Finite Group Databasebeta

← #49161824

Subgroups — #49161824

Order 37504.

688 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627597C211yesmaximal, characteristic
2C2×C4×C22⋊D293C211yesmaximal, characteristic, complemented
3C2×#22627581C211yesmaximal, characteristic, complemented
4#34036031C211yesmaximal, characteristic, complemented
5#34036029C211yesmaximal, characteristic, complemented
6#34036029C211yesmaximal, characteristic, complemented
7#34036026C211yesmaximal, characteristic, complemented
8#34036022C211yesmaximal, characteristic, complemented
9#34036020C211yesmaximal, characteristic, complemented
10#34036008C211yesmaximal, characteristic, complemented
11#34036007C211yesmaximal, characteristic, complemented
12C4×#17989781C211yesmaximal, characteristic, complemented
13#34035976C211yesmaximal, characteristic, complemented
14#34035976C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C293C211yesmaximal, characteristic, complemented
16C22×D1172C2211yescharacteristic
17C22×C22⋊D293C2211yescharacteristic
18C22×C22⋊D293C2211yescharacteristic
19C22×C4×D293C2211yescharacteristic
20C22×C4×D293C2211yescharacteristic
21#22627597C418yescomplemented
22C4×C22⋊D293—24no—
23#22627581—24no—
24C2×C4⋊Dic293C2211yescharacteristic
25C2×#17989781C2211yescharacteristic
26C2×#17989781C2211yescharacteristic
27C2×#17989781C2211yescharacteristic
28C2×#17989781C2211yescharacteristic
29C2×C22⋊Dic293C2211yescharacteristic
30C2×C22⋊Dic293C2211yescharacteristic
31C2×#17989779C2211yescharacteristic
32C2×#17989779C2211yescharacteristic
33C2×#17989779C2211yescharacteristic
34C2×#17989779C2211yescharacteristic
35C2×#17989779C2211yescharacteristic
36C2×#17989779C2211yescharacteristic
37C2×#17989779C2211yescharacteristic
38C2×#17989779C2211yescharacteristic
39C2×C4×Dic293C2211yescharacteristic
40C2×C4×Dic293C2211yescharacteristic
41C2×C4×Dic293C2211yescharacteristic
42C2×C4×Dic293C2211yescharacteristic
43#22627531C2211yescharacteristic
44#22627531C2211yescharacteristic
45#22627531C2211yescharacteristic
46#22627531C2211yescharacteristic
47C23×C4×C293C2211yescharacteristic
48C2×C22⋊C4×C293C2211yescharacteristic
49C2×C22⋊C4×C293C2211yescharacteristic
50C2×C42×C293C2211yescharacteristic
51C2×C42×C293C2211yescharacteristic
52#95×C293C2211yescharacteristic
53#95×C293C2211yescharacteristic
54C23×D293C2311yescharacteristic
55C23×D293C2311yescharacteristic
56C2×D1172—41no—
57C2×D1172—41no—
58C2×D1172C2×C414yescomplemented
59C2×C22⋊D293—22no—
60C2×C22⋊D293—22no—
61C2×C22⋊D293—41no—
62C2×C22⋊D293—41no—
63C2×C22⋊D293C2×C414yes—
64C2×C22⋊D293C2×C414yescomplemented
65C22×Dic293C2311yescharacteristic
66C22×Dic293C2311yescharacteristic
67C22×Dic293C2311yescharacteristic
68C22×Dic293C2311yescharacteristic
69C22×Dic293C2311yescharacteristic
70C22×Dic293C2311yescharacteristic
71C2×C4×D293—22no—
72C2×C4×D293—22no—
