharacteristic, complemented
28#63788—32no—
29#63780—32no—
30C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
31C22×C3×C3⋊D3C2×C411yescharacteristic
32C22×C32×D3C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
35C4×C3⋊(C3⋊D3)—21no—
36C4×C3⋊(C3⋊D3)—21no—
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3D411yescharacteristic
42C2×C32×Dic3C2×C411yescharacteristic
43C32×C4×D3—21no—
44C32×C4×D3—21no—
45#6293—21no—
46#6293—21no—
47#6292—21no—
48#6292—21no—
49#6291—21no—
50#6291—21no—
51#6290—21no—
52#6290—21no—
53C3⋊#622—22no—
54C3×#622—22no—
55#5132—91nomaximal
56C22×C32×C4—31no—
57C2×#638—31no—
58C2×#638—31no—
59#3828—61no—
60#3828—61no—
61#3828—61no—
62#3828—61no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—62no—
66C2×C4×C3⋊D3—122no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×#634—31no—
69C2×C3×C4×D3—22no—
70C2×C3×C4×D3—62no—
71C2×#627—62no—
72C2×#626—62no—
73C2×#626—62no—
74C2×#625—62no—
75#3798—64no—
76#3798—64no—
77#3798—64no—
78#3798—64no—
79#3792—32no—
80#3792C2×D312yes—
81#3788—21no—
82#3788—61no—
83#3787—21no—
84#3787—61no—
85C22×C33C22×C411yescharacteristic
86C2×C3⋊(C3⋊D3)—21no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3×C3⋊D3—41no—
89C2×C3×C3⋊D3C22⋊C411yescharacteristic
90C2×C3×C3⋊D3C22⋊C411yescharacteristic
91C2×C32×D3—21no—
92C2×C32×D3—41no—
93C33×C4C22⋊C411yescharacteristic
94C33×C4C22⋊C411yescharacteristic
95C3⋊(C3⋊Dic3)—21no—
96C3×C3⋊Dic3C2×D411yescharacteristic
97C3×C3⋊Dic3C2×D411yescharacteristic
98C32×Dic3—21no—
99C2×#235—91no—
100C2×#235—182no—
101C2×C3×M16—91no—
102C2×#208—91no—
103#824—92no—
104#823—92no—
105C23×C32—31no—
106C22×C3⋊D3—31no—
107C22×C3⋊D3—62no—
108C22×C3⋊D3—122no—
109C22×C3⋊D3C4×D311yescharacteristic
110C22×C3×D3—22no—
111C22×C3×D3—62no—
112C2×C32×C4—22no—
113C2×C32×C4—31no—
114C2×C32×C4—42no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117C2×C32×C4C4×D311yescharacteristic
118#638—61no—
119#638—61no—
120#638—61no—
121#638—61no—
122C2×C3⋊Dic3—31no—
123C2×C3⋊Dic3—62no—
124C2×C3⋊Dic3—122no—
125C2×C3⋊Dic3C22×D311yescharacteristic
126C4⋊(C3⋊D3)—61no—
127C4⋊(C3⋊D3)—61no—
128C4×C3⋊D3—61no—
129C4×C3⋊D3—61no—
130C4×C3⋊D3—64no—
131C4×C3⋊D3—64no—
132C4×C3⋊D3—124no—
133C4×C3⋊D3—124no—
134C4×C3⋊D3D1211yescharacteristic
135C4×C3⋊D3D1211yescharacteristic
136C4×C3⋊D3#6711yescharacteristic
137C4×C3⋊D3#6711yescharacteristic
138#634—61no—
139#634—61no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—62no—
142C3×C4×D3—24no—
143C3×C4×D3—24no—
144C3×C4×D3—62no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×C4×D3—62no—
148#627—64no—
149#627—64no—
150#626—64no—
151#626—64no—
152#626—64no—
153#626—64no—
154#625—64no—
