Finite Group Databasebeta

← C23×#7756 #19155382

Subgroups — C23×#7756 #19155382

Order 5888.

504 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C23×#2732C211yesmaximal, characteristic
2C23×#2057C211yesmaximal, characteristic, complemented
3C23×#2023C211yesmaximal, characteristic
4C23×#2016C211yesmaximal, characteristic, complemented
5C23×#1272C211yesmaximal, characteristic, complemented
6C23×#1271C211yesmaximal, characteristic, complemented
7C23×#1270C211yesmaximal, characteristic
8C22×Q16×C23—21no—
9C2×C23×C4⋊Q8C2211yescharacteristic
10C2×C23×C22⋊Q8—21no—
11C2×C23×#107—21no—
12C2×C23×#103—21no—
13C2×C23×#102—21no—
14C23×#399—21no—
15C23×#398—21no—
16C23×#396—21no—
17C23×#391C2211yescharacteristic
18C23×#390C2211yescharacteristic
19C23×#389C2211yescharacteristic
20C23×#386—21no—
21C23×#385—21no—
22C23×#340—21no—
23C23×#340—21no—
24C23×#336C2211yescharacteristic
25C23×#336C2211yescharacteristic
26C23×#336C2211yescharacteristic
27C22×Q8×C23—21no—
28C22×Q8×C23D411yescharacteristic
29C22×D4×C23D411yescharacteristic
30C23×C4×C23—21no—
31C2×Q16×C23—42no—
32C2×Q16×C23—42no—
33C2×Q16×C23—42no—
34C22×C8×C23—21no—
35C22×C8×C23—21no—
36C22×C8×C23—21no—
37C23×C4⋊Q8—42no—
38C23×C22⋊Q8—42no—
39C23×C22⋊Q8—42no—
40C2×C4⋊C4×C23—21no—
41C2×C4⋊C4×C23—21no—
42C2×C4⋊C4×C23—21no—
43C2×C4⋊C4×C23—21no—
44C2×C4⋊C4×C23—21no—
45C2×C4⋊C4×C23—21no—
46C2×C4⋊C4×C23D411yescharacteristic
47C2×C4⋊C4×C23D411yescharacteristic
48C2×C4⋊C4×C23D411yescharacteristic
49C2×C4⋊C4×C23D411yescharacteristic
50C2×C22⋊C4×C23—21no—
51C2×C22⋊C4×C23—21no—
52C2×C22⋊C4×C23—21no—
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—21no—
55C2×C42×C23C2311yescharacteristic
56C23×#107—42no—
57C23×#103—42no—
58C23×#102—42no—
59C23×#95—21no—
60C23×#95—21no—
61C23×#95—21no—
62C23×#95—21no—
63C24×C23—21no—
64C2×Q8×C23—21no—
65C2×Q8×C23—22no—
66C2×Q8×C23—41no—
67C2×Q8×C23—41no—
68C2×Q8×C23—41no—
69C2×Q8×C23—41no—
70C2×Q8×C23—41no—
71C2×Q8×C23—41no—
72C2×Q8×C23—41no—
73C2×Q8×C23D812yes—
74C2×D4×C23—21no—
75C2×D4×C23—41no—
76C2×D4×C23—41no—
77C2×D4×C23Q1612yescomplemented
78C22×C4×C23—21no—
79C22×C4×C23—21no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—42no—
84C22×C4×C23—41no—
85C22×C4×C23—81no—
86C22×C4×C23—81no—
87C22×C4×C23C2×D411yescharacteristic
88C22×C4×C23C2×D411yescharacteristic
89C22×C4×C23C2×D411yescharacteristic
90Q16×C23—48no—
91C2×C8×C23—22no—
92C2×C8×C23—22no—
93C2×C8×C23—41no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C2×C8×C23—41no—
97C2×C8×C23—41no—
98C2×C8×C23—41no—
99C2×C8×C23—41no—
100C4⋊C4×C23—22no—
101C4⋊C4×C23—22no—
102C4⋊C4×C23—21no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—41no—
105C4⋊C4×C23—41no—
106C4⋊C4×C23—41no—
107C4⋊C4×C23—42no—
108C4⋊C4×C23—42no—
109C4⋊C4×C23—41no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23—41no—
112C4⋊C4×C23—41no—
113C4⋊C4×C23—81no—
114C22⋊C4×C23—42no—
115C22⋊C4×C23—41no—
116C22⋊C4×C23—41no—
117C22⋊C4×C23—41no—
118C22⋊C4×C23—41no—
119C22⋊C4×C23—81no—
120C22⋊C4×C23—81no—
121C42×C23—41no—
122#7756C2311yesmaximal, characteristic, direct product factor, Sylow
123C23×C23—22no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23—41no—
128C23×C23—41no—
129C23×C23—41no—
130C23×C23#12011yescharacteristic
131Q8×C23—42no—
132Q8×C23—42no—
133Q8×C23—42no—
134Q8×C23—81no—
135Q8×C23—81no—
136Q8×C23—81no—
137D4×C23—42no—
138D4×C23—81no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—22no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—42no—
160C2×C4×C23—41no—
161C2×C4×C23—84no—
162C2×C4×C23—81no—
163C2×C4×C23—81no—
164C2×C4×C23C2×D811yescharacteristic
165C2×C4×C23C2×Q1611yescharacteristic
166C2×C4×C23#13511yescharacteristic
167C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C23#13711yescharacteristic
169C2×C4×C23#13711yescharacteristic
170C8×C23—42no—
171C8×C23—42no—
172C8×C23—81no—
173#2732C2×C2311yescharacteristic
174#2057C2×C2311yescharacteristic, complemented
175#2023C2×C2311yescharacteristic
176#2016C2×C2311yescharacteristic, complemented
177#1272C2×C2311yescharacteristic, complemented