73C2×C4×D293—22no—
74C2×C4×D293—22no—
75C2×C4×D293—41no—
76C2×C4×D293C2×C414yescomplemented
77C4⋊Dic293—41no—
78#17989781—22no—
79#17989781—41no—
80#17989781C2×C414yescomplemented
81#17989781D414yescomplemented
82C22⋊Dic293—22no—
83C22⋊Dic293—41no—
84#17989779—22no—
85#17989779—22no—
86#17989779—22no—
87#17989779—22no—
88#17989779—22no—
89#17989779—22no—
90#17989779—41no—
91#17989779C2×C414yescomplemented
92C4×Dic293—22no—
93C4×Dic293—22no—
94C4×Dic293—22no—
95C4×Dic293—41no—
96C24×C293C2311yescharacteristic
97C22×C4×C293—22no—
98C22×C4×C293—22no—
99C22×C4×C293—41no—
100C22×C4×C293C2311yescharacteristic
101C22×C4×C293C2311yescharacteristic
102C22×C4×C293C2311yescharacteristic
103C22×C4×C293C2311yescharacteristic
104C22×C4×C293C2311yescharacteristic
105C22×C4×C293C2311yescharacteristic
106C22⋊C4×C293—22no—
107C22⋊C4×C293C2×C414yescomplemented
108C42×C293—22no—
109C42×C293—41no—
110C22×D293—21no—
111C22×D293—21no—
112C22×D293—21no—
113C22×D293—21no—
114C22×D293—21no—
115C22×D293—21no—
116C22×D293—21no—
117C22×D293—21no—
118C22×D293—21no—
119C22×D293—21no—
120C22×D293—21no—
121C22×D293—21no—
122C22×D293C22×C412yes—
123C22×D293C22×C412yes—
124D1172—44no—
125C22⋊D293—28no—
126C22⋊D293—44no—
127C2×Dic293—21no—
128C2×Dic293—21no—
129C2×Dic293—21no—
130C2×Dic293—21no—
131C2×Dic293—21no—
132C2×Dic293—21no—
133C2×Dic293—21no—
134C2×Dic293—21no—
135C2×Dic293—21no—
136C2×Dic293—21no—
137C2×Dic293—21no—
138C2×Dic293—21no—
139C2×Dic293—21no—
140C2×Dic293C22×C412yes—
141C2×Dic293C22×C412yes—
142C2×Dic293C2×D412yes—
143C2×Dic293C2×D412yes—
144C2×Dic293#4112yes—
145C4×D293—28no—
146C4×D293—44no—
147C23×C293—21no—
148C23×C293—21no—
149C23×C293—21no—
150C23×C293—21no—
151C23×C293—21no—
152C23×C293—21no—
153C23×C293C22×C412yes—
154C23×C293C2411yescharacteristic
155C2×C4×C293—21no—
156C2×C4×C293—21no—
157C2×C4×C293—21no—
158C2×C4×C293—21no—
159C2×C4×C293—21no—
160C2×C4×C293—21no—
161C2×C4×C293—21no—
162C2×C4×C293—21no—
163C2×C4×C293—21no—
164C2×C4×C293—21no—
165C2×C4×C293—21no—
166C2×C4×C293—21no—
167C2×C4×C293—21no—
168C2×C4×C293—44no—
169C2×C4×C293C22×C412yes—
170C2×C4×C293C22×C412yes—
171C2×C4×C293C2×D412yes—
172C2×C4×C293#4112yes—
173C2×C4×C293#4112yes—
174C22×C293—22no—
175C22×C293—22no—
176C22×C293—22no—
177C22×C293—41no—
178C22×C293—41no—
179C22×C293—41no—
180C22×C293—41no—
181C22×C293C23×C411yescharacteristic
182C22×C293C22×D411yescharacteristic
183C22×C293C2×#4111yescharacteristic
184C22×C293C2×#4111yescharacteristic
185C22×C293C2×#4111yescharacteristic
186C22×C293H(5,𝔽2)11yescharacteristic
187C22×C293H(5,𝔽2)11yescharacteristic
188C2×D293—22no—
189C2×D293—22no—
190C2×D293—22no—
191C2×D293—22no—
192C2×D293—22no—
193C2×D293—22no—
194C2×D293—41no—
195C2×D293—41no—
196C2×D293—41no—
197C2×D293—41no—
198C2×D293—41no—
199C2×D293—41no—
200C2×D293—41no—
201C2×D293—41no—
202C4×C293—22no—
203C4×C293—22no—