155#625—64no—
156#622—22no—
157#622—62no—
158#413—271noSylow
159C2×C33—21no—
160C2×C33C2×C22⋊C411yescharacteristic
161C3⋊(C3⋊D3)—41no—
162C3×C3⋊D3—21no—
163C3×C3⋊D3—21no—
164C32×D3—41no—
165C22×C3×C4—62no—
166C22×C3×C4—91no—
167C2×#67—91no—
168C2×#67—91no—
169C2×#67—182no—
170C2×#67—182no—
171#235—181no—
172#235—181no—
173#235—181no—
174#235—181no—
175#235—184no—
176#235—184no—
177#235—184no—
178#235—184no—
179C2×D12—91no—
180C2×D12—182no—
181C2×C4×D3—62no—
182C2×C4×D3—91no—
183C2×C4×D3—182no—
184C2×C4×D3—362no—
185C2×C4×D3#15711yescharacteristic
186C2×Dic6—91no—
187C2×Dic6—182no—
188C3×M16—92no—
189C3×M16—181no—
190C2×C3×C8—181no—
191#208—92no—
192#208—181no—
193C2×C3⋊C8—181no—
194C22×C32—22no—
195C22×C32—42no—
196C22×C32—61no—
197C22×C32—121no—
198C22×C32C2×C4×D311yescharacteristic
199C2×C3⋊D3—41no—
200C2×C3⋊D3—61no—
201C2×C3⋊D3—64no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—124no—
204C2×C3⋊D3—128no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
207C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
208C2×C3×D3—24no—
209C2×C3×D3—44no—
210C2×C3×D3—62no—
211C2×C3×D3—62no—
212C2×C3×D3—124no—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—42no—
216C32×C4—42no—
217C32×C4—61no—
218C32×C4(C2×D3)⋊C411yescharacteristic
219C32×C4(C2×D3)⋊C411yescharacteristic
220C3⋊Dic3—61no—
221C3⋊Dic3—64no—
222C3⋊Dic3—124no—
223C3⋊Dic3C2×D1211yescharacteristic
224C3⋊Dic3C2×#6711yescharacteristic
225C3×Dic3—24no—
226C3×Dic3—62no—
227C3×Dic3—62no—
228C2×#41—271no—
229C2×M16—91no—
230C2×M16—271no—
231#101—272no—
232#100—272no—
233C33#41311yescharacteristic, complemented, Sylow
234C23×C3—62no—
235C23×C3—91no—
236C22×D3—62no—
237C22×D3—91no—
238C22×D3—182no—
239C22×D3—362no—
240C22×D3C2×#15711yescharacteristic
241C2×C3×C4—22no—
242C2×C3×C4—42no—
243C2×C3×C4—62no—
244C2×C3×C4—64no—
245C2×C3×C4—64no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4—91no—
249C2×C3×C4—91no—
250C2×C3×C4—91no—
251C2×C3×C4C2×#15711yescharacteristic
252#67—181no—
253#67—181no—
254#67—184no—
255#67—184no—
256#67—181no—
257#67—181no—
258#67—184no—
259#67—184no—
260C2×Dic3—62no—
261C2×Dic3—91no—
262C2×Dic3—182no—
263C2×Dic3—362no—
264C2×Dic3C2×#15711yescharacteristic
265D12—184no—
266D12—184no—
267D12—181no—
268D12—181no—
269C4×D3—21no—
270C4×D3—21no—
271C4×D3—62no—
272C4×D3—62no—
273C4×D3—62no—
274C4×D3—62no—
275C4×D3—184no—
276C4×D3—184no—
277C4×D3—181no—
278C4×D3—181no—
279C4×D3—364no—
280C4×D3—364no—
281Dic6—181no—
282Dic6—181no—
283Dic6—184no—
284Dic6—184no—
285C3×C8—182no—
286C3⋊C8—182no—
287C2×C32—21no—
288C2×C32—24no—
289C2×C32—22no—
290C2×C32—44no—
291C2×C32—42no—
292C2×C32—121no—