178#1271C2×C2311yescharacteristic, complemented
179#1270C2×C2311yescharacteristic
180C22×C23—42no—
181C22×C23—42no—
182C22×C23—42no—
183C22×C23—81no—
184C22×C23—81no—
185C22×C23—81no—
186C22×C23—81no—
187C22×C23#44711yescharacteristic
188C22×C23#45111yescharacteristic
189C22×C23#45211yescharacteristic
190C22×C23#45311yescharacteristic
191C22×C23#45511yescharacteristic
192C22×C23#45611yescharacteristic
193C22×C23B(4,𝔽2)11yescharacteristic
194C4×C23—21no—
195C4×C23—21no—
196C4×C23—21no—
197C4×C23—21no—
198C4×C23—21no—
199C4×C23—21no—
200C4×C23—42no—
201C4×C23—81no—
202C4×C23—82no—
203C4×C23—81no—
204C4×C23—81no—
205C4×C23—81no—
206C22×Q16—21no—
207C2×C4⋊Q8C22×C2311yescharacteristic
208C2×C22⋊Q8—21no—
209C2×#107—21no—
210C2×#103—21no—
211C2×#102—21no—
212#399—21no—
213#398—21no—
214#396—21no—
215#391C22×C2311yescharacteristic
216#390C22×C2311yescharacteristic
217#389C22×C2311yescharacteristic
218#386—21no—
219#385—21no—
220#340—21no—
221#340—21no—
222#336C22×C2311yescharacteristic
223#336C22×C2311yescharacteristic
224#336C22×C2311yescharacteristic
225C2×C23—82no—
226C2×C23#159511yescharacteristic
227C2×C23#160211yescharacteristic
228C2×C23#160311yescharacteristic
229C2×C23#163211yescharacteristic
230C2×C23#164111yescharacteristic
231C2×C23#164911yescharacteristic
232C2×C23#166111yescharacteristic
233C22×Q8—21no—
234C22×Q8D4×C2311yescharacteristic
235C22×D4D4×C2311yescharacteristic
236C23×C4—21no—
237C2×Q16—42no—
238C2×Q16—42no—
239C2×Q16—42no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C4⋊Q8—42no—
244C22⋊Q8—42no—
245C22⋊Q8—42no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4D4×C2311yescharacteristic
253C2×C4⋊C4D4×C2311yescharacteristic
254C2×C4⋊C4D4×C2311yescharacteristic
255C2×C4⋊C4D4×C2311yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C42C23×C2311yescharacteristic
262#107—42no—
263#103—42no—
264#102—42no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269C23#775611yescharacteristic, direct product factor, Sylow
270C24—21no—
271C2×Q8—21no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8D8×C2312yes—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4Q16×C2312yescomplemented
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4C2×D4×C2311yescharacteristic
295C22×C4C2×D4×C2311yescharacteristic
296C22×C4C2×D4×C2311yescharacteristic
297Q16—48no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—81no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C42—41no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23C23×#12011yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—42no—
340Q8—81no—
341Q8—81no—
342Q8—81no—
343D4—42no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—84no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4C2×D8×C2311yescharacteristic
371C2×C4C2×Q16×C2311yescharacteristic
372C2×C4C23×#13511yescharacteristic
373C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
374C2×C4C23×#13711yescharacteristic
375C2×C4C23×#13711yescharacteristic
376C8—42no—
377C8—42no—
378C8—81no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C23×#44711yescharacteristic
387C22C23×#45111yescharacteristic
388C22C23×#45211yescharacteristic
389C22C23×#45311yescharacteristic
390C22C23×#45511yescharacteristic
391C22C23×#45611yescharacteristic
392C22C23×B(4,𝔽2)11yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—82no—
405C2—82no—
406C2C23×#159511yescharacteristic
407C2C23×#160211yescharacteristic
408C2C23×#160311yescharacteristic
409C2C23×#163211yescharacteristic
410C2C23×#164111yescharacteristic
411C2C23×#164911yescharacteristic
412C2C23×#166111yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2057C2semidirect
4. C23×#2016C2semidirect
5. C23×#1272C2semidirect
6. C23×#1271C2semidirect
77. C2×D4×C23Q16semidirect
122. #7756C23direct (N × H)
174. #2057C2×C23semidirect
176. #2016C2×C23semidirect
177. #1272C2×C23semidirect
178. #1271C2×C23semidirect
269. C23#7756direct (N × H)
284. C2×D4Q16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#27322
C23×#20572
C23×#20232
C23×#20162
C23×#12722
C23×#12712
C23×#12702
#775623