204C4×C293—22no—
205C4×C293—22no—
206C4×C293—22no—
207C4×C293—41no—
208Dic293—22no—
209Dic293—22no—
210Dic293—41no—
211Dic293—41no—
212Dic293—41no—
213Dic293C4×D414yes—
214C2×C293—42no—
215C2×C293C2×C4×D411yescharacteristic
216C2×C293C4×#4111yescharacteristic
217C2×C293#51811yescharacteristic
218C2×C293#54611yescharacteristic
219C2×C293#55011yescharacteristic
220C2×C293#55311yescharacteristic
221C2×C293#55511yescharacteristic
222D293—42no—
223D293—42no—
224C293#236311yescharacteristic, complemented, Sylow
225#2363—2931nomaximal, Sylow
226C2×#121—2931no—
227C2×C4×D4—2931no—
228C2×#117—2931no—
229#390—2931no—
230#390—2931no—
231#390—2931no—
232#389—2931no—
233#388—2931no—
234#388—2931no—
235#386—2931no—
236#386—2931no—
237C4×C4⋊C4—2931no—
238C4×C22⋊C4—2931no—
239C4×C22⋊C4—2931no—
240C4×C22⋊C4D29311yescharacteristic, complemented
241C22×D4—2931no—
242C22×D4—2931no—
243C22×D4—2931no—
244C23×C4—2931no—
245C23×C4—2931no—
246C23×C4C2×D29311yescharacteristic
247#121—2938no—
248C4×D4—5864no—
249#117—5864no—
250C2×C4⋊C4—2931no—
251C2×C4⋊C4—2931no—
252C2×C4⋊C4—2931no—
253C2×C4⋊C4—2931no—
254C2×C4⋊C4—2931no—
255C2×C22⋊C4—2931no—
256C2×C22⋊C4—2931no—
257C2×C22⋊C4—2931no—
258C2×C22⋊C4—2931no—
259C2×C22⋊C4—2931no—
260C2×C22⋊C4—2931no—
261C2×C22⋊C4—2931no—
262C2×C22⋊C4—2931no—
263C2×C22⋊C4—2931no—
264C2×C22⋊C4—2931no—
265C2×C22⋊C4C2×D29311yescharacteristic
266C2×C22⋊C4C2×D29311yescharacteristic
267C2×C42—2931no—
268C2×C42—2931no—
269C2×C42—2931no—
270C2×C42—2931no—
271C2×C42C2×D29311yescharacteristic
272C2×C42C2×D29311yescharacteristic
273#95—2931no—
274#95—2931no—
275#95—2931no—
276#95—2931no—
277#95C2×D29311yescharacteristic
278#95C2×D29311yescharacteristic
279C24—2931no—
280C24—2931no—
281C24C22×D29311yescharacteristic
282C2×D4—2934no—
283C2×D4—2934no—
284C2×D4—2934no—
285C2×D4—5862no—
286C2×D4—5862no—
287C2×D4—11721no—
288C2×D4—11721no—
289C2×D4—11721no—
290C2×D4—11721no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—2931no—
295C22×C4—2931no—
296C22×C4—2931no—
297C22×C4—2934no—
298C22×C4—2931no—
299C22×C4—2931no—
300C22×C4—2931no—
301C22×C4—5862no—
302C22×C4—5862no—
303C22×C4—5862no—
304C22×C4—5862no—
305C22×C4—11721no—
306C22×C4C22×D29311yescharacteristic
307C22×C4C22×D29311yescharacteristic
308C22×C4C22×D29311yescharacteristic
309C22×C4C22×D29311yescharacteristic
310C22×C4C22×D29311yescharacteristic
311C22×C4C22×D29311yescharacteristic
312C4⋊C4—2934no—
313C4⋊C4—2934no—
314C4⋊C4—5862no—
315C4⋊C4—11721no—
316C4⋊C4—11721no—
317C22⋊C4—22no—
318C22⋊C4—2934no—
319C22⋊C4—5862no—
320C22⋊C4—5862no—
321C22⋊C4—5862no—
322C22⋊C4—5862no—
323C22⋊C4—5862no—
324C22⋊C4—5862no—
325C22⋊C4—5862no—
326C22⋊C4—11721no—
327C22⋊C4—11721no—
328C22⋊C4C4×D29314yescomplemented
329C42—22no—
330C42—41no—
331C42—5862no—
332C42—5862no—
333C42—5862no—
334C42—11721no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2932no—