293C2×C32C2×(C2×D3)⋊C411yescharacteristic
294C3⋊D3—21no—
295C3⋊D3—21no—
296C3⋊D3—124no—
297C3⋊D3—128no—
298C3⋊D3—121no—
299C3×D3—44no—
300C3×D3—122no—
301C3×D3—122no—
302#41—541no—
303#41—541no—
304#41—541no—
305#41—541no—
306C2×Q8—271no—
307C2×D4—271no—
308C2×D4—271no—
309C2×D4—271no—
310C22×C4—31no—
311C22×C4—91no—
312C22×C4—271no—
313M16—92no—
314M16—181no—
315M16—272no—
316M16—541no—
317C2×C8—181no—
318C2×C8—541no—
319C22×C3—22no—
320C22×C3—42no—
321C22×C3—64no—
322C22×C3—91no—
323C22×C3—91no—
324C22×C3—124no—
325C22×C3—361no—
326C22×C3C22×#15711yescharacteristic
327C2×D3—21no—
328C2×D3—41no—
329C2×D3—62no—
330C2×D3—62no—
331C2×D3—124no—
332C2×D3—184no—
333C2×D3—181no—
334C2×D3—364no—
335C2×D3—364no—
336C2×D3—368no—
337C2×D3—361no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—42no—
341C3×C4—42no—
342C3×C4—64no—
343C3×C4—91no—
344C3×C4—91no—
345C3×C4C22⋊#15711yescharacteristic
346C3×C4C22⋊#15711yescharacteristic
347Dic3—21no—
348Dic3—62no—
349Dic3—62no—
350Dic3—184no—
351Dic3—181no—
352Dic3—364no—
353C32—22no—
354C32—42no—
355C32#513211yescharacteristic, complemented
356C23—31no—
357C23—91no—
358C23—271no—
359Q8—541no—
360Q8—541no—
361D4—541no—
362D4—541no—
363D4—541no—
364D4—541no—
365D4—541no—
366D4—541no—
367C2×C4—31no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—91no—
371C2×C4—91no—
372C2×C4—91no—
373C2×C4—91no—
374C2×C4—91no—
375C2×C4—271no—
376C2×C4—541no—
377C2×C4—541no—
378C2×C4D3×#15711yescharacteristic
379C8—182no—
380C8—542no—
381C2×C3—24no—
382C2×C3—21no—
383C2×C3—22no—
384C2×C3—44no—
385C2×C3—42no—
386C2×C3—124no—
387C2×C3—181no—
388C2×C3—181no—
389C2×C3C2×C22⋊#15711yescharacteristic
390D3—41no—
391D3—122no—
392D3—122no—
393D3—364no—
394D3—368no—
395D3—361no—
396C22—61no—
397C22—91no—
398C22—91no—
399C22—121no—
400C22—361no—
401C22—541no—
402C22—1081no—
403C22C2×D3×#15711yescharacteristic
404C4—61no—
405C4—91no—
406C4—91no—
407C4—541no—
408C4(C2×D3)⋊#15711yescharacteristic
409C4(C2×D3)⋊#15711yescharacteristic
410C3—22no—
411C3—42no—
412C3#1059380811yescharacteristic, complemented
413C2—21no—
414C2—121no—
415C2—181no—
416C2—181no—
417C2—1081no—
418C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC4(C2×D3)⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89020C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724511C2semidirect
3. C3×#64283C2semidirect
4. #11722730C2semidirect
5. #11722729C2semidirect
27. #64283D3semidirect
233. C33#413semidirect
355. C32#5132semidirect
412. C3#10593808semidirect

Maximal subgroups

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

subgroupindex
C2×#890202
#117245112
C3×#642832
#117227302
#117227292
#105938083
#51329