342C23—2932no—
343C23—5861no—
344C23—5861no—
345C23—5861no—
346C23—5861no—
347C23—5861no—
348C23—5861no—
349C23—5861no—
350C23—5861no—
351C23—5861no—
352C23—5861no—
353C23—5861no—
354C23—5861no—
355C23C2×C4×D29312yes—
356C23C23×D29311yescharacteristic
357D4—5868no—
358D4—11724no—
359D4—11724no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—2932no—
375C2×C4—2932no—
376C2×C4—2932no—
377C2×C4—2932no—
378C2×C4—2932no—
379C2×C4—5868no—
380C2×C4—5861no—
381C2×C4—5861no—
382C2×C4—5861no—
383C2×C4—5861no—
384C2×C4—5861no—
385C2×C4—5861no—
386C2×C4—5861no—
387C2×C4—5861no—
388C2×C4—5861no—
389C2×C4—5861no—
390C2×C4—5861no—
391C2×C4—5861no—
392C2×C4—5861no—
393C2×C4—11724no—
394C2×C4C2×C4×D29312yes—
395C2×C4C2×C4×D29312yes—
396C2×C4D4×D29312yes—
397C2×C4#1798980212yes—
398C2×C4#1798980212yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—5862no—
407C22—5862no—
408C22—5862no—
409C22—5862no—
410C22—5862no—
411C22—5862no—
412C22—11721no—
413C22—11721no—
414C22—11721no—
415C22—11721no—
416C22—11721no—
417C22—11721no—
418C22—11721no—
419C22—11721no—
420C22C22×C4×D29311yescharacteristic
421C22C2×D4×D29311yescharacteristic
422C22#41×D29311yescharacteristic
423C22C2×#1798980211yescharacteristic
424C22C2×#1798980211yescharacteristic
425C22#2262766911yescharacteristic
426C22#2262767011yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—2934no—
434C4—5862no—
435C4—5862no—
436C4—11721no—
437C4—11721no—
438C4—11721no—
439C2—42no—
440C2—11722no—
441C2—11722no—
442C2C4×D4×D29311yescharacteristic
443C2C4×#1798980211yescharacteristic
444C2#3403660111yescharacteristic
445C2#3403676911yescharacteristic
446C2#3403680011yescharacteristic
447C2#3403682811yescharacteristic
448C2#3403684211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D293
commutatorC22×C293C23×C4
FittingC4×C22⋊C4×C293C2
socle, cosocleC23×C293C24
prime radicalC2×#22627597C2
hypercenterC4×C22⋊C4D293
hypocenterC293#2363
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D293C2semidirect
3. C2×#22627581C2semidirect
4. #34036031C2semidirect
5. #34036029C2semidirect
6. #34036029C2semidirect
7. #34036026C2semidirect
8. #34036022C2semidirect
9. #34036020C2semidirect
10. #34036008C2semidirect
11. #34036007C2semidirect
12. C4×#17989781C2semidirect
13. #34035976C2semidirect
14. #34035976C2semidirect
15. C4×C22⋊C4×C293C2semidirect
21. #22627597C4semidirect
58. C2×D1172C2×C4semidirect
64. C2×C22⋊D293C2×C4semidirect
76. C2×C4×D293C2×C4semidirect
80. #17989781C2×C4semidirect
81. #17989781D4semidirect
91. #17989779C2×C4semidirect
107. C22⋊C4×C293C2×C4semidirect
224. C293#2363semidirect
240. C4×C22⋊C4D293semidirect
328. C22⋊C4C4×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226275972
C2×C4×C22⋊D2932
C2×#226275812
#340360312
#340360292
#340360262
#340360222
#340360202
#340360082
#340360072
C4×#179897812
#340359762
C4×C22⋊C4×C2932
#2